Graphing Calculator and Spreadsheet Masters · 2017. 6. 15. · and be used solely in conjunction...
Transcript of Graphing Calculator and Spreadsheet Masters · 2017. 6. 15. · and be used solely in conjunction...
Graphing Calculatorand
Spreadsheet Masters
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240
Glencoe Algebra 2ISBN: 0-07-828020-6 Graphing Calculator and Spreadsheet Masters
2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
RESOURCE MATERIALSTI-83 Plus Graphing Calculator Template ........................................................ 1Spreadsheet Template ....................................................................................... 3
TI-83 Plus Graphing Calculator HandbookIntroduction to the Graphing Calculator ................................................................ 5Graphing Functions .............................................................................................. 8Analyzing Functions..............................................................................................11Graphing Inequalities............................................................................................16Matrices ................................................................................................................19Graphing Trigonometric Functions .......................................................................21Graphing Special Functions..................................................................................22Statistics and Statistical Graphs ...........................................................................23
GRAPHING CALCULATOR INVESTIGATIONSUse with Lesson Title Page
1-3 Solving Equations and Checking Solutions...........................272-6 Transformations: Greatest Integer Function..........................293-4 Linear Programming..............................................................314-4 Matrices for 30°, 45°, and 60° Rotations ...............................335-4 Using Tables to Factor by Grouping......................................356-7 Quadratic Inequalities and the Test Menu.............................377-6 Rational Root Theorem .........................................................398-3 Matrices and Equations of Circles.........................................419-3 Horizontal Asymptotes and Tables........................................43
10-1 Regression Equation Lab ......................................................4511-6 Recursion and Iteration .........................................................4712-5 Probabilities...........................................................................4913-4 Law of Sines: Ambiguous Case ............................................5114-2 Sinusoidal Equations.............................................................53
SPREADSHEET INVESTIGATIONS Use with Lesson Title Page
1-4 Absolute Value Statements ...................................................282-4 Using Linear Equations .........................................................303-1 Break-Even Point ..................................................................324-5 Cramer’s Rule .......................................................................345-7 Appreciation and Depreciation ..............................................366-5 Approximating the Real Zeros of Polynomials ......................387-7 Operations on Functions .......................................................408-2 Parabolas ..............................................................................429-4 Variation ................................................................................44
10-6 Net Present Value .................................................................4611-2 Sequences and Series ..........................................................4812-2 Permutations And Combinations...........................................5013-1 Cofunctions ...........................................................................5214-3 Trigonometric Identities .........................................................54
Answers ..............................................................................................................55
iii
CONTENTS
iv
Teacher’s Guide to Using the Graphing Calculator and
Spreadsheet Masters
This booklet contains a TI-83 Plus Graphing CalculatorHandbook, which summarizes many of the graphing calculatorskills students might use in pre-algebra. This is a referencetool and does not contain additional exercises for students.
This booklet also includes a Graphing Calculator Investigationand a Spreadsheet Investigation for each chapter in GlencoeAlgebra 2.• The graphing calculator activities are written with TI-83
Plus keystrokes provided. If your students use anothercalculator, they will need to modify their keystrokes tocomplete the activity.
• The spreadsheet masters were developed for use withMicrosoft® Excel. If you use a different spreadsheetapplication software, you may need to alter the commandsgiven in the activity.
When to Use Each activity should be used as an extension ofthe lesson to which it is referenced. Because these activitiescan be done independently of classroom time, they may beused as extra credit or as enrichments for those students whohave completed their assignments in a timely manner.
vGlencoe Division, Macmillan/McGraw-Hill Geometry
TI–83 Plus Graphing Calculator Template
NAME ______________________________________________ DATE ____________ PERIOD _____
© Glencoe/McGraw-Hill 1 Glencoe Algebra 2
Spreadsheet Template
NAME ______________________________________________ DATE ____________ PERIOD _____
© Glencoe/McGraw-Hill 3 Glencoe Algebra 2
TI–83 Plus Graphing Calculator HandbookIntroduction to the Graphing Calculator
NAME ______________________________________________ DATE ____________ PERIOD _____
11
© Glencoe/McGraw-Hill 5 Glencoe Algebra 2
This section introduces you to some commonly-used keys and menus ofthe calculator.
MODE The key allows you to select your preferences in manyaspects of calculation and graphing. Many of these settings are rarelychanged in common usage. This screen shows the default mode settings.
← type of numeric notation← number of decimal places in results← unit of angle measure used← type of graph (function, parametric, polar, sequence)← whether to connect graphed points
← real, rectangular complex, or polar complex number system← graph occupies full screen, top of screen with
HOME screen below, or left side of screen with TABLE onright
To change the preferences, use the arrow keys to highlight your choiceand press .
FORMAT The FORMAT menu is the second function of andsets preferences for the appearance of your graphing screen. The defaultscreen is shown below.
← rectangular or polar coordinate system← whether to display the cursor coordinates on screen← whether to show a grid pattern on screen← whether to show the axes← whether to label the axes← whether to show the equation being graphed
You can change your preferences in the FORMAT menu in the sameway you change settings.
Many keys on the calculator access menus from which you can select afunction, command, or setting. Some keys access multiple menus. Youcan use the right and left arrow keys to scroll through the differentmenu names located at the top of the screen. As each menu name ishighlighted, the choices on the screen change. The screens on the nextpage show various menus accessed by using .MATH
MODE
ZOOM
ENTER
MODESettingPreferences
Using Menus
To select a choice in a menu, either use the arrow keys to highlight yourchoice and press or simply press the number or letter of yourselection. Notice that entry 7 in the first screen has a down arrow insteadof a colon after the 7. This signifies there are more entries in the menu.
Above most keys are one or two additional labels representing commands,menus, letters, lists, or operational symbols. These are accessed by using
or .
• accesses the commands on the left above the key. Note thatthese commands and are the same color.
• accesses the commands on the right above each key. These commands and are also the same color.
• Pressing engages the [A-LOCK] or Alpha Lock command.This enables you to select consecutive commands without pressing
before each command. This is especially useful when enteringprograms.
Each letter accessed by using can be used to enter words or labelson the screen, but can also be used as a variable. A value can be storedto each variable.
A graphing calculator is also a scientific calculator. That is, it follows the order of operations when evaluating entries. Unlike some scientificcalculators, the graphing calculator displays every entry in the expression.
Before pressing to evaluate the expression, you can use the arrowkeys to scroll through the expression to make corrections. Correctionscan be made in three ways.
• Use to delete any unwanted entries.
• Use [INS] to insert omitted entries.
• “Type” over an incorrect entry. This overprints any entries and doesnot shift the entries to the right as a word processor does.
2nd
DEL
ENTER
ALPHA
ALPHA
ALPHA
ALPHA2nd
ALPHA
ALPHA
2nd
2nd
ALPHA2nd
ENTER
© Glencoe/McGraw-Hill 6 Glencoe Algebra 2
AlternateFunction Keys
Whenever analternate functionis indicated in thekeystrokes of thisappendix, we willuse brackets toshow that thefunction is listedabove a key.
Computation
Math menu Number menu
Complex Number menu Probability menu
If you have an expression that you wish to evaluate repeatedly with a change in one part of the expression, you can press [ENTRY]after you have pressed and the expression will reappear. You can edit it for your next computation. The ENTRY command always repeatsthe last entered expression. You cannot scroll back through previousexpressions you have evaluated.
Evaluate .
Press: [��] 3 4 6
5 12 3 6
Note that the square root function automaticallyincludes a left parenthesis. You must enter theright parenthesis to indicate the end of theexpression under the radical sign. If you havethe decimal in the Float mode, as many as 10digits may appear in the answer.
Evaluate each expression if a � 4, b � �5, c � 2, d � �23�, and e � �1.5.
a. abc � 3de4 b.�ce
2��
48ab�
For a series of expressions that use the same values for the variables, itis often helpful to store the value for each variable into the calculator.You can combine several commands in one line by using the colon aftereach command. The following commands save the values for variablesa, b, c, d and e.
Press: 4 [A] [:] 5 [B] [:] 2
[C] [:] 2 3 [D] [:] 1.5
[E]
a. Method 1: Using stored values
[A] [B] [C] 3
[D] [E] 4
Method 2: Entering computations
4 5 2 3 2 3
1.5 4
b. Method 1: Using stored values
[E] 4 [A]
[C] 8 [B]
Method 2: Entering computations
1.5 4 4 2
8 5 ENTER)(–)�
+x 2(�)�+(–)(
ENTER)ALPHA+x 2ALPHA
(�)ALPHA+ALPHA(
ENTER)
(–)(���—�(–)�
ENTERALPHA
ALPHA—ALPHAALPHAALPHA
ENTERALPHA
STO(–)ALPHAALPHASTO�ALPHAALPHA
STOALPHAALPHASTO(–)ALPHAALPHASTO
ENTER�))(–)—
(+)(—x 22nd
�32 � 4�(6) ��[5 � (��12)]�3�����6
ENTER
2nd
© Glencoe/McGraw-Hill 7 Glencoe Algebra 2
The minus keyand the negativekey are differentkeys.
Example 1
2
© Glencoe/McGraw-Hill 8 Glencoe Algebra 2
Most functions can be graphed by using the key. The viewing window most often used for non-trigonometric functions is the standardviewing window [�10, 10] scl:1 by [�10, 10] scl:1, which can be accessedby selecting 6:ZStandard on the ZOOM menu. Then the window can beadjusted so that a complete graph can be viewed. A complete graph isone that shows the basic characteristics of the parent graph.
Linear Functions A complete linear graph shows the x- and y-intercepts.
a. Graph y � 3x � 4 in the standard viewing window.
Press: 3 4 6
If your calculator is already set for thestandard viewing window, press instead of 6.
Both the x- and y-intercepts of the lineargraph are viewable in this window, so thegraph is complete.
b. Graph y � �2(x � 5) � 2.
Press: 2 5 2
When this equation is graphed in the standard viewing window(Figure 1), a complete graph is not visible. The graph indicates thatthe y-intercept is less than �10. You can experiment with the Yminsetting or you can rewrite the equation in y � mx � b form, whichwould be y � �2x � 12. The y-intercept is �12, so Ymin should beless than �12. Remember that Xmax and Ymax can be less than 10so that your screen is less compressed. Use the WINDOW menu tochange the parameters, or settings, and press to view theresult. There are many windows that will enable you to view thecomplete graph (Figure 2).
Quadratic Functions When graphing quadratic functions, acomplete graph includes the vertex of the parabola and enough of thegraph to determine if it opens upward or downward.
(continued on the next page)
GRAPH
GRAPH—)+((–)Y=
ZOOM
GRAPH
ZOOM—Y=
Y=
TI–83 Plus Graphing Calculator HandbookGraphing Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
22
Example 1
2
Figure 1
[�10, 10] scl:1 by [�10, 10] scl:1
Figure 2
[�10, 10] scl:1 by [�15, 5] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 9 Glencoe Algebra 2
Graph y � 4(x � 3)2 � 4.
Press: 4 3 4
While the standard viewing window shows a complete graph, you may want to change the viewing window to see more of the graph.
Polynomial Functions The graphs of other polynomial functionsare complete when their maxima, minima, and x-intercepts are visible.
a. Graph y � 5x3 � 4x2 � 2x � 4.
Press: 5 3 4 2
4
A complete graph is shown in the standard viewing window. You may want to redefine your window to observe the intercepts,maximum, and minimum points more closely.
b. Graph y � x4 � 13x2 � 36.
Press: 4 13 36
The standard viewing window (Figure 1) does not show a completegraph. It seems that only the y parameters need to be adjusted.Experiment to find a window that is suitable. Figure 2 shows asample.
Exponential Functions A complete graph of an exponentialfunction shows the curvature of the graph and the y-value that itapproaches.
Graph y � 92 � x.
Press: 9 2
Note that you must use parentheses to group theterms that make up the exponent.
A complete graph seems to appear in the secondquadrant of the standard viewing window. Vary the WINDOW settings to view the graph moreclosely.
GRAPH)+(Y=
GRAPH+x 2—Y=
GRAPH+
—x 2+Y=
GRAPH+x 2)—(Y=
Example 3
4
[�7, 1] scl:1 by [�1, 9] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
Figure 1
[�10, 10] scl:1 by [�10, 10] scl:1
Figure 2
[�10, 10] scl:1 by [�8, 40] scl:2
© Glencoe/McGraw-Hill 10 Glencoe Algebra 2
Logarithmic Functions A complete graph of a logarithmic functionshows the curvature of the graph and the values, or locations, of theasymptotes that the curve approaches.
a. Graph y � log (x � 6).
Press: 6
An entire graph appears in the standardviewing window, but is very small. Redefine the WINDOW parameters for y, so that the graph is more visible.
b. Graph y � log4 x.
To graph a logarithmic function with a base other than 10, you must first change the function by using the change of base formula,loga x � �
lloogg
ax
�.
Press:4
An entire graph appears in the standardviewing window, but is very small. Redefine the WINDOW parameters, so that the graph is more visible.
You can graph multiple functions on a single screen. Each function isdenoted by Y1�, Y2�, Y3�, and so on, in the Y� menu. To graph morethan one function, press at the end of each function you are entering and the cursor will move to the next function to be entered.
Systems of Equations
Graph y � 0.5x � 4 and y � 2x2 � 5x � 1.
Press: 0.5 4 2
5 1
The standard viewing window shows that the line and parabola intersect in two points.
GRAPH+—
x 2ENTER+Y=
ENTER
GRAPH)
LOG�)LOGY=
GRAPH
)+LOGY=
Example 5
Example 6
[�10, 10] scl:1 by [�2, 2] scl:1
[�1, 5] scl:1 by [�3, 3] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 11 Glencoe Algebra 2
In addition to graphing a function, you can use other tools on a graphingcalculator to analyze functions. One of those tools is a function table.
How to Use a Table You may complete a table manually orautomatically. To create a table for one or more functions, you mustfirst enter each function into the Y� list. Then set up and create thetable.
a. Use a table to evaluate the function y � 4x2 � 2x � 7 for {�9, �4, 0, 1, 5}.
In this case you only need to evaluate the function for selectedvalues, so use the TBLSET menu to have the calculator ask for thevalues of the independent variable (domain) and find the functionvalue (range) automatically.
Press: 4 27 [TBLSET]
[TABLE] 9 4
0 1 5
b. Use a table to evaluate the functions y � 5x2 � x � 1 and y � 6 � x3 for the integers from �3 to 3, inclusive.
When you want to evaluate a function for a range of values, have thecalculator find both the values of the independent variable and thefunction values automatically. In Table Setup, enter the initialnumber of the domain as the TblStart value and the incrementbetween the values of the independent variable as �Tbl. Enteringmore than one function in the Y� list allows you to evaluate all ofthe functions in one table.
Press: 5 1 6 3[TBLSET] 3 1
[TABLE]
Once you create a table, you can scroll through the values using the arrow keys.
2ndENTER
ENTERENTER(–)
2nd—ENTER
+—x 2Y=
ENTERENTERENTERENTER
(–)ENTER(–)2nd
ENTER2nd
+—x 2Y=
TI–83 Plus Graphing Calculator HandbookAnalyzing Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
33
The functionvalues are thedependentvariable values.
Example 1
allows you to quickly adjust the viewing window of a graph indifferent ways. The effect of each choice on the ZOOM menu is shown onthe next page.
1: ZBox Allows you to draw a box to define the viewing window2: Zoom In Magnifies the graph around the cursor3: Zoom Out Views more of a graph around the cursor4: ZDecimal Sets �X and �Y to 0.15: ZSquare Sets equal-sized pixels on the x-and y-axes6: ZStandard Sets the standard viewing window, [�10, 10] scl:1 by
[�10, 10] scl:17: ZTrig Sets the built-in trig window, ���
4274�
π, �4274�
π� scl: �π2� by [�4, 4]
scl:1 for radians or [�352.5. 352.5] scl:90 by [�4, 4] scl: 1for degrees
8: ZInteger Sets integer values on both the x-and y-axes9: ZoomStat Sets values for displaying all of the data in the current
stat lists0: ZoomFit Fits Ymin and Ymax to show all function values for Xmin
to Xmax
Using to Graph in the Standard and Square Windows
Graph the circle with equation x2 � y2 � 16 in the standardviewing window. Then use ZSquare to view the graph in asquare screen.
First solve the equation for y in order to enter it into the Y� list.
x2 � y2 � 16 → y � � �16 � x�2��
The two pieces of the graph can be entered at one time using {�1, 1}.This expression tells the calculator to graph �1 and 1 times thefunction.
Press: [ { ] 1 1 [ } ] [√__
] 16
6
The circle is distorted when viewed in the standard viewing window.
Press: 5
Using ZSquare makes the circle appear as a circle.
ZOOM
ZOOM
)x 2—2nd2nd,(–)2ndY=
ZOOM
ZOOM
Example 2
© Glencoe/McGraw-Hill 12 Glencoe Algebra 2
[�10, 10] scl:1 by [�10, 10] scl:1
[�15.16, 15.16] scl:1 by [�10, 10] scl:1
Make sure thatCoordOn ishighlighted in theFORMAT menuto display thecursor coordinatesas you trace.
Using to Zoom In and Out Graph y � 0.5x3 � 3x2 � 12in the standard viewing. Zoom out to view a complete graph.Then zoom in to approximate the y-intercept of the graph tothe nearest whole number.
Press: 0.5 3 3 12 6
The complete graph is not shown in the standard viewing window.(Figure 1) When you zoom out or in, the calculator allows you to choosethe point around which it will zoom. Zooming out around the originonce allows a complete graph to be shown. (Figure 2)
Press: 3
Now zoom in to approximate the y-intercept. Choose a point close to theintercept by using the arrow keys.
Press: 2
The y-intercept appears to be about �12.Zooming in again may allow you to makea closer approximation.
The feature allows you to move the cursor along a graph anddisplay the coordinates of the points on the graph.
Using Graph y � 4x � 2 and y � �3x2 � x � 5. Use theTRACE feature to approximate the coordinates of theintersection of the graphs in the first quadrant. Then evaluatey � �3x2 � x � 5 for x � 1.7.
Press: 4 2 3 5
Move the cursor along the graphs using and .
Pressing or moves the cursor more quickly. If your cursor moves offof the screen, the calculator will automaticallyupdate the viewing window so that the cursoris visible. Use and to move from onefunction to the other. The intersection is atabout (0.4, 4).
To evaluate a function for a value and move tothat point, place the cursor on the function graph.Then enter the value and press . When x � 1.7, y � �5.37 for y � �3x2 � x � 5.
ENTER
2nd2nd
TRACE+—x 2(–)ENTER+Y=
TRACE
TRACE
ENTERZOOM
ENTERZOOM
ZOOM—x 2—Y=
ZOOMExample 3
Example 4
© Glencoe/McGraw-Hill 13 Glencoe Algebra 2
Figure 1
[�10, 10] scl:1 by [�10, 10] scl:1
Figure 2
[�40, 40] scl:1 by [�40, 40] scl:1
[�10, 10] scl:1 by [�24.19, �4.19] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 14 Glencoe Algebra 2
The intersectionof the graphsmust appear onthe screen to findthe coordinateswhen usingintersect.
Using to locate the intersection points of the graphs of twofunctions gives you an approximation of the coordinates. For more accuratecoordinates, you can use the intersect option on the CALC menu.
Finding Intersection Points Use 5:intersection on the CALCmenu to find the coordinates of the intersection of the graphs ofy � 4x � 2 and y � �3x2 � x � 5.
If you do not have the functions graphed, enter the functions into theY� list and press . Then find the coordinates of the intersection.
Press: [CALC] 5
Place the cursor on one graph and press . Then move the cursor to the other
graph and press . To guess at theintersection or enter an x-value and press
. If there is more than one intersectionpoint, the caculator will find the one closestto your guess. The cursor will move to theintersection point and the coordinates will be displayed.
The CALC menu also allows you to find the zeros of a function.
Finding Zeros Find the zeros of f(x) � �2x4 � 3x2 � 2x � 5.
Press: 2 4 3 2 5 [CALC] 2
The calculator can find one zero at a time. Usethe arrow keys or enter a value to choose the left bound for the interval in which the calculator will search for the zero and press
. Choose the right bound and press
. Select a point near the zero using the arrow keys or by entering a value and press
. Repeat with another interval to find the other zero. The zeros of this function are about �1.42 and 1.71.
Real-world application problems often require you to find the relativeminimum or maximum of a function. You can use 3:minimum or4:maximum features on the CALC menu of a graphing calculator tosolve these problems.
ENTER
ENTER
ENTER
2nd++x 2+(–)Y=
ENTER
ENTER
ENTER
2nd
GRAPH
TRACE
Example 5
Example 6
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 15 Glencoe Algebra 2
Finding Maxima and Minima Determine the relativeminimum and the relative maximum for the graph of f(x) � 4x3 � 6x � 5.
First graph the function.
Press: 4 3 6 5 6
To find the relative minimum press [CALC] 3.
Similar to finding a zero, choose the left and right bound of the interval and guess theminimum or the maximum. The point at about(0.71, 2.17) is a relative minimum.
Use a similar method to find the relative maximum, by pressing [CALC] 4. The point at about (�0.71, 7.83) is a relativemaximum.
2nd
2nd
ZOOM+—Y=
Example 7
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 16 Glencoe Algebra 2
NAME ______________________________________________ DATE ____________ PERIOD _____
44 TI–83 Plus Graphing Calculator HandbookGraphing Inequalities
Most linear and nonlinear inequalities can be graphed using the key and selecting the appropriate graph style in the Y� editor. To select theappropriate graph style, select the graph style icon in the first column of theY� editor and press repeatedly to rotate through the graph styles.
• To shade the area above a graph, select the Above style icon, .
• To shade the area below a graph, select the Below style icon, .
Before graphing an inequality, clear any functions in the Y� list bypressing and then using the arrow keys and the key to selectand clear all functions. If you do not wish to clear a function, you can turnthat particular graph off by using the arrow keys to position the cursorover that function’s � sign and then pressing to change theselection status.
Linear Inequalities
a. Graph y � 2x � 3 in the standard viewing window.
First enter the boundary equation y � 2x � 3 into the Y� list.
Press: 2 3
Next, press the key until the icon before � flashes. Press until the icon changes to the Below style icon, , for “y �”.Finally, if your calculator is not already set forthe standard viewing window, press 6.Otherwise, press .
b. Graph y � �4x � 5 in the standard viewing window.
Press: 4 5
Next, press the key until the icon before � flashes.Then press until the icon changes to the Above style icon, , since the inequality asks for “y ”. Finally, press .GRAPH
ENTER
+(–)Y=
GRAPH
ZOOM
ENTER
—Y=
ENTER
CLEARY=
ENTER
Y=
Example 1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
Nonlinear Inequalities The procedure for graphing nonlinearinequalities is the same as that of graphing linear inequalities.
a. Graph y � 0.25x2 � 4 in the standard viewingwindow.
Press: 0.25 4
Next, select the Below style icon, , since theinequality asks for “y �”, and then press
.
b. Graph y � 0.2x4 � 3x2 � 4.
Press: 0.2 4
2 4
Next, select the Above style icon, , since the inequality asks for “y ”. Then press
.
c. Graph y � �x � 2� � 4.
Press: [�� ] 2 4
Next, select the Below style icon, , since the inequality asks for “y �”. Then press
.
d. Graph y � 3x � 5.
Press: 3 5
Next, select the Above style icon, , since the inequality asks for “y ”. Then press
.GRAPH
—Y=
GRAPH
+)+2ndY=
GRAPH
+
—Y=
GRAPH
—x 2Y=
Example 2
© Glencoe/McGraw-Hill 17 Glencoe Algebra 2
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
The Shade(command canonly be used withtwo inequalitieswhich can bewritten with “y �”in one inequalityand “y ” in theother.
Graphing systems of inequalities on a graphing calculator is similar tographing systems of equations.
Graph the system of inequalities.y � 2x � 5 y � x2 � 4x � 1
Method 1: Shading Options in Y�
Press: 2 5
4 1
Select the Above style icon, , for y 2x � 5 and the Below style icon, , for y � x2 � 4x � 1.Then press .
Notice that the first inequality is indicated using vertical lines and the second inequalityuses horizontal lines. The solution to the systemis shown by the intersection of the shaded areas.
Method 2: Using the Shade Command
Some systems of inequalities can be graphed by using the Shade(command and entering a function for a lower boundary and a functionfor the upper boundary of the inequality. The calculator first graphsboth functions and then shades above the first function entered andbelow the second function entered.
Before graphing an inequality using the Shade( command, clear any graphics from the viewing window by pressing [DRAW] 1 .Also clear any equations in the Y� list. If not already there, return tothe home screen by pressing [QUIT].
Press: [DRAW] 7 2 5
4 1 ENTER)+—x 2
,—2nd
2nd
ENTER2nd
GRAPH
+—
x 2ENTER—Y=
Example 3
© Glencoe/McGraw-Hill 18 Glencoe Algebra 2
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
TI–83 Plus Graphing Calculator HandbookMatrices
NAME ______________________________________________ DATE ____________ PERIOD _____
55
© Glencoe/McGraw-Hill 19 Glencoe Algebra 2
A graphing calculator can perform operations with matrices. It can alsofind determinants and inverses of matrices. The MATRX menus areaccessed using [MATRX].
There are three menus in the MATRX menu.
• The NAMES menu lists the matrix locations available. There are tenmatrix variables, [A] through [J].
• The MATH menu lists the matrix functions available.
• The EDIT menu allows you to define matrices.
A matrix with dimension 2 3 indicates a matrix with 2 rows and 3columns. Depending on available memory, a matrix may have up to 99rows or columns.
Entering a Matrix Enter matrix A � � �.To enter a matrix into your calculator, choose the EDIT menu andselect the matrix name. Then enter the dimensions and elements of thematrix.
Press: [MATRX] 22 1 3 2 (�) 2
Press [QUIT] to return to the HOME screen. Then press [MATRX]
to display the matrix.
You can find the determinant and inverse of a matrix very quickly witha graphing calculator.
Determinant and Inverse of a Matrix
a. Find the determinant of matrix A.
Press: [MATRX] 1 [MATRX]1
The determinant of matrix A is �8.
b. Find the inverse of matrix A.
Press: [MATRX] 1
A�1� � �0.375�0.125
0.250.25
ENTERx –12nd
ENTER
2nd2nd
ENTER
ENTER2nd
2nd
ENTERENTERENTERENTER
ENTERENTER2nd
3�2
12
2nd
Example 1
Example 2
Operations with Matrices
Enter matrix B � � �. Then perform each operation.
a. �12
�B
First, enter matrix B.
Press: [MATRX] 2 2
3 2 3 64 8 5 [QUIT]
Then find �12
�B.
Press: .5 [MATRX] 2
�12
�B � � �
b. AB
Press: [MATRX] 1 [MATRX]
2
AB � � �
c. A2
Press: [MATRX] 1
A2 � � �
d. AB � B
Press: [MATRX] 1 [MATRX]
2 [MATRX] 2
AB � B � � �277
�1814
160
ENTER2nd+
2nd2nd
�310
7�2
ENTERx 22nd
212
�2122
14�4
ENTER
2nd2nd
32.5
1.5�4
12
ENTER2nd
2ndENTER(–)ENTER
ENTERENTERENTERENTER
ENTER2nd
65
3�8
24
Example 3
© Glencoe/McGraw-Hill 20 Glencoe Algebra 2
TI–83 Plus Graphing Calculator HandbookGraphing Trigonometric Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
66
© Glencoe/McGraw-Hill 21 Glencoe Algebra 2
Trigonometric functions and the inverses of trigonometric functions canbe graphed using . The functions and their inverses can be graphedin degrees or radians. You must set the calculator in Radian or Degreemode. The standard viewing window for trigonometric functions can beset by pressing 7:Trig, which automatically adjusts the x- and y-axes scales for degrees or radians.
a. Using Degrees Graph y � cos x.
First, set the calculator in degree modeby pressing .
Now enter and graph the function. Press 7.
b. Using Radians Graph y � sin x.
Change to radian mode by pressing
. Press to delete
the function entered in part a. Then press
to enter the new function.
Next, press 7 to set the viewing window to accommodate radian mode and graph the function.
c. Amplitude, Period, and Phase Shift Graph y � 2 sin ��12
� x � 60°�using the viewing window [�540, 540] scl:90 by [�3, 3] scl:1.Then state the amplitude, period, and phase shift.
Make sure the calculator is in degree mode.
Press: 2 2 60
The amplitude of the function is �|�22� 2|�
or 2. The period is or 720°, and
the phase shift is or �120°.��
160�
�2
360��
�12�
GRAPH
)—�SINY=
ZOOM
)SIN
CLEARY=ENTER
MODE
ZOOM)COSY=
ENTERMODE
ZOOM
Y=
Example 1
[�352.5, 352.5] scl:90 by [�4, 4] scl:1
[�2 , 2 ] scl:�2
� by [�4, 4] scl:1
[�540, 540] scl:90 by [�3, 3] scl:1
© Glencoe/McGraw-Hill 22 Glencoe Algebra 2
Most special functions can be graphed using the key. The absolutevalue function abs( and the greatest integer function int( can be found inthe MATH NUM menu.
Absolute Value Graph y � 2|x � 4|.
Press 2 1 4
6 to graph the function in the standard viewing window.
Greatest Integer Function Graph y � [[x � 1.5]].
First, make sure the calculator is set for dotplotting rather than the connected plotting usedin most other functions. Press , highlightDot, and press .
Then, enter the function. Press
5 1.5 6. If your
calculator is already set for the standard
viewing window, press instead of 6.
The TEST menu allows you to graph other piecewise functions. Enter thepieces of the function as a sum of the products of each piece of the
function and its domain. For example, y � � is entered as
(5)(X<2) � (4X)(X>2) in the Y� menu.
Piecewise Function Graph y � � .
Place the calculator in Dot mode. Then enter the function in the Y�list using the TEST menu options.
Press: 3 [TEST] 6 3 1
3 [TEST] 5
[TEST] 6 29 2
[TEST] 3 2 6ZOOM)
2nd()—(
+)2nd(
)2nd(–)(
)+(+)(–)
2nd()(Y=
3 if x � �31 � x if �3 x � 29 � 2x if x 2
5 if x � 24x if x 2
ZOOM
GRAPH
ZOOM)—
MATHY=
ENTER
MODE
ZOOM
)—MATHY=
Y=
TI–83 Plus Graphing Calculator HandbookGraphing Special Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Example 1
Example 3
2
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 23 Glencoe Algebra 2
TI–83 Plus Graphing Calculator HandbookStatistics and Statistical Graphs
NAME ______________________________________________ DATE______________ PERIOD _____
99
A graphing calculator allows you to enter a set of data and generatestatistics and statistical graphs. Before you enter data values, make sureyou clear the Y� list, L1 and L2, and the graphics screen. Clear the Y�list by pressing . Use the key to select additional equations and clear them also. To clear L1 and L2, press 4[L1] [L2] . If you need to clear the graphics screen, press [DRAW] 1 .
Enter Data into Lists Enter the following data into a graphingcalculator.
49 53 54 54 56 55 57 61 51 58 41 59 54 50 60 44
Press: 1 49 53 54 54
56 55 57 61
51 58 41 59
54 50 60 44
You can use the up and down arrow keys to scroll through the list.
Find Mean, Median, and Mode Find the mean, median, andmode of the data in Example 1.
Press 1 . This function displays many statistics aboutthe data. X–– denotes the mean. Scroll down to find the median.
The calculator does not have a function to determine the mode. You canfind the mode by examining the data. First sort the data to write themin order from least to greatest.
Press: 2 [L1]
Then scroll through the data by pressing 1 and using the and keys. You will find that the mode is 54.
STAT
ENTER)2ndSTAT
ENTERSTAT
ENTERENTERENTERENTER
ENTERENTERENTERENTER
ENTERENTERENTERENTER
ENTERENTERENTERENTERSTAT
ENTER
2ndENTER,2ndSTAT
CLEARY=
Example 1
2
mean
median
ENTER
© Glencoe/McGraw-Hill 24 Glencoe Algebra 2
Box-and-Whisker Plots
a. Draw a box-and-whisker plot for the data.
30.2 29.0 26.2 25.8 23.8 43.0 19.8 19.4 26.0 46.6 26.8 22.8 35.4 25.2 12.2 31.4
Set the viewing window. Next, set the plottype. Press [STAT PLOT] 1
to highlight and press. Make sure L1 is entered in Xlist:.
If not, move the cursor to highlight andpress [L1] . Then, enter the data into L1. Press 1 30.229 ... 31.4 .
b. Draw a box-and-whisker plot with outliers using the data.
Without clearing the lists or graphic screen,press [STAT PLOT] 2
, to highlight , and press .
Make sure L1 is entered in Xlist:.Press .
c. Find the upper and lower quartiles, the median, and the outliers.
Press and use and to move the cursor along the graph.The values will be displayed. For this data, the upper quartile is30.8, the lower quartile is 23.3, the median is 26.1, and the outliersare 43 and 46.6.
Histograms
a. Use the data on the number of public libraries in each stateand Washington, D.C., to make a histogram.
273 102 159 196 1030 235 244 30 27 428 366 49 141772 427 554 372 188 322 273 187 491 659 361 243 346110 283 78 238 455 92 1067 352 86 684 192 201 64074 180 134 284 753 96 204 308 309 174 451 74
Enter the data in L1. Press 1 273 102 ... 74 .
Set the viewing window. Choose Xmin, Xmax, and Xscl to determinethe number of bars in the histogram. For this data, the least value is27 and the greatest is 1067. If Xmin � 0, Xmax � 1100, and Xscl � 100, the histogram will have 11 bars each representing aninterval of 100.
Choose the type of graph. Press
[STAT PLOT] 1
[L1] 1 . Then press
to draw the histogram.GRAPH
ENTERENTER2nd
ENTERENTER
2nd
ENTERENTERENTERSTAT
TRACE
GRAPH
ENTER
ENTER2nd
GRAPHENTERENTER
ENTERSTAT
ENTER2nd
ENTER
ENTER2nd
Examples 3
4
[10, 50] scl:2 by [0, 10] scl:1
[0, 1100] scl:100 by [0, 12] scl:1
[10, 50] scl:2 by [0, 10] scl:1
© Glencoe/McGraw-Hill 25 Glencoe Algebra 2
Make sure youhave cleared theY= list, L1 andL2, and thegraphic screen.
b. Use the data in the table to draw ahistogram and its frequency polygram.
Enter the class marks as L1. Press
1 162.5 177.5 ... 312.5 .
Move the cursor to L2. Enter thefrequencies.
Press 7 15 ... 3[QUIT].
Set the viewing window. Use the minimum and maximum of theclass limits for Xmin and Xmax. Use the size of the intervals forXscl. Choose the y-axis values to show the complete histogram.
Set the plot type. Press [STAT PLOT] 1[L1] [L2] . Then press
.(Figure 1) Without clearing the lists or graphic screen,press [STAT PLOT] 1 to highlight , and press . Make sure L1 is entered in Xlist: and L2 is the Ylist.Choose as the mark. Press .
Scatter Plot, Connected Line Scatter Plot, and Regression Line
a. Use these data to draw a scatter plot:(20.0, 5.2), (10.2, 1.9), (7.3, 1.6), (6.8, 2.6),(5.9, 1.0), (2.6, 0.7), (2.8, 0.35), (2.7, 0.15).
Clear previous data and graphs and setthe viewing window. Enter the x-valuesinto L1 and the y-values into L2. Thendraw the scatter plot by pressing [STAT PLOT] 1 to highlight and press . Make sure L1 is the Xlist: and L2 is the Ylist:. Then press
.
b. Use the data to draw a line graph.
Press [STAT PLOT] 1 to highlight and press .GRAPHENTER
2nd
GRAPH
ENTER
ENTER
2nd
GRAPH
ENTER
ENTER2nd
GRAPH
ENTER2ndENTER2ndENTER
ENTER2nd
2ndENTERENTERENTER
ENTERENTERENTER
STAT
Example 5
Class Limits Frequency
155–170 7170–185 15185–200 34200–215 38215–230 42230–245 35245–260 33260–275 21275–290 18290–305 6305–320 3
[155, 320] scl:15 by [0, 50] scl:5
Figure 1
[155, 320] scl:15 by [0, 50] scl:5
Figure 2
[0, 25] scl:5 by [0, 6] scl:1
[0, 25] scl:5 by [0, 6] scl:1
© Glencoe/McGraw-Hill 26 Glencoe Algebra 2
c. Draw a regression line for the data in the table.
Set the plot to display a scatter plot by pressing [STAT PLOT] 1
[QUIT].
To calculate the coefficients of regression press 4 .
Then, write the equation of the regressionline. You can automatically enter theregression equation in the Y� list.
Press 5 1. Finally, graph the regression line by pressing .
There are also regression models for analyzing data that are not linearbuilt into the calculator.
Nonlinear Regression Find a sine regression equation tomodel the data in the table. Graph the data and the regressionequation.
Enter the data into lists L1 and L2. Press 1 2 ...12 39 42 ... 40 .
Find the regression statistics.
Press: [C]
Enter the regression equation into the Y� list.
Press: 5 1
Then format the scatter plot to graph the data by pressing [STAT PLOT] 1 .Make sure that L1 is chosen as the Xlist and L2 is chosen as the Ylist. Set the viewingwindow. Press to see the scatter plot and the graph of the regression equation.
GRAPH
ENTER2nd
VARSY=
ENTERALPHASTAT
ENTERENTERENTERENTER
ENTERENTERSTAT
GRAPH
VARSY=
ENTERSTAT
2nd
ENTER2nd
Example 6
[0, 25] scl:5 by [0, 6] scl:1
x 1 2 3 4 5 6 7 8 9 10 11 12
y 39 42 45 48 54 59 63 64 59 52 44 40
[0, 13] scl:1 by [30, 70] scl:5
Graphing Calculator InvestigationSolving Equations and Checking Solutions(Use with Lesson 1-3.)
NAME ______________________________________________ DATE ____________ PERIOD _____
11
© Glencoe/McGraw-Hill 27 Glencoe Algebra 2
When solving equations, checking the solutions is an important process. Agraphing calculator can be used to check the solution of an equation.
Solve �2(5y � 1) � y � �4(y � 3).
Graph the expression on the left side of the equation in Y1 and the expression on the right side of the equation in Y2. Choose anappropriate view window so that the intersection of the graphs isvisible. Then use the intersect command to find the coordinates ofthe common point.
Keystrokes: 2 5 1 4 3 6 8 [CALC] 5
[QUIT] .
The x-coordinate, ��
710�, is the solution to the equation. The y-coordinate
is the value of both sides of the equation when x � ��
710�.
ENTERENTERMATH2ndENTERENTERENTER
2ndENTERZOOMZOOM)—(
(–)ENTER—)—((–)Y=
Solve each equation.
1. �3(2w � 7) � 9 � 2(5w � 4) 2. 1.5(4 � x) � 1.3(2 � x) 3. �14�(a � 2) � �
16�(5 � a)
w � ��52
� x � 17 a � �45
�
4. 3(2z � 25) � 2(z � 1) � 78 5. �m
3� 4� � �
3m5� 1� � 1 6. �
x �2
5� � �
12� � 2x � �
x �8
3�
z � �14
� m � �8 x � �2111�
Example 1Example 1
Solve �x5� � �
x4� � �
12�(x � 2).
Graph the expression on the left side of the equation in Y1 and theexpression on the right side of the equation in Y2. Enter Y1 - Y2 inY3. Then graph the function in Y3. Use the zero function under theCALC menu to determine where the graph of Y3 equals zero. Thispoint will be the solution.
Keystrokes: 5 4 1 2 2
2 6 [CALC] 2.
Use arrow keys and enter to set the bound prompts. The solution is x � �2101�
.
2ndZOOMENTER
ENTERENTERVARS—ENTERENTER
VARSENTER)—()�(ENTER
)�(—)�(Y=
Example 2Example 2
ExercisesExercises
[�47, 47] scl:10 by [�31, 31] scl:10
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 28 Glencoe Algebra 2
You can use a spreadsheet to try several different values in an equation tohelp you determine whether the statement is sometimes, always, or nevertrue. Remember that showing that a statement is true for some values doesnot prove that it is true for all values. However, finding one value for which astatement is false proves that it is not true for all values.
Spreadsheet InvestigationAbsolute Value Statements (Use with Lesson 1-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
11
Use a spreadsheet to determine whether each absolute value statement is sometimes, always, or never true.
1. For all real numbers a and b, a � 0, |ax � b| � 0. sometimes
2. If a and b are real numbers, then |a � b| � |a| � |b|. sometimes
3. If a and b are real numbers, then |a � b| � �x. never
4. If a and b are real numbers, then |a| � |b| � a � b. sometimes
5. If a and b are real numbers, then c|a � b| � c|a| � |b|. sometimes
ExercisesExercises
Determine whether c|a � b| � |ca � cb| is sometimes,always, or never true.
Try a number of values for a, b, and c to determine whether the statement istrue or false for each set of values.
Step 1 Use Columns A, B, and Cfor the values of a, b, andc. Choose several sets ofvalues including positiveand negative numbers,and zero.
Step 2 Use Column D to testthe equation. A formulasuch as C2*ABS(A2�B2)� ABS(C2*A2�C2*B2)in cell D2 returns TRUEif the equation is true.
Through observation of Column D, when c is negative the statement is nottrue. The absolute value statement, c|a � b| � |ca � cb| is sometimes true;it is true only if c � 0.
ExampleExample
Graphing Calculator InvestigationTransformations: Greatest Integer Function(Use with Lesson 2-6.)
NAME ______________________________________________ DATE ____________ PERIOD _____
22
© Glencoe/McGraw-Hill 29 Glencoe Algebra 2
A graphing calculator can be used to display transformations to the greatestinteger function. This is done by using the int( command under the MATH:NUM menu. When graphing the greatest integer function, it is important toset the calculator to Dot mode.
Graph each function. Evaluate it for x � 1, x � 1.3, and x � 2.Compare the graph of the function to the graph of f(x) � [[x]].
1. g(x) � [[x]] � 3 2. g(x) � [[x � 2]] 3. g(x) � [[x + 4]]
[�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1
g(1)� �2, g(1.3) � �2, g(2) � �1 g(1) � �1, g(1.3) � �1, g(2) �0 g(1) �5, g(1.3) �5, g(2) �6shifted down 3 units shifted right 2 units shifted left 4 units
4. g(x) � [[�x]] 5. g(x) � [[2x]] 6. g(x) � 3[[x]]
[�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1
g(1)� �1, g(1.3) � �2, g(2) � �2 g(1) � 2, g(1.3) � 2, g(2) � 4 g(1) � 3, g(1.3) � 3, g(2) � 6reflected across the y-axis compressed by �
12
� horizontally expanded by 3 vertically
ExercisesExercises
Graph f(x) � [[x]] and g(x) � [[x]] � 2 on the same axes.Evaluate each function for x � 0.6, x � 1, x � 1.3, x � 2,
x � �0.5, and x � �1.1. Compare the graphs of the functions.Enter f(x) in Y1 and g(x) in Y2. Graph the functions.Keystrokes: 5 5 2
6 4.
Use TRACE to evaluate each function for the given values.Keystrokes: 0.6 1 1.3 2 0.5
1.1 0.6 1 1.3 2
0.5 1.1 .
f(0.6) � 0, f(1) � 1, f(1.3) � 1, f(2) � 2, f(�0.5) � �1, and f(�1.1) � �2g(0.6) � 2, g(1) � 3, g(1.3) � 3, g(2) � 4, g(�0.5) � �1, and g(�1.1) � 0
The graph of g(x) � [[x � 2]] is the same as the graph of f(x) � [[x]]shifted 2 units up.
ENTER(–)ENTER
(–)ENTERENTERENTERENTERENTER(–)ENTER
(–)ENTERENTERENTERENTERTRACE
ZOOMZOOM
ENTER+)MATHENTER)MATHY=
ExampleExample
[�4.7, 4.7] scl:1 by [�3.1, 3.1] scl:1
© Glencoe/McGraw-Hill 30 Glencoe Algebra 2
The slope intercept form of a linear equation is y � mx � b, where m is theslope and b is the y-intercept. Recall that the formula for the slope of a line through (x1, y1) and (x2, y2) is m � �
yx
22
��
yx
11
�.You can use the formula for slope and the slope-intercept form to find the value of b.
Spreadsheet InvestigationUsing Linear Equations (Use with Lesson 2-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
22
Use a spreadsheet to find the slope and y-intercept of the line through each pairof points. Then write an equation of the line in slope-intercept form.
1. (0, �5), (2, 5) 2. (4, 2), (�3, �5)5, �5; y � �5x � 5 1, �2; y � �x � 2
3. (�1, �4), (1, 3) 4. (�4, �9), (8, 3)
�72
�, ��12
�; y � ��72
�x � �12
� �1121�, �5�
13
�; y � �1121�x � 5�
13
�
5. (12, 9), (10, 10) 6. (�1.5, 3.1), (0.9, 1.9)
��12
�, 15; y � ��12
�x � 15 �0.5, 2.35; y � �0.5x � 2.35
7. Does the spreadsheet work when two points have the same x-coordinates? Explain.No; The slope is undefined.
ExercisesExercises
State the slope and y-intercept of the graph of the linethrough (5, 2) and (4, 1). Then write an equation of the
line in slope-intercept form.
Step 1 Use Columns A and B to represent the first point, and Columns Cand D to represent the second point on the line. Enter the formulafor slope in Column E.
Step 2 Substitute one of the given points into the slope-intercept from andsolve for b. Since we know the slope of the line, we can solve for b.
y � mx � b Slope-intercept formy1 � mx1 � b Substitute (x1, y1).
y1 � mx1 � b Solve for b.
Enter this formula into Column F using the names of the spreadsheet cells.
The slope of the line through (5, 2) and (4, 1) is 1 andthe y-intercept is �3. Thus, the equation of the lineslope-intercept form is y � 1x � (�3) or y � x � 3.
ExampleExample
Graph the system x � 3y � �7, 5x � y � 13, x � 6y � �9,3x � 2y � �7, and f(x, y) � 4x � 3y. Find the coordinates
of the feasible region. Then find the maximum and minimum valuesfor the system.Solve each inequality for y. Enter each boundary equation in the Y�screen. Find the vertices of the feasible region. Then find the valuesof f(x, y) to determine the maximum and minimum values.Keystrokes: 1 3 7 3
5 13 1 6 3
2 3 2 7 2 6
[CALC] 5 [QUIT] [ { ]
[Y] 4 3 [Y] [ } ]
[CALC] 5 [QUIT]
[ENTRY] [CALC] 5
[QUIT] [ENTRY]
[CALC] 5 [QUIT]
[ENTRY] .
The maximum value of the system is 18 and the minimum value is �10.
ENTER
2nd2ndENTERENTERENTER
2ndGRAPHENTER2nd2ndENTERENTER
ENTER2ndGRAPHENTER2nd
2ndENTERENTERENTER2ndGRAPH
ENTER2ndALPHA—,ALPHA,2nd2ndENTERENTERENTER2nd
ZOOM)�(+)�(ENTER)
�(—)�((–)ENTER+(–)
ENTER)�(+)�(Y=
ExampleExample
Graphing Calculator InvestigationLinear Programming (Use with Lesson 3-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
© Glencoe/McGraw-Hill 31 Glencoe Algebra 2
A graphing calculator can store the x- and y-coordinates when using theintersect command in the [CALC] menu. This can be displayed on thehome screen and used to evaluate an expression with x and y variables. Thisprocess is useful in finding the vertices of the feasible region and determiningthe maximum or minimum value for f(x, y).
[�10, 10] scl:1 by [�10, 10] scl:1
Graph each system. Find the coordinates of the vertices of the feasibleregion. Then find the maximum and minimum values for the system.
1. 2x � 3y � 6 2. y � 4x � 6 3. y � 16 � x3x � 2y � �4 x � 4y � 7 0 � 2y � 175x � y � 15 2x � y � 7 2x � 3y � 11f(x, y) � x � 3y x � 6y � 10 y � 3x � 1
f(x, y) � 2x � y y � 2x � 13y � 7 � 2xf(x, y) � 5x � 6y
[�10, 10] scl:1 by [�10, 10] scl: 1 [�10, 10] scl:1 by [�10, 10] scl: 1 [�10, 10] scl:1 by [�10, 10] scl: 1
(0, 2), (3, 0), (2, 5); (�1, 2), (�2, �2), (3, 1), (4, �1); (5.5, 0), (6.5, 0), (7.5, 8.5),min. � 3, max. � 17 min. � �4, max. � 9 (1.2, 4.6), (2.5, 2), (9.66, 6.33);
min. � 24.5, max. � 88.5
ExercisesExercises
© Glencoe/McGraw-Hill 32 Glencoe Algebra 2
Spreadsheet InvestigationBreak-Even Point (Use with Lesson 3-1.)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
1. If Carly could decrease her annual overhead to $14,000, what would thebreak-even point be? between 1400 and 1500 candles
2. Suppose Carly decreases her annual overhead to $14,000 and increasesthe price of a candle to $14.00. What is the new break-even point?between 1200 and 1300 candles
ExercisesExercises
Carly Ericson is considering opening a candle business. She estimates that she will have an annual
overhead of $15,000. It costs Carly $3.00 to make a jar candle, whichshe sells for $12.50. What is Carly’s break-even point?
Use Column A for the number of candles. Columns Band C are the cost and the income, respectively.
Extend the rows of the spreadsheet to find the point atwhich the income first exceeds the cost. The break-evenpoint occurs between this point and the previous point.In this case, the break even point occurs between 1500and 1600 candles.
The chart tool of the spreadsheet allows you to graphthe data. The graph verifies the solution.
ExampleExample
You have learned that the break-even point is the point at which the incomeequals the cost. You can use the formulas and charts in a spreadsheet to finda break-even point.
Graphing Calculator InvestigationMatrices for 30°, 45°, 60° Rotations (Use with Lesson 4-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
© Glencoe/McGraw-Hill 33 Glencoe Algebra 2
The rotation matrix for 90° counterclockwise about the origin is [0 �11 0 ].
The general rotation matrix for any angle counterclockwise about the
origin is [cos �sin].
Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(4, 6), and D(0, 6).Find the coordinates of the vertices of the image after each counterclockwise rotation. Round to the nearest tenth.
1. 45° 2. 30° 3. 60°A(0, 0), B(2.8, 2.8), A(0, 0), B(3.5, 2), A(0, 0), B(2, 3.5),C(�1.4, 7.1), D(�4.2, 4.2) C(0.5, 7.2), D(�3, 5.2) C(�3.2, 6.5), D(�5.2, 3)
4. 120° 5. 75° 6. 225°A(0, 0), B(�2, 3.5), A(0, 0), B(1, 3.9), A(0, 0), B(�2.8, 2.8),C(�7.2, 0.5),D(-5.2, �3) C(�4.8, 5.4), D(�5.8, 1.6) C(�7.1, �1.4), D(�4.2, �4.2)
ExercisesExercises
Find the coordinates of the image of �ABC with vertices A(0, 0), B(6, 0) and C(3, 4) after a
counterclockwise rotation of 30° about the origin.Enter the coordinates of the vertices of the in vertex matrix [A] andthe rotation matrix in matrix [B]. Be sure the calculator is set inDegree mode. Keystrokes: [MATRX] 2 3 0 6 3 0 0 4 [MATRX] 2 2 2 30
30 30 30 [QUIT] [MATRX] 2 [MATRX] 1 . Hold to scroll across to see the other coordinates.
The coordinates of the image of �ABC are A(0, 0), B(5.2, 3), andC(0.6, 5.0).
ENTER2nd2nd
2ndENTER)COS)SINENTER)SIN
(–)ENTER)COSENTERENTER
2ndENTERENTERENTERENTERENTERENTERENTER
ENTERENTER2nd
Example 1Example 1
Find the coordinates of the image of �ABC with vertices A(0, 0), B(6, 0) and C(3, 4) after a two
rotations of 45° counterclockwise about the origin.In order to rotate the image twice, store the vertex matrix of the first image.Keystrokes: [MATRX] 2 45
45 45 45
[QUIT] [MATRX] 2 [MATRX] 1
[ENTRY] [MATRX] 6 [ENTRY]
[ENTRY] [MATRX] 6 .
The vertices of �ABC are A(0, 0), B(0, 6), and C(�4, 3).
ENTER2nd
2nd2ndENTER2ndSTO
2ndENTER2nd2nd2nd
ENTER)COS)SINENTER)SIN(–)ENTER
)COSENTERENTER2nd
Example 2Example 2
sin cos
© Glencoe/McGraw-Hill 34 Glencoe Algebra 2
Spreadsheet InvestigationCramer’s Rule (Use with Lesson 4-5.)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
You have learned to solve systems of linear equations by using matrix equations and the inverse matrix. Another way to solve systems is to useCramer’s Rule. Study the spreadsheet below to discover Cramer’s Rule.
To use the spreadsheet to solve asystem of equations, write eachequation in the form below.
ax � by � c
The values for the system 6x � 3y� �12 and 5x � y � 8 are shown.In the spreadsheet, the values of a,b, and c for the first equation areentered in cells A1, B1, and C1,respectively. The values of a, b, andc for the second equation areentered in cells A2, B2, and C2,respectively.
The values in cells B10 and B11represent the solution for the system.
1. Study the formula in cell A4. Write a matrix whose determinant is found using this formula.
[A1 B1]A2 B22. Write matrices whose determinants are found using the formulas in cells A6 and A8.
[C1 B1] ; [A1 C1]C2 B2 A2 C23. Explain how the values of x and y are found using Cramer’s rule.
|C1 B1| |A1 C1|x �C2 B2
; y �A2 C2
|A1 B1| |A1 B1|A2 B2 A2 B2Use the spreadsheet to solve each system of equations.
4. 6x � 3y � �12 5. 5x � 3y � 19 6. 8x � 3y � 115x � y � 8 7x � 2y � 8 6x � 9y � 15(4, �12) (2, �3) (1.6, 0.6)
7. 0.3x � 1.6y � 0.44 8. 3y � 4x + 28 9. y � �0.5x � 40.4x � 2.5y � 0.66 5x � 7y � 8 y � 4x � 5(0.4, 0.2) (�4, 4) (2, 3)
ExercisesExercises
Graphing Calculator InvestigationUsing Tables to Factor by Grouping (Use with Lesson 5-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
© Glencoe/McGraw-Hill 35 Glencoe Algebra 2
The TABLE feature of a graphing calculator can be used to help factor apolynomial of the form ax2 � bx � c.
Factor 10x2 � 43x � 28 by grouping.
Make a table of the negative factors of 10 � 28 or 280. Look for a pairof factors whose sum is �43.
Enter the equation y � �28
x0
� in Y1 to find the factors of 280. Then,
find the sum of the factors using y � �28
x0
� � x in Y2. Set up the table
to display the negative factors of 280 by setting �Tbl = to �1.Examine the results.
Keystrokes: 280 [TBLSET] 1 1
[TABLE].
The last line of the table shows that �43x may be replaced with �8x +(�35x).
10x2 � 43x � 28 � 10x2 � 8x � (�35x) � 28� 2x(5x � 4) � (�7)(5x � 4)� (5x � 4)(2x � 7)
Thus, 10x2 � 43x � 28 � (5x � 4)(2x � 7).
2ndENTER(–)ENTER(–)2ndENTER
+ENTERENTERVARSENTER�Y=
Factor each polynomial.
1. y2 � 20y � 96 2. 4z2 � 33z � 35 3. 4y2 � y �18 4. 6a2 � 2a � 15(y � 4)(y � 24) (4z � 5)(z � 7) (4y � 9)(y � 2) prime
5. 6m2 � 17m � 12 6. 24z2 � 46z � 15 7. 36y2 � 84y � 49 8. 4b2 � 36b � 403(2m � 3)(3m � 4) (12z � 5)(2z � 3) (6y � 7)2 (2b � 31)(2b � 13)
Example 1Example 1
Factor 12x2 � 7x � 12.
Look at the factors of 12 � �12 or �144 for a pair whose sum is �7.Enter an equation to determine the factors in Y1 and an equation tofind the sum of factors in Y2. Examine the table to find a sum of �7.Keystrokes: 144
[TBLSET] 1 1 [TABLE].
12x2 � 7x � 12 � 12x2 � 9x � (�16x) � 12� 3x(4x � 3) � 4(4x � 3)� (4x � 3)(3x � 4)
Thus, 12x2 � 7x � 12 � (4x � 3)(3x � 4).
2ndENTERENTER2ndENTER+
ENTERENTERVARSENTER�(–)Y=
ExercisesExercises
Example 2Example 2
© Glencoe/McGraw-Hill 36 Glencoe Algebra 2
Spreadsheet InvestigationAppreciation and Depreciation (Use with Lesson 5-7.)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
1. If Mr. Blackstock chooses another property in the neighborhood that costs$99,900, what are the expected values of that home in the same periods of time?$103,896.00, $105,953.55, $116,868.87, $130,178.88
2. What would Mr. Blackstock’s profit be on the $99,900 home if he sold itafter 9 years and 3 months? $143,589.89
3. If an antique chair worth $165.00 increases in value an average of 3�12�%
every year, how much will it be worth next year? $170.78
4. Often assets like cars decrease in value over time. This asset is said todepreciate. If the value decreases by a fixed percent each year, or otherperiod of time, the amount y of that quantity after t years is given by y � a(1 � r)t, where a is the initial amount and r is the percent of decreaseexpressed as a decimal. Use a spreadsheet to find the value of a car purchased for $18,500 after 2 years, 2 years and 6 months, and 4 yearsand 3 months if the car depreciates at a rate of 12% per year.$14,326.40, $13,439.35, $10,745.41
ExercisesExercises
Michael Blackstock is considering buying a piece ofinvestment property for $95,000. The homes in the
area are appreciating at an average rate of 4% per year. Find theexpected value of the home in 1 year, 1 year and 6 months, 4 years,and 6 years and 9 months.Use rows 1 and 2 to enter the initial amount and the rate ofincrease. Then use Column A to enter the amounts of time.Enter the numbers of months as a fraction of a year since tis measured in years. Column B contains the formulas forthe value of the home.
Format the cells containing the values as currency so thatthey are displayed as dollars and cents. The expected valueof the home after each amount of time is shown in thespreadsheet.
ExampleExample
When an asset such as a house increases in value over time, it is said toappreciate. If the value increases by a fixed percent each year, or other period of time, the amount y of that quantity after t years is given by
y � a(1 � r)t,
where a is the initial amount and r is the percent of increase expressed as adecimal. You can use a spreadsheet to investigate future values of an asset.
Graphing Calculator InvestigationQuadratic Inequalities and the Test Menu(Use with Lesson 6-7.)
NAME ______________________________________________ DATE ____________ PERIOD _____
66
© Glencoe/McGraw-Hill 37 Glencoe Algebra 2
The inequality symbols, called relational operators, in the TEST menu can beused to display the solution of a quadratic inequality. Another method that canbe used to find the solution set of a quadratic inequality is to graph each sideof an inequality separately. Examine the graphs and use the intersect functionto determine the range of values for which the inequality is true.
Solve each inequality.
1. �x2 � 10x � 21 � 0 2. x2 � 9 � 0 3. x2 � 10x � 25 � 0{x | x �7 or x � �3} {x | �3 x 3} {x | x � �5}
4. x2 � 3x � 28 5. 2x2 � x � 3 6. 4x2 � 12x � 9 0{x | �7 � x � �4 } {x | x � �1.5 or x � 1} {x | x �1.5 or x � �1.5}
7. 23 �x2 � 10x 8. x2 � 4x � 13 � 0 9. (x � 1)(x �3) 0{x | x � 3.58 or x � 6.41} {x | �2.12 � x � 6.12} {x | x �1 or x � 3}
ExercisesExercises
Solve x2 � x � 6.
Place the calculator in Dot mode. Enter the inequality into Y1.Then trace the graph and describe the solution as an inequality.Keystrokes: [TEST] 4 6 4.
Use TRACE to determine the endpoints of the segments.Theses values are used to express the solution of the inequality,{ x | x � � 3 or x � 2 }.
ZOOM2nd+x 2Y=
Example 1Example 1
Solve 2x2 � 4x � 5 � 3.
Place the left side of the inequality in Y1 and the right side in Y2.Determine the points of intersection. Use the intersection points to express the solution set of the inequality. Be sure to set the calculator to Connected mode.Keystrokes: 2 4 5 3
6.
Press [CALC] 5 and use the key to move the cursor to the left of the first intersection point. Press . Then move the cursor to the right of the intersection point and press
. One of the values used in the solution set is displayed.Repeat the procedure on the other intersection point.
The solution is { x | �3.24 � x � 1.24}.
ENTER
ENTER
ENTER
2nd
ZOOM
ENTERENTER—+x 2Y=
Example 2Example 2
[�4.7, 4.7] scl:1 by [�3.1, 3.1] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
[�10, 10] scl:1 by [�10, 10] scl:1
© Glencoe/McGraw-Hill 38 Glencoe Algebra 2
You have learned the Location Principle, which can be used to approximatethe real zeros of a polynomial.
In the spreadsheet above, the positive real zero of ƒ(x) � x2 � 2 can beapproximated in the following way. Set the spreadsheet preference to manualcalculation. The values in A2 and B2 are the endpoints of a range of values.The values in D2 through J2 are values equally in the interval from A2 toB2. The formulas for these values are A2, A2 � (B2 � A2)�6, A2 � 2*(B2 � A2)/6,A2 � 3*(B2 � A2)/6, A2 � 4*(B2 � A2)/6, A2 � 5*(B2 � A2)/6, and B2,respectively.
Row 3 gives the function values at these points. The function ƒ(x) � x2 � 2 isentered into the spreadsheet in Cell D3 as D2^2 � 2. This function is thencopied to the remaining cells in the row.
You can use this spreadsheet to study the function values at the points incells D2 through J2. The value in cell F3 is positive and the value in cell G3is negative, so there must be a zero between �1.6667 and 0. Enter these values in cells A2 and B2, respectively, and recalculate the spreadsheet. (Youwill have to recalculate a number of times.) The result is a new table fromwhich you can see that there is a zero between 1.41414 and 1.414306.Because these values agree to three decimal places, the zero is about 1.414.This can be verified by using algebra.
By solving x2 � 2 � 0, we obtain x � ��2�. The positive root is x � ��2� � 1.414213. . . , which verifies the result.
Spreadsheet InvestigationApproximating the Real Zeros of Polynomials(Use with Lesson 6-5.)
NAME ______________________________________________ DATE ____________ PERIOD _____
66
1. Use a spreadsheet like the one above to approximate the zero of ƒ(x) � 3x � 2 to threedecimal places. Then verify your answer by using algebra to find the exact value of theroot. The spreadsheet gives x � 0.667. By solving for x algebraically,x � �
23
�. So, the approximation is correct.
2. Use a spreadsheet like the one above to approximate the real zeros of f(x) � x2 � 2x � 0.5.Round your answer to four decimal places. Then, verify your answer by using the quadratic formula. The process gives �1.7071 and �0.2929 to the nearest ten-thousandth. The quadratic formula gives x � �1 � �
�22�
�. �1 � ��22�
� ��1.7071 and �1 � �
�22�
� � �0.2929.3. Use a spreadsheet like the one above to approximate the real zero of ƒ(x) � x3 � �
32�x2 � 6x � 2
between � 0.4 and � 0.3. �0.3781 to the nearest ten-thousandth
ExercisesExercises
Graphing Calculator InvestigationRational Root Theorem (Use with Lesson 7-6.)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill 39 Glencoe Algebra 2
The following program performs synthetic division and displays thedepressed polynomial coefficients in rational form. The program will allowthe testing of possible rational zeros of a polynomial function.
PROGRAM:SYNTHDIVDisp "DEGREE OF DIVIDEND" P�1→P Q→L2(P)Input M Disp "COEFFICIENT" P+1→PDisp "COEFFICIENTS?" Input A If P�M+1Disp "0�SAME" A→L1(P) Goto 3Disp "1�QUOTIENT" If PM�1 StopDisp "2�NEW" Goto 1 Lbl 4Input U Lbl 2 0→PDisp "POSSIBLE ROOT" 1→P Lbl 5Input R 0→S 1�P→PIf U�0 Lbl 3 L2(P)→L1(P)Goto 2 L1(P) →F If PM�1If U�1 F�S→Q Goto 5Goto 4 Disp Q � Frac Goto 20→P Pause Lbl 1 RQ→S
Find all of the rational zeros of f(x) � 2x3 � 11x2 � 12 x � 9.
Use the program to test possible zeros.Keystrokes: [SYNTHDIV] 3 2
1 2 11 12 9 .Press until the screen displays Done.
The column of numbers are the coefficients of the depressed polynomial.Since the last number is not zero, press 3 . Choose 0 for the same coefficients. Press 1 then until finished. Repeat this until a zero is found. Then press 2 for the degree of the depressed polynomial and 1 for the quotient.
The zeros are 3, 3, and �12�.
ENTER
ENTER
ENTER(–)ENTER
ENTERENTER
ENTER
ENTERENTERENTER(–)ENTERENTERENTER
ENTERENTERENTERPRGM
Find all the zeros of each function.
1. f(x) � x3 � 8x2 � 23x � 30 1, �3, 10 2. f(x) � x3 �7x2 � 2x � 40 �2, 4, 5
3. f(x) � 2x3 � x2 � 32x � 16 4, �4, �12
� 4. f(x) � x4 � x3 � 11x2 � 9x � 18 1, �2, 3, �3
5. p(x) � 3x4 � 11x3 � 11x2 � x � 2 �1,�2, �13
� 6. p(x) � x4 � 2x3 � x2 � 8x � 12 �1, 3, �2i
7. p(x) � 3x5 � x4 � 243x � 81 3, �3,��13
� 8. p(x) � 3x4 � 13x3 � 15x2 � 4 �2,��1 �6
�13��
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 40 Glencoe Algebra 2
Spreadsheet InvestigationOperations on Functions (Use with Lesson 7-7.)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Study and use the spreadsheet above.
1. Find k(x) � (3x � 2) � (x2 � 2x). How does it compare to h(x)?k(x) � x2 � x � 2 � h(x)
2. Change the functions in the spreadsheet to f(x) � �2x
�, g(x) � 1 � x2, and
h(x) � 1 � �2x
� � x2. How are these functions related? Is it true that
f(x) � g(x) � h(x)? (f � g)(x) � h(x); yes
3. Make a conjecture about (f � g)(x) for any functions f(x) and g(x).(f � g)(x) � f(x) � g(x)
4. Make a conjecture about (f � g)(x) for any functions f(x) and g(x). Use thespreadsheet to test your conjecture. Does it appear to be true? Explainyour answer. (f � g)(x) � f(x) � g(x); See students’ work.
Find (f � g)(x), (f � g)(x), for each f(x) and g(x). Use the spreadsheetto find function values to verify your solutions. 5-7. See students’spreadsheets.
5. f(x) � 6x � 8 6. f(x) � x2 � 1 7. f(x) � 10x2
g(x) � 9 � x g(x) � 3x � 4 g(x) � 6 � x2
7x � 17; 5x � 1 x2 � 3x � 3; x2 � 3x � 5 9x2 � 6; 11x2 � 6
ExercisesExercises
It is possible to perform operations on functions such as addition, subtrac-tion, multiplication and division. You can use a spreadsheet to investigatethe relationships among functions.
Consider the functions f(x) � 3x � 2, g(x) � x2 � 2x, and h(x) � x2 � x + 2.Find the function values of each function for several values of x.Does it appear that f(x) � g(x) � h(x)?
Use Column A for the chosen values of x.Columns B, C, and E are f(x), g(x), and h(x)respectively. Use Column D for f(x) � g(x).
For every value of x, f(x) � g(x) � h(x).
Graphing Calculator InvestigationMatrices and Equations of Circles (Use with Lesson 8-3.)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
© Glencoe/McGraw-Hill 41 Glencoe Algebra 2
A graphing calculator can be used to write the equation of a circle in theform x2 � y2 � Dx � Ey � F � 0 given any three points on the circle.
Write the equation of the circle that passes through the given points. Identify the center and
radius of each circle.
a. A(5, 3), B(�2, 2), and C(�1, �5)
Substitute each ordered pair for (x, y) in x2 � y2 � Dx � Ey � F � 0 to formthe a system of equations.5D � 3E � F � �34 �2D � 2E � F � �8 �D � 5E � F � �26Solve the system using a matrix equation to find D, E, and F. Replace thecoefficients in the expanded form. Then, complete the square to write theequation in standard form to identify the center and radius.Keystrokes: [MATRX] 3 3 5
3 1 2 2 1 1 5 1 [MATRX] [EDIT] 2
3 1 34 8 26
[QUIT] [MATRX] [MATRX] 2
.Thus, D � �4, E � 2, and F � �20.The expanded form is x2 � y2 � 4x � 2y � 20 � 0.After completing the square, the standard form is (x � 2)2 � (y � 1)2 � 25.The center is ( 2, �1), and the radius is 5.
b. A(�2, 3), B(6, �5), and C(0, 7)
Find a system of equations. Then enter the equations into an augmentedmatrix. Reduce the matrix to row reduced echelon form using the rref(command. The row reduced echelon form of an augmented matrix willdisplay the solution to the system.�2D � 3E + F � �13 6D � 5E � F � �61 7E � F � �49Keystrokes: Enter the system of equations as [A], a 3 � 4 augmented matrix. Then use the reduced row echelon form by pressing [MATRX] [B] [MATRX] .The solution is D = �10, E = �4, and F = �21. The expanded formis x2 � y2 � 10x � 4y � 21 = 0, standard form is (x � 5)2 � (y � 2)2
� 50. The center is (5, 2) and the radius is 5�2�.
ENTER)ENTER2ndALPHA
2nd
ENTER
2nd x –1ENTER2nd2nd
ENTER(–)ENTER(–)ENTER(–)ENTERENTERENTER
2ndENTERENTER(–)ENTER
(–)ENTERENTERENTER(–)ENTERENTERENTER
ENTERENTERENTER2nd
ExampleExample
Write the equation of the circle that passes through the givenpoints. Identify the center and radius of each circle.
1. (0, �1), (�3, �2), and (�6, �1) 2. (7, �1), (11, �5), and (3, �5) 3. (�2, 7), (�9, 0), and (�10, �5)x2 � y2 � 6x � 6y � 7 � 0; x2 � y2 � 14x � 10y �58 � 0; x2�y2� 6x � 10y � 135 � 0;C(�3, 3), R � 5 C(7, �5), R � 4 C(3, �5), R � 13
ExercisesExercises
© Glencoe/McGraw-Hill 42 Glencoe Algebra 2
Spreadsheet InvestigationParabolas (Use with Lesson 8-2.)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
The spreadsheet below uses the equation of a parabola in the form y � a(x � h)2 � k or x � a(y � k)2 � h to find information about theparabola. x or y is entered in Column D and the values of a, h, and kare entered into Columns A, B, and C respectively.
1. Which row represents the equation y � 3x2 � 24x � 50? row 3
2. Write the standard form of the equation represented by row 2.
x � �14
� (y � 1)2 � 33. What formula should be used in cell F2? 1/ABS(A2)
4. Find the vertex, length of latus rectum, axis of symmetry, focus, directrix,and direction of opening of a parabola with equation (y � 8)2 � �4(x � 4).(8, 4); 4; y � 4; (7, 4); x � 9; left
ExercisesExercises
You have learned many of thecharacteristics of parabolaswith vertical and horizontalaxes of symmetry. The information is summarized in the table at the right. Youcan use what you know to create a spreadsheet to analyze given equations ofparabolas.
form of equation y � a(x � h)2 � k x � a(y � k)2 � hvertex (h, k) (h, k)axis of symmetry x � h y � kfocus (h, k � �
41a�) (h � �
41a�, k)
directrix y � k � �41a� x � h � �
41a�
direction of opening upward if a 0, right if a 0, leftdownward if a � 0 if a � 0
length of latus |�1a
�| units |�1a
�| unitsrectum
Graphing Calculator InvestigationHorizontal Asymptotes and Tables(Use with Lesson 9-3.)
NAME ______________________________________________ DATE ____________ PERIOD _____
99
© Glencoe/McGraw-Hill 43 Glencoe Algebra 2
The line y � b is a horizontal asymptote for the rational function f(x) if f(x) → b as x → � or as x → � �. The horizontal asymptote can be found byusing the TABLE feature of the graphing calculator.
ExercisesExercises
Find the horizontal asymptote for each function.
a. f(x) � �x2 � 41x � 5�
Enter the function into Y1. Place [TblSet] in the Ask mode. Enter thenumbers 10,000, 100,000, 1,000,000, and 5,000,000 and their opposites inthe x-list.Keystrokes: 1 4 5 [TBLSET] [TABLE]. Then enter thevalues for x.
Notice that as x increases, y approaches 0. Thus, when y � 0 is thehorizontal asymptote.
b. f(x) � �2x2 �3x
52
x � 6�
Enter the equation into Y1. Enter the numbers 10,000, 100,000,1,000,000, and 5,000,000 and their opposites in the x-list. Note thepattern. As x increases, y approaches 1.5. Thus, y � 1.5 is thehorizontal asymptote.
2ndENTER
2nd)—+x 2(�Y=
ExampleExample
Find the horizontal asymptote for each function.
1. f(x) � �x2�x
1� y � 2 2. f(x) � �2x2x�
2
7�x
1� 12� y � �
12
� 3. f(x) � �2x3 �62xx3
2 � 2� y � 3
4. f(x) � �3x2 �25xx � 1� y � 0 5. f(x) � �
15x2 �x3
3x � 7� y � 0 6. f(x) � y � 0
7. f(x) � �5xx2
��
23
� none 8. f(x) � �2x2 �6x
33
x � 6� none 9. f(x) � �2x
2� 4� none
x3 � 8x2 � 4x � 11���x4 � 3x3� 4x � 6
You have learned to solve problems involving direct, inverse, and joint variation.Many physical situations involve at least one of these types of variation. Forexample, according to Newton’s law of universal gravitation, the weight of amass near Earth depends on the distance between the mass and the centerof Earth. Study the spreadsheet below to determine the type of variationthat exists between the quantity of an astronaut’s weight and the distance of the astronaut from the center of Earth.
In the spreadsheet, the values for the astronaut’s weight in newtons areentered in the cells in column A, and the values for the astronaut’s distancein meters from the center of Earth are entered in cells in column B. Column Ccontains the astronaut’s distance from Earth’s surface.
© Glencoe/McGraw-Hill 44 Glencoe Algebra 2
Spreadsheet InvestigationVariation (Use with Lesson 9-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
99
1. Use the values in the spreadsheet to make a graph of the astronaut’s weight plotted against the astronaut’s distance from Earth’s center.
2. Based on your graph, is this an inverse or direct variation? inverse
3. Write an equation that represents this situation. LetW represent the astronaut’s weight, k the constant ofvariation, and R the distance from Earth’s center.
W � �RK
2�
4. Use the equation to find the weight of the astronaut at these distances from Earth’s surface. (Hint: Remember to add these values to the value in cell B2 to find the distance from Earth’s center.)a. 145,300,000 m b. 65 m c. 25,600 m
1.299615 N 734.5494 N 728.7047 N
d. 300,800,700 m e. 6580 m f. 180,560 m0.316872 N 733.0515 N 694.6873 N
ExercisesExercises
Graphing Calculator InvestigationRegression Equation Lab (Use with Lesson 10-1.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
© Glencoe/McGraw-Hill 45 Glencoe Algebra 2
A graphing calculator can be used to determine a regression equation thatbest fits a set of data. This activity requires tiles labeled on one side, and acontainer.
Collect the DataStep 1 Place the tiles on the desktop and count the total number. Record
the total number. Then place the tiles in the container and gentlyshake.
Step 2 Pour the tiles onto the desktop, remove all the tiles with a labelshowing, and set these aside. Count the remaining tiles without thelabels showing and return them to the container.
Step 3 Record the data in atable like this one.
Step 4 Repeat step 2 and 3 until the number of tiles without labels is zeroor the number remains constant.
Step 5 Take the tiles that were set aside in Step 2 and pour them out ofthe container onto the desktop. Remove the tiles without the labelshowing and count the tiles with the label showing. Repeat thisprocess until all the tiles have been removed.
Step 6 Record the data in atable like this one.
Analyze the Data 1-6. Answers will vary.
1. Enter trials in L1 and number of tiles without label showing in L2. Entertrials in L3 and number of tiles with the label showing in L4.
2. Use [STATPLOT] to make a scatter plot. Make a graph on paper for eachplot. Record the window used. Describe the pattern of the points.
3. From the [CALC] menu find the regression equation that best fitsthe data. Record the two closest equations, rounding values to the nearesthundredths. List and discuss the r and/or r2 values. Also include thegraphs in determining the best-fitting regression equation.
4. Sketch your best-fit regression equation choice for each scatter-plot on paper.
5. Describe any problems with the data or the regression equations.
6. Insert (0, total number of tiles) in the tables and the lists. Describe theeffect on the graphs. What happens with [PwrReg] and [ExpReg] whenthis ordered pair is inserted? Explain why this occurs?
STAT
Trials Number of tiles without label showingx y12
Trials Number of tiles with the label showingx y12
© Glencoe/McGraw-Hill 46 Glencoe Algebra 2
Spreadsheet InvestigationNet Present Value (Use after Lesson 10-6.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
1. If the NPV is greater than the cost, the investment will pay for itself.Based on the spreadsheet shown above, would it be cost-effective for thecompany to buy the van? Explain. The cost is actually about $75greater than the NPV, so it would not be cost-effective to buythe van.
2. Four times a year, Josey and Drew publish a magazine. They want to buy acolor printer that costs $1750. The cost of capital for this purchase wouldbe 6%. They are planning to raise the price of their magazine from $1 to$2. Create a spreadsheet to determine the NPV for this purchase.a. The last issue of the magazine sold 500 copies. If each issue of the magazine
printed in color sells 100 copies more than the previous issue, is theprinter a good investment after one year? Explain. No, after oneyear the NPV is only about $1682.14.
b. If the sales of the magazine continue to rise at the same rate, is theprinter a good investment after two years? Yes, after two years theNPV is about $5210.28. The NPV is about $3460.28 greaterthan the cost.
3. a. Calculate the NPV for an investment over a period of six years if thecost of capital is 4.5% and the investment will bring a cash flow of $750every year. The NPV would be about $3868.40.
b. Would this be a good investment of $3000? Explain? Yes, the NPV is$1131.60 greater than the cost.
ExercisesExercises
You have learned how to use exponential and logarithmic functions to performa number of financial analyses. Spreadsheets can be used to perform manytypes of analyses, such as calculating the Net Present Value of expendituresor investments. For example, when a businessowner is considering a major purchase, it is agood idea to find out whether the investmentwill be profitable in the future. Consider theexample of a local restaurant-delivery servicethat is debating whether to buy a van for$8000. The owners of the company estimatethat the van will bring in $2500 per year overfour years. They can use the following formulato find the present value of the future cash flowto find the Net Present Value (NPV), that is,how much the profits would be worth in today’s
dollars. NPV � �(1C�Fn
r)n�, where CFn � the cash
flow in period n and r � the cost of capital, which is eitherthe interest that will be paid on a loan or the interest thatthe money would earn if it were invested.
Graphing Calculator InvestigationRecursion and Iteration (Use with Lesson 11-6.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1111
© Glencoe/McGraw-Hill 47 Glencoe Algebra 2
A graphing calculator can be used to perform iterations and recursions.
Find the first 3 iterates of f(x) = 4x +15 if x0 = 5.
Store x0 in X. Then enter the expression on the home screen. Storethe result to X. Repeat the calculation for each iterate.Keystrokes: 5 4 15
.
x1 � 35, x2 � 155, and x3 � 635
ENTERENTERENTER
STO+ENTERSTO
Find the first three iterates of each function.
1. f(x) � 6x � 12 if x0 � 5 2. f(x) � 2x2 � 3 if x0 � �1
x1 � 42, x2 � 264, x3 � 1596 x1 � �1,x2 � �1,x3 � �1
3. f(x) � x2 � 4x � 5 if x0 � 1 4. f(x) � 2x2 � 2x � 1 if x0 � �12�
x1 � 2, x2 � 1, x3 � 2 x1 � �52
�, x2 � �327�, x3 � �
14245�
A bank account has an initial balance of $11,250.00. Interest is paidat the end of each year. Find the account balance under the giveninterest rate after the stated time period.
5. 3.8%, 2 years 6. 4.75%, 5 years 7. 6.05%, 10 years 8. 7.44%, 15 years$12,121.25 $14,188.05 $20,242.27 $33,009.77
Example 1Example 1
A savings account has an initial balance of $3000.00.At the end of each year, the bank pays 6% interest and
charges a $20 annual fee. Find the account balance after 6 years.
Store the initial value and enter an expression to calculate the balance at the end of a year.Keystrokes: 3000 1.06 20
.
At the end of six years, the account has a balance of $4116.05.
ENTERENTERENTERENTERENTERENTER
STO—ENTERSTO
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 48 Glencoe Algebra 2
You have learned about the characteristics of numbers in a sequence. Aspreadsheet can calculate a sequence and enable you to find the sum ofterms in the series.
Spreadsheet InvestigationSequences and Series (Use after Lesson 11-2.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1111
1. Create a spreadsheet like the one in the example above. Record the initialsequence as �4, �1, and 2. Repeat the process you followed in the example.What are the next six numbers in the sequence?5, 8, 11, 14, 17, and 20
2. Describe the steps the spreadsheet program completes to find the nextterm in the sequence. First, the program calculates the commondifference by subtracting any term from its succeeding term.Then, it adds the common difference to the last term to findthe next term in the sequence.
3. Use the spreadsheet to find the value for the 16th term in the sequence.41
4. Find the sum of the 3rd through 13th terms in the sequence. 187
ExercisesExercises
Create a spreadsheet like the one below and enter the first three terms of a sequence. Find the first ten
terms of the sequence. Then find the sum of the first ten terms of theseries.
Highlight cells B2 through D2 and move your cursor to any corner of thehighlighted cells until a black cross appears. Drag across the row and releaseit at cell K2. The next values in the sequence will appear in the cells.
To find the sum of the first 10 terms in the series, highlight the cells containing the terms, then click the � symbol on the toolbar. The sum willappear in the next cell. Note that this will work for arithmetic series only.The sum of the first ten terms of this series is 7.5
ExampleExample
Graphing Calculator InvestigationProbabilities (Use with Lesson 12-5.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1212
© Glencoe/McGraw-Hill 49 Glencoe Algebra 2
A graphing calculator can be used to perform calculations involving permu-tations, combinations, and probability.
There are 5 girls and 3 boys on a class committee. Asubcommittee of 3 people is being chosen at random.
What is the probability that the subcommittee will have at least 2 girls?
P(at least 2 girls) � P(2 girls) � P(3 girls). Each probability is theproduct of the combinations of girls and boys divided by the combinations of all the students taken 3 at a time.Keystrokes: 5 3 2 3 3 1 5
3 3 3 3 0 8 3 3
.
The probability that the subcommittee has at least 2 girls is �57�.
ENTER
ENTERMATHMATH�)MATH
MATH+MATH MATH(
Find each probability.
1. There are 5 girls and 4 boys on the school publications committee. A group of 5 membersis being chosen at random to attend a workshop on school newspapers. Find each probability.a. at least 3 girls b. 4 girls or 4 boys c. at least 2 boys
�1201� �
12256
� �1201�
2. Two cards are drawn from a standard deck of cards. Find each probability.a. both queens or both black b. both kings or both aces c. both face cards or both black
�25251
� �2221� �
168683
�
3. Find the probability that a committee of 6 U.S. Representatives selected at random from
7 Democrats and 7 Republicans will have at least 3 Republicans on the committee. �340029
�
4. Three CDs are randomly selected from a collection of 6 rock and 5 rap CDs. Find the
probability that at least 2 are rock. �1393�
Example 1Example 1
Two cards are randomly selected from a standarddeck of cards. Find the probability that both cards
are kings or that both cards are red.Since these events are mutually inclusive find the combinations of 4kings taken 2 at a time plus 26 red cards taken 2 at a time minus 2red kings taken 2 at a time divided by the combinations of 52 cardstaken 2 at a time.Keystrokes: 4 3 2 26 3 2 2
3 2 52 3 2 .
The probability of choosing 2 kings or two red cards is �25251�
.
ENTERENTERMATHMATH�)
MATH—MATH+MATH(
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 50 Glencoe Algebra 2
You have learned the formulas for the number of permutations of n objectstaken r at a time, P(n, r), and the number of combinations of n objects takenr at a time, C(n, r). You are going to set up a spreadsheet like the one shownbelow to perform analyses of these functions.
In the spreadsheet, the values in row 1 represent n, the values in row 2 represent r, and the formulas for P(n, r) and C(n, r) are in rows 3 and 4,respectively.
The formula to calculate P(n, r) is �FACT(B1)/FACT(B1-B2).
FACT is a special function from the function list and should not be enteredfrom the letters on the keyboard. Enter the formula in B3. Then drag thecursor across the row to copy the formula into cells C3 through G3.
The formula for C(n, r) is �FACT(B1)/(FACT(B1�B2)*FACT(B2)) and shouldbe entered in cell B4. Copy the formula into cells C4 through G4.
Spreadsheet InvestigationPermutations and Combinations (Use after Lesson 12-2.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1212
1. Compare the values of P(n, r) and C(n, r) for n � 5 and r � 0 through 5, aswell as for two other choices of n and r. Most of the values of P(n, r)are much larger than the corresponding values of C(n, r). Thevalues of P(n, r) tend to increase, while the values of C(n, r)tend to increase and then decrease.
2. Several identities hold for P(n, r) and C(n, r). Use the spreadsheet to verifythe following identities by finding three examples of each. 2a-2c. Seestudents’ work.a. P(n, n) � P(n, n � 1)
b. C(n � 1, r) � C(n, r � 1) � C(n, r)
c. C(n, 0) � C(n, 1) � C(n, 2) � . . . � C(n, n) � 2n
ExercisesExercises
Graphing Calculator InvestigationLaw of Sines: Ambiguous Case (Use with Lesson 13-4.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1313
© Glencoe/McGraw-Hill 51 Glencoe Algebra 2
A graphing calculator can be used to illustrate the Ambiguous Case for theLaw of Sines. This program constructs a visual representation of given information. From the drawing, the number of solutions can be determined.LAWSINES:Lbl 1 bcos(A→D:bsin(A→EDisp "A�" max(1,int((D�a�.999))→XmaxInput A min(�1,D-int((D�a�.999))→XminDisp "a�" int((E�a�.999)→YmaxInput M min(�1,Ymax�2(Xmax�Xmin)/3) →YminDisp "b�" ZsquareInput B Line(0,0,D,E){0,1,2}→L1:ML1�B→L2 Line(0,0,Xmax,0)LinReg(ax�b) L1,L2 Circle(D,E,a)AxesOff:ClrDraw
In �ABC, A � 35°, a � 34, and b � 45. Determinewhether �ABC has one, two, or no solutions.
Run the program and enter the given information. Examine theresulting figure for intersection points.Keystrokes: to highlight the LAWSINES then press
. Follow the prompts. A � 35 a � 34 b� 45 . Notice that the circle whose radius is a units intersectsthe horizontal segment twice. This indicates there are two solutionsor two triangles are possible.
ENTER
ENTERENTERENTERENTER
PRGM
Determine whether each triangle has one, two, or no possible solutions.
1. A � 44.3°, a � 22, and b � 20.1 1 2. A � 126°, a � 12, and b � 7 2
3. A � 21°, a � 2, and b � 3 2 4. A � 55°, a � 11, and b � 15 0
5. A � 112°, a � 5, and b � 7 0 6. B � 38.6°, b � 22.9, and c � 33.7 2
7. C � 30°, c � 20.2, and b � 40.4 1 8. B � 50°, b � 13, and c � 15 2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 52 Glencoe Algebra 2
Spreadsheet InvestigationCofunctions (Use with Lesson 13-1.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1313
1. Use the spreadsheet to make a graph of the sine values for the angles from 0° to 360°. Then make a graph of the cosine values.
2. If f(x) and g(x) are cofunctions, then f(x) � g(90° � x). Compare the shapesof the graphs. How can you tell that sine and cosine are cofunctions by their shapes? The graphs have a similar shape, butthe cosine graph is shifted 90° compared with the sine graphbecause they are cofunctions with sin � � cos (90 � �).
ExercisesExercises
The functions of sine and cosine are cofunctions. Set up a spreadsheet likethe one shown below to investigate the relationships between cofunctions.
In the spreadsheet, the values in row 1 are the angle values in degrees. Thevalues in rows 2 and 3 are the calculated values for the sine and cosine foreach angle, respectively. To use the spreadsheet to find the functions for anyangle, first enter each function into the spreadsheet in the form shown below.
�SIN(B1*PI()/180) �COS(B1*PI()/180)
This form of the formula contains additional information that is necessaryfor the spreadsheet to use degrees to calculate the answer. Without it, thespreadsheet cannot recognize that the angle is measured in degrees and willreturn the wrong answer.
Then, complete the spreadsheet by entering the angles up to 360° whosemeasures are mulitples of 30° and 45°.
Graphing Calculator InvestigationSinusoidal Equations (Use with Lesson 14-2.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1414
© Glencoe/McGraw-Hill 53 Glencoe Algebra 2
A graphing calculator can be used to verify a sinusoidal regression equationin the form y � a sin (bx � c) � d given four data points. The sinusoidalregression is found under [CALC] [C].STAT
As a person rides a Ferris wheel, the person’s distance fromthe ground varies sinusoidally with time. Let t be the
number of seconds that have elapsed since the Ferris wheel started. Therider’s position when the last seat is filled and the Ferris wheel starts iswhen t � 0. Suppose it takes 3 seconds to reach the top of the Ferris wheel,43 feet above the ground. The diameter of the wheel is 40 feet, and it makesa revolution every 8 seconds. Create a table of values and write the sinusoidalequation.
Keystrokes: Enter the data in L1 and L2. Choose an appropriatewindow. Use [STATPLOT] to graph the points. [C]
[L1] [L2] .
a � 20, b � �4π
�, c � 1, and d � 23
h(t) � 20 sin �4π
�(t � 1) � 23
ENTERENTERENTERVARS,2nd,2nd
ALPHASTAT
As the paddlewheel of a steamboat turns, a point on the paddleblade moves so that its distance, h, from the water’s surface is asinusoidal function of time. The wheel’s diameter is 18 feet, and itcompletes a revolution every 10 seconds. The height of the point atvarious times is shown in the table.
1. Why is the height the same after 14 seconds as it is after 4 seconds? The wheel completes a revolution every 10 seconds.
2. What are the values of a, b, c, and d?
a � 9, b � �5π
�, c � �32
�, and d � 7
3. Write a regression equation.
h(t) � 9 sin [�5π
�(t � �32
�)] � 7
ExampleExample
ExercisesExercises
t sec. 1 3 5 7 9 11h(t) ft. 23 43 23 3 23 43
t 1.5 4 6.5 9 11.5 14(seconds)
h(t)7 16 7 �2 7 16(feet)
© Glencoe/McGraw-Hill 54 Glencoe Algebra 2
Spreadsheet InvestigationTrigonometric Identities (Use after Lesson 14-3.)
NAME ______________________________________________ DATE ____________ PERIOD _____
1414
1. Study the values in Columns D and E. What identity seems to be possiblefrom this pattern? tan (�B) � �tan B
2. Enter the formula SIN(B) in Column F. Then enter the formulaCOS(B)*TAN(B) in Column G. What identity do these two new columnssuggest? sin B � cos B tan B
3. Make a column with the formula SIN(PI()-B). What identity do you discover? sin (π� B) � sin B or sin (π� B) � cos B tan B
ExercisesExercises
A trigonometric identity holds for all values of where each expression isdefined. For example, sin � cos � tan . You have learned to provealgbraically that an equation is an identity. You can use a spreadsheet to testequations for specific values to see if an equation might be an identity.
To use the spreadsheet to test the values of expressions for different angles,enter the angle measures in the cells in Column A, and enter the expressionsfrom the equations you want to test in the columns to the right. First, enterthe formula �RADIANS(A) in Column B to convert degrees to radians.(Recall that you can do this by entering the formula RADIANS(A2) in cellB2, copying cell B2, and pasting to fill the rest of Column B.) In the spreadsheet shown, the formula SIN(B)/COS(B) is in the cells in Column C.The cells in Column D contain the formula TAN(B). Column E contains theformula TAN(�B). Notice that the values in Columns C and D agree withthe identity stated above.
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Exercises
Exercises
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© Glencoe/McGraw-Hill 55 Glencoe Algebra 2
©G
lenc
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Gle
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Alg
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2
You
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.
Spre
adsh
eet
Invest
igati
on
Ab
solu
te V
alu
e S
tate
men
ts(U
se w
ith L
esso
n 1
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
11
Use
a s
pre
adsh
eet
to d
eter
min
e w
het
her
eac
h a
bso
lute
val
ue
stat
emen
t is
som
etim
es,a
lwa
ys,o
r n
ever
tru
e.
1.F
or a
ll r
eal
nu
mbe
rs a
and
b,a
�0,
|ax
�b|
�0.
som
etim
es
2.If
aan
d b
are
real
nu
mbe
rs,t
hen
|a
�b|
�|a
| �
|b|.
som
etim
es
3.If
aan
d b
are
real
nu
mbe
rs,t
hen
|a
�b|
��
x.n
ever
4.If
aan
d b
are
real
nu
mbe
rs,t
hen
|a|
�|b
| �
a�
b.so
met
imes
5.If
aan
d b
are
real
nu
mbe
rs,t
hen
c|a
�b|
�c|
a|�
|b|.
som
etim
es
Exercises
Exercises
Det
erm
ine
wh
eth
er c
|a�
b|�
|ca
�cb
| is
som
etim
es,
alw
ays
,or
nev
ertr
ue.
Try
a n
um
ber
of v
alu
es f
or a
,b,a
nd
cto
det
erm
ine
wh
eth
er t
he
stat
emen
t is
tru
e or
fal
se f
or e
ach
set
of
valu
es.
Ste
p1
Use
Col
um
ns
A,B
,an
d C
for
the
valu
es o
f a,
b,an
dc.
Ch
oose
sev
eral
set
s of
valu
es i
ncl
udi
ng
posi
tive
and
neg
ativ
e n
um
bers
,an
d ze
ro.
Ste
p2
Use
Col
um
n D
to
test
the
equ
atio
n.A
for
mu
lasu
ch a
s C
2*A
BS
(A2�
B2)
�A
BS
(C2*
A2�
C2*
B2)
in c
ell
D2
retu
rns
TR
UE
if t
he
equ
atio
n i
s tr
ue.
Th
rou
gh o
bser
vati
on o
f C
olu
mn
D,w
hen
cis
neg
ativ
e th
e st
atem
ent
is n
ottr
ue.
Th
e ab
solu
te v
alu
e st
atem
ent,
c|a
�b|
�|c
a�
cb|
is s
omet
imes
tru
e;it
is
tru
e on
ly i
f c
�0.
Example
Example
A
© Glencoe/McGraw-Hill 56 Glencoe Algebra 2
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Tran
sfo
rmat
ion
s: G
reat
est
Inte
ger
Fu
nct
ion
(Use
wit
h L
esso
n 2
-6.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
22
©G
lenc
oe/M
cGra
w-H
ill29
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
dis
play
tra
nsf
orm
atio
ns
to t
he
grea
test
inte
ger
fun
ctio
n.T
his
is
don
e by
usi
ng
the
int(
com
man
d u
nde
r th
e M
AT
H:
NU
Mm
enu
.Wh
en g
raph
ing
the
grea
test
in
tege
r fu
nct
ion
,it
is i
mpo
rtan
t to
set
the
calc
ula
tor
to D
ot m
ode.
Gra
ph
eac
h f
un
ctio
n.E
valu
ate
it f
or x
�1,
x�
1.3,
and
x�
2.C
omp
are
the
grap
h o
f th
e fu
nct
ion
to
the
grap
h o
f f(
x) �
[[x]
].
1.g(
x)�
[[x]]
�3
2.g(
x) �
[[x�
2]]
3.g(
x) �
[[x+
4]]
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
g(1
)��
2, g
(1.3
) ��
2, g
(2) �
�1
g(1
) ��
1, g
(1.3
) ��
1, g
(2) �
0g
(1) �
5, g
(1.3
) �5,
g(2
) �6
shift
ed d
own
3 un
itssh
ifted
rig
ht 2
uni
tssh
ifted
left
4 un
its
4.g(
x) �
[[�x]
]5.
g(x)
�[[2
x]]
6.g(
x) �
3[[x
]]
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
[�4.
7, 4
.7] s
cl: 1
by
[�3.
1, 3
.1] s
cl: 1
g(1
)��
1, g
(1.3
) ��
2, g
(2) �
�2
g(1
) �2,
g(1
.3) �
2, g
(2) �
4g
(1)�
3, g
(1.3
)�3,
g(2
)�6
refle
cted
acr
oss
the
y-ax
isco
mpr
esse
d by
�1 2�ho
rizo
ntal
lyex
pand
ed b
y 3
vert
ical
ly
Exercises
Exercises
Gra
ph
f(x
)�
[[x]]
and
g(x
) �
[[x]]
�2
on t
he
sam
e ax
es.
Eva
luat
e ea
ch f
un
ctio
n f
or x
�0.
6,x
�1,
x�
1.3,
x�
2,x
��
0.5,
and
x�
�1.
1.C
omp
are
the
grap
hs
of t
he
fun
ctio
ns.
En
ter
f(x)
in
Y1
and
g(x)
in
Y2.
Gra
ph t
he
fun
ctio
ns.
Key
stro
kes:
5 5
2 6
4.
Use
TR
AC
Eto
eva
luat
e ea
ch f
un
ctio
n f
or t
he
give
n v
alu
es.
Key
stro
kes:
0.6
1 1.
3 2
0.5
1.1
0.6
1 1.
3 2
0.5
1.1
.
f(0.
6)�
0,f(
1) �
1,f(
1.3)
�1,
f(2)
�2,
f(�
0.5)
��
1,an
d f(
�1.
1) �
�2
g(0.
6) �
2,g(
1) �
3,g(
1.3)
�3,
g(2)
�4,
g(�
0.5)
��
1,an
d g(
�1.
1) �
0
Th
e gr
aph
of
g(x)
�[[x
�2]
]is
th
e sa
me
as t
he
grap
h o
f f(
x)�
[[x]]
shif
ted
2 u
nit
s u
p.
EN
TER
(–)
EN
TER
(–)
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
(–)
EN
TER
(–)
EN
TER
EN
TER
EN
TER
EN
TER
TR
AC
E
ZO
OM
ZO
OM
EN
TER
+)
MA
TH
EN
TER
)M
AT
HY
=
Example
Example
[�4.
7, 4
.7] s
cl:1
by
[�3.
1, 3
.1] s
cl:1
Answers (Chapter 2)
©G
lenc
oe/M
cGra
w-H
ill30
Gle
ncoe
Alg
ebra
2
Th
e sl
ope
inte
rcep
t fo
rm o
f a
lin
ear
equ
atio
n i
s y
�m
x�
b,w
her
e m
is t
he
slop
e an
d b
is t
he
y-in
terc
ept.
Rec
all
that
th
e fo
rmu
la f
or t
he
slop
e of
a l
ine
thro
ugh
(x 1
,y1)
an
d (x
2,y 2
) is
m�
�y x2 2
� �y x1 1
�.Y
ou c
an u
se t
he
form
ula
for
slo
pe
and
the
slop
e-in
terc
ept
form
to
fin
d th
e va
lue
of b
.
Spre
adsh
eet
Invest
igati
on
Usi
ng
Lin
ear
Eq
uat
ion
s(U
se w
ith L
esso
n 2
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
22
Use
a s
pre
adsh
eet
to f
ind
th
e sl
ope
and
y-i
nte
rcep
t of
th
e li
ne
thro
ugh
eac
h p
air
of p
oin
ts.T
hen
wri
te a
n e
qu
atio
n o
f th
e li
ne
in s
lop
e-in
terc
ept
form
.
1.(0
,�5)
,(2,
5)2.
(4,2
),(�
3,�
5)5,
�5;
y�
�5x
�5
1, �
2; y
��
x�
2
3.(�
1,�
4),(
1,3)
4.(�
4,�
9),(
8,3)
�7 2�, �
�1 2�; y
��
�7 2�x �
�1 2�� 11 21 �
, �5 �
1 3�; y
�� 11 21 �
x�
5 �1 3�
5.(1
2,9)
,(10
,10)
6.(�
1.5,
3.1)
,(0.
9,1.
9)
��1 2�,
15;
y�
��1 2�x
�15
�0.
5, 2
.35;
y�
�0.
5x�
2.35
7.D
oes
the
spre
adsh
eet
wor
k w
hen
tw
o po
ints
hav
e th
e sa
me
x-co
ordi
nat
es?
Exp
lain
.N
o;
Th
e sl
op
e is
un
def
ined
.
Exercises
Exercises
Sta
te t
he
slop
e an
d y
-in
terc
ept
of t
he
grap
h o
f th
e li
ne
thro
ugh
(5,
2) a
nd
(4,
1).T
hen
wri
te a
n e
qu
atio
n o
f th
eli
ne
in s
lop
e-in
terc
ept
form
.
Ste
p1
Use
Col
um
ns
A a
nd
B t
o re
pres
ent
the
firs
t po
int,
and
Col
um
ns
Can
d D
to
repr
esen
t th
e se
con
d po
int
on t
he
lin
e.E
nte
r th
e fo
rmu
lafo
r sl
ope
in C
olu
mn
E.
Ste
p2
Su
bsti
tute
on
e of
th
e gi
ven
poi
nts
in
to t
he
slop
e-in
terc
ept
from
an
dso
lve
for
b.S
ince
we
know
th
e sl
ope
of t
he
lin
e,w
e ca
n s
olve
for
b.
y�
mx
�b
Slo
pe-i
nte
rcep
t fo
rmy 1
�m
x 1�
bS
ubs
titu
te (
x 1,y
1).
y 1�
mx 1
�b
Sol
ve f
or b
.
En
ter
this
for
mu
la i
nto
Col
um
n F
usi
ng
the
nam
es o
f th
e sp
read
shee
t ce
lls.
Th
e sl
ope
of t
he
lin
e th
rou
gh (
5,2)
an
d (4
,1)
is 1
an
dth
e y-
inte
rcep
t is
�3.
Th
us,
the
equ
atio
n o
f th
e li
ne
slop
e-in
terc
ept
form
is
y�
1x�
(�3)
or
y�
x�
3.
Example
Example
Answers (Chapter 3)
© Glencoe/McGraw-Hill 57 Glencoe Algebra 2
A
Gra
ph
th
e sy
stem
x�
3y�
�7,
5x�
y�
13,x
�6y
��
9,3x
�2y
��
7,an
d f
(x,y
)�4x
�3y
.Fin
d t
he
coor
din
ates
of t
he
feas
ible
reg
ion
.Th
en f
ind
th
e m
axim
um
an
d m
inim
um
val
ues
for
the
syst
em.
Sol
ve e
ach
in
equ
alit
y fo
r y.
En
ter
each
bou
nda
ry e
quat
ion
in
th
e Y
�sc
reen
.Fin
d th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n.T
hen
fin
d th
e va
lues
of f
(x,y
) to
det
erm
ine
the
max
imu
m a
nd
min
imu
m v
alu
es.
Key
stro
kes:
1 3
7 3
5 13
1
6 3
2 3
2 7
2 6
[CA
LC
] 5
[QU
IT]
[ {
]
[Y]
4 3
[Y]
[ }
]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
.
The
max
imum
val
ue o
f the
sys
tem
is 1
8 an
d th
e m
inim
um v
alue
is�
10.
EN
TER
2nd
2nd
EN
TER
EN
TER
EN
TER
2nd
GR
AP
HE
NTE
R2n
d2n
dE
NTE
RE
NTE
R
EN
TER
2nd
GR
AP
HE
NTE
R2n
d
2nd
EN
TER
EN
TER
EN
TER
2nd
GR
AP
H
EN
TER
2nd
ALP
HA
—,
ALP
HA
,2n
d2n
dE
NTE
RE
NTE
RE
NTE
R2n
d
ZO
OM
)�
(+
)�
(E
NTE
R)
�(
—)
�(
(–)
EN
TER
+(–
)
EN
TER
)�
(+
)�
(Y
=
Example
ExampleG
raph
ing C
alc
ula
tor
Invest
igati
on
Lin
ear
Pro
gra
mm
ing
(Use
wit
h L
esso
n 3
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
33
©G
lenc
oe/M
cGra
w-H
ill31
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
sto
re t
he
x- a
nd
y-co
ordi
nat
es w
hen
usi
ng
the
inte
rsec
tco
mm
and
in t
he
[CA
LC
]m
enu
.Th
is c
an b
e di
spla
yed
on t
he
hom
e sc
reen
an
d u
sed
to e
valu
ate
an e
xpre
ssio
n w
ith
xan
d y
vari
able
s.T
his
proc
ess
is u
sefu
l in
find
ing
the
vert
ices
of
the
feas
ible
reg
ion
and
dete
rmin
ing
the
max
imu
m o
r m
inim
um
val
ue
for
f(x,
y).
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl:1
Gra
ph
eac
h s
yste
m.F
ind
th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
e fe
asib
lere
gion
.Th
en f
ind
th
e m
axim
um
an
d m
inim
um
val
ues
for
th
e sy
stem
.
1.2x
�3y
�6
2.y
�4x
�6
3.y
�16
�x
3x�
2y�
�4
x�
4y�
70
�2y
�17
5x�
y�
152x
�y
�7
2x�
3y�
11f(
x,y)
�x
�3y
x�
6y�
10y
�3x
�1
f(x,
y) �
2x�
yy
�2x
�13
y�
7 �
2xf(
x,y)
�5x
�6y
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
(0, 2
), (
3, 0
), (
2, 5
);(�
1, 2
), (
�2,
�2)
, (3,
1),
(4,
�1)
;(5
.5, 0
), (
6.5,
0),
(7.
5, 8
.5),
m
in.�
3, m
ax. �
17m
in. �
�4,
max
. �9
(1.2
, 4.6
), (
2.5,
2),
(9.
66, 6
.33)
;m
in. �
24.5
, max
. �88
.5
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill32
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Bre
ak-E
ven
Po
int
(Use
wit
h L
esso
n 3
-1.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
33
1.If
Car
ly c
ould
dec
reas
e h
er a
nn
ual
ove
rhea
d to
$14
,000
,wh
at w
ould
th
ebr
eak-
even
poi
nt
be?
bet
wee
n 1
400
and
150
0 ca
nd
les
2.S
upp
ose
Car
ly d
ecre
ases
her
an
nu
al o
verh
ead
to $
14,0
00 a
nd
incr
ease
sth
e pr
ice
of a
can
dle
to $
14.0
0.W
hat
is
the
new
bre
ak-e
ven
poi
nt?
bet
wee
n 1
200
and
130
0 ca
nd
les
Exercises
Exercises
Car
ly E
rics
on i
s co
nsi
der
ing
open
ing
a ca
nd
le
bu
sin
ess.
Sh
e es
tim
ates
th
at s
he
wil
l h
ave
an a
nn
ual
over
hea
d o
f $1
5,00
0.It
cos
ts C
arly
$3.
00 t
o m
ake
a ja
r ca
nd
le,w
hic
hsh
e se
lls
for
$12.
50.W
hat
is
Car
ly’s
bre
ak-e
ven
poi
nt?
Use
Col
um
n A
for
th
e n
um
ber
of c
andl
es.C
olu
mn
s B
and
C a
re t
he
cost
an
d th
e in
com
e,re
spec
tive
ly.
Ext
end
the
row
s of
th
e sp
read
shee
t to
fin
d th
e po
int
atw
hich
the
inco
me
firs
t ex
ceed
s th
e co
st.T
he b
reak
-eve
npo
int
occu
rs b
etw
een
th
is p
oin
t an
d th
e pr
evio
us
poin
t.In
th
is c
ase,
the
brea
k ev
en p
oin
t oc
curs
bet
wee
n 1
500
and
1600
can
dles
.
Th
e ch
art
tool
of
the
spre
adsh
eet
allo
ws
you
to
grap
hth
e da
ta.T
he
grap
h v
erif
ies
the
solu
tion
.
Example
Example
You
hav
e le
arn
ed t
hat
th
e br
eak-
even
poi
nt
is t
he
poin
t at
wh
ich
th
e in
com
eeq
ual
s th
e co
st.Y
ou c
an u
se t
he
form
ula
s an
d ch
arts
in
a s
prea
dsh
eet
to f
ind
a br
eak-
even
poi
nt.
© Glencoe/McGraw-Hill 58 Glencoe Algebra 2
Answers (Chapter 4)
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Mat
rice
s fo
r 30
°, 4
5°, 6
0°R
ota
tio
ns
(Use
wit
h L
esso
n 4
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
44
©G
lenc
oe/M
cGra
w-H
ill33
Gle
ncoe
Alg
ebra
2
Th
e ro
tati
on m
atri
x fo
r 90
°co
un
terc
lock
wis
e ab
out
the
orig
in i
s [0
�1
10
].
Th
e ge
ner
al r
otat
ion
mat
rix
for
any
angl
e
cou
nte
rclo
ckw
ise
abou
t th
e
orig
in i
s [co
s�
sin
].
Qu
adri
late
ral
AB
CD
has
ver
tice
s A
(0,0
),B
(4,0
),C
(4,6
),an
d D
(0,6
).F
ind
th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
e im
age
afte
r ea
ch
cou
nte
rclo
ckw
ise
rota
tion
.Rou
nd
to
the
nea
rest
ten
th.
1.45
°2.
30°
3.60
°A
(0,
0),
B(
2.8,
2.8
),A
(0,
0),
B(
3.5,
2),
A(
0, 0
), B
(2,
3.5
),C
(�
1.4,
7.1
), D
(�
4.2,
4.2
)C
(0.
5, 7
.2),
D(
�3,
5.2
)C
(�
3.2,
6.5
), D
(�
5.2,
3)
4.12
0°5.
75°
6.22
5°A
(0,
0),
B(
�2,
3.5
),A
(0,
0),
B(
1, 3
.9),
A(
0, 0
), B
(�
2.8,
2.8
),C
(�
7.2,
0.5
),D(
-5.2
, �3)
C(
�4.
8, 5
.4),
D(
�5.
8, 1
.6)
C(
�7.
1, �
1.4)
, D(
�4.
2, �
4.2)
Exercises
Exercises
Fin
d t
he
coor
din
ates
of
the
imag
e of
�A
BC
wit
h
vert
ices
A(0
,0),
B(6
,0)
and
C(3
,4)
afte
r a
cou
nte
rclo
ckw
ise
rota
tion
of
30°
abou
t th
e or
igin
.E
nte
r th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e in
ver
tex
mat
rix
[A]
and
the
rota
tion
mat
rix
in m
atri
x [B
].B
e su
re t
he
calc
ula
tor
is s
et i
nD
egre
e m
ode.
Key
stro
kes:
[MA
TR
X]
2 3
0 6
3 0
0 4
[MA
TR
X]
2 2
2 30
30
30
30
[Q
UIT
] [M
AT
RX
] 2
[MA
TR
X]
1 .H
old
to
scro
ll a
cros
s to
see
th
e ot
her
coo
rdin
ates
.
Th
e co
ordi
nat
es o
f th
e im
age
of �
AB
Car
e A
(0,
0),B
(5.
2,3)
,an
dC
(0.
6,5.
0).
EN
TER
2nd
2nd
2nd
EN
TER
)C
OS
)S
INE
NTE
R)
SIN
(–)
EN
TER
)C
OS
EN
TER
EN
TER
2nd
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
2nd
Example
1Example
1
Fin
d t
he
coor
din
ates
of
the
imag
e of
�A
BC
wit
h
vert
ices
A(0
,0),
B(6
,0)
and
C(3
,4)
afte
r a
two
rota
tion
s of
45°
cou
nte
rclo
ckw
ise
abou
t th
e or
igin
.In
ord
er t
o ro
tate
th
e im
age
twic
e,st
ore
the
vert
ex m
atri
x of
th
e fi
rst
imag
e.K
eyst
roke
s:[M
AT
RX
] 2
45
45
45
45
[QU
IT]
[MA
TR
X]
2 [M
AT
RX
] 1
[EN
TR
Y]
[MA
TR
X]
6 [E
NT
RY
]
[EN
TR
Y]
[MA
TR
X]
6 .
Th
e ve
rtic
es o
f �
AB
C
are
A(
0,0)
,B(
0,6)
,an
d C
(�
4,3)
.
EN
TER
2nd
2nd
2nd
EN
TER
2nd
ST
O
2nd
EN
TER
2nd
2nd
2nd
EN
TER
)C
OS
)S
INE
NTE
R)
SIN
(–)
EN
TER
)C
OS
EN
TER
EN
TER
2nd
Example
2Example
2
sin
co
s
©G
lenc
oe/M
cGra
w-H
ill34
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Cra
mer
’s R
ule
(Use
wit
h L
esso
n 4
-5.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
44
You
hav
e le
arn
ed t
o so
lve
syst
ems
of l
inea
r eq
uat
ion
s by
usi
ng
mat
rix
equ
atio
ns
and
the
inve
rse
mat
rix.
An
oth
er w
ay t
o so
lve
syst
ems
is t
o u
seC
ram
er’s
Ru
le.S
tudy
th
e sp
read
shee
t be
low
to
disc
over
Cra
mer
’s R
ule
.
To
use
th
e sp
read
shee
t to
sol
ve a
syst
em o
f eq
uat
ion
s,w
rite
eac
heq
uat
ion
in
th
e fo
rm b
elow
.
ax�
by�
c
Th
e va
lues
for
th
e sy
stem
6x
�3y
��
12 a
nd
5x�
y�
8 ar
e sh
own
.In
th
e sp
read
shee
t,th
e va
lues
of
a,b,
and
cfo
r th
e fi
rst
equ
atio
n a
reen
tere
d in
cel
ls A
1,B
1,an
d C
1,re
spec
tive
ly.T
he
valu
es o
f a,
b,an
dc
for
the
seco
nd
equ
atio
n a
reen
tere
d in
cel
ls A
2,B
2,an
d C
2,re
spec
tive
ly.
Th
e va
lues
in
cel
ls B
10 a
nd
B11
repr
esen
t th
e so
luti
on f
or t
he
syst
em.
1.S
tudy
the
for
mul
a in
cel
l A4.
Wri
te a
mat
rix
who
se d
eter
min
ant
is f
ound
usi
ng t
his
form
ula.
[A1
B1 ]
A2
B2
2.W
rite
mat
rice
s w
hos
e de
term
inan
ts a
re f
oun
d u
sin
g th
e fo
rmu
las
in c
ells
A6
and
A8.
[C1
B1 ] ; [A
1C
1 ]C
2B
2A
2C
23.
Exp
lain
how
th
e va
lues
of
xan
d y
are
fou
nd
usi
ng
Cra
mer
’s r
ule
.
|C1
B1
||A
1C
1|
x�
C2
B2
; y
�A
2C
2
|A1
B1
||A
1B
1|
A2
B2
A2
B2
Use
th
e sp
read
shee
t to
sol
ve e
ach
sys
tem
of
equ
atio
ns.
4.6x
�3y
��
125.
5x�
3y�
196.
8x�
3y�
115x
�y
�8
7x�
2y�
86x
�9y
�15
(4,�
12)
(2, �
3)(1
.6, 0
.6)
7.0.
3x�
1.6y
�0.
448.
3y�
4x+
289.
y�
�0.
5x�
40.
4x�
2.5y
�0.
665x
�7y
�8
y
�4x
�5
(0.4
, 0.2
)(�
4, 4
)(2
, 3)
Exercises
Exercises
Answers (Chapter 5)
© Glencoe/McGraw-Hill 59 Glencoe Algebra 2
A
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Usi
ng
Tab
les
to F
acto
r b
y G
rou
pin
g(U
se w
ith L
esso
n 5
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
55
©G
lenc
oe/M
cGra
w-H
ill35
Gle
ncoe
Alg
ebra
2
Th
e T
AB
LE
feat
ure
of
a gr
aph
ing
calc
ula
tor
can
be
use
d to
hel
p fa
ctor
apo
lyn
omia
l of
th
e fo
rm a
x2�
bx�
c.
Fac
tor
10x2
�43
x�
28 b
y gr
oup
ing.
Mak
e a
tabl
e of
th
e n
egat
ive
fact
ors
of 1
0�
28 o
r 28
0.L
ook
for
a pa
irof
fac
tors
wh
ose
sum
is
�43
.
En
ter
the
equ
atio
n y
��28 x0 �
in Y
1to
fin
d th
e fa
ctor
s of
280
.Th
en,
fin
d th
e su
m o
f th
e fa
ctor
s u
sin
g y
��28 x0 �
�x
in Y
2.S
et u
p th
e ta
ble
to d
ispl
ay t
he
neg
ativ
e fa
ctor
s of
280
by
sett
ing
�T
bl
= to
�1.
Exa
min
e th
e re
sult
s.
Key
stro
kes:
280
[TB
LS
ET
] 1
1 [T
AB
LE
].
Th
e la
st l
ine
of t
he
tabl
e sh
ows
that
�43
xm
ay b
e re
plac
ed w
ith
�
8x+(
�35
x).
10x2
�43
x�
28�
10x2
�8x
�(�
35x)
�28
�2x
(5x
�4)
�(�
7)(5
x�
4)�
(5x
�4)
(2x
�7)
Th
us,
10x2
�43
x�
28 �
(5x
�4)
(2x
�7)
.
2nd
EN
TER
(–)
EN
TER
(–)
2nd
EN
TER
+E
NTE
RE
NTE
RV
AR
SE
NTE
R�
Y=
Fac
tor
each
pol
ynom
ial.
1.y2
�20
y�
962.
4z2
�33
z�
353.
4y2
�y
�18
4.6a
2�
2a�
15(y
�4)
(y�
24)
(4z
�5)
(z�
7)(4
y �
9)(y
�2)
prim
e
5.6m
2�
17m
�12
6.24
z2�
46z
�15
7.36
y2�
84y
�49
8.4b
2�
36b
�40
3(2
m�
3)(3
m�
4)(1
2z�
5)(2
z�
3)(6
y�
7)2
(2b
�31
)(2b
�13
)
Example
1Example
1
Fac
tor
12x2
�7x
�12
.
Loo
k at
th
e fa
ctor
s of
12
��
12 o
r�
144
for
a pa
ir w
hos
e su
m i
s�
7.E
nte
r an
equ
atio
n t
o de
term
ine
the
fact
ors
in Y
1an
d an
equ
atio
n t
ofi
nd
the
sum
of
fact
ors
in Y
2.E
xam
ine
the
tabl
e to
fin
d a
sum
of�
7.K
eyst
roke
s:14
4
[TB
LS
ET
] 1
1 [T
AB
LE
].
12x2
�7x
�12
� 1
2x2
�9x
�(�
16x)
�12
�3x
(4x
�3)
�4(
4x�
3)�
(4x
�3)
(3x
�4)
T
hu
s,12
x2�
7x�
12 �
(4x
�3)
(3x
�4)
.
2nd
EN
TER
EN
TER
2nd
EN
TER
+
EN
TER
EN
TER
VA
RS
EN
TER
�(–
)Y
=
Exercises
Exercises
Example
2Example
2
©G
lenc
oe/M
cGra
w-H
ill36
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Ap
pre
ciat
ion
an
d D
epre
ciat
ion
(Use
wit
h L
esso
n 5
-7.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
55
1.If
Mr.
Bla
ckst
ock
choo
ses
anot
her
pro
pert
y in
th
e n
eigh
borh
ood
that
cos
ts$9
9,90
0,w
hat
are
the
expe
cted
val
ues
of t
hat
hom
e in
the
sam
e pe
riod
s of
tim
e?$1
03,8
96.0
0, $
105,
953.
55, $
116,
868.
87, $
130,
178.
88
2.W
hat
wou
ld M
r.B
lack
stoc
k’s
prof
it b
e on
th
e $9
9,90
0 h
ome
if h
e so
ld i
taf
ter
9 ye
ars
and
3 m
onth
s?$1
43,5
89.8
9
3.If
an
an
tiqu
e ch
air
wor
th $
165.
00 i
ncr
ease
s in
val
ue
an a
vera
ge o
f 3 �
1 2�%ev
ery
year
,how
mu
ch w
ill
it b
e w
orth
nex
t ye
ar?
$170
.78
4.O
ften
ass
ets
like
car
s de
crea
se i
n v
alu
e ov
er t
ime.
Th
is a
sset
is
said
to
dep
reci
ate.
If t
he
valu
e de
crea
ses
by a
fix
ed p
erce
nt
each
yea
r,or
oth
erpe
riod
of
tim
e,th
e am
oun
t y
of t
hat
qu
anti
ty a
fter
tye
ars
is g
iven
by
y�
a(1
�r)
t ,w
her
e a
is t
he
init
ial
amou
nt
and
ris
th
e pe
rcen
t of
dec
reas
eex
pres
sed
as a
dec
imal
.Use
a s
prea
dsh
eet
to f
ind
the
valu
e of
a c
ar
purc
has
ed f
or $
18,5
00 a
fter
2 y
ears
,2 y
ears
an
d 6
mon
ths,
and
4 ye
ars
and
3 m
onth
s if
th
e ca
r de
prec
iate
s at
a r
ate
of 1
2% p
er y
ear.
$14,
326.
40, $
13,4
39.3
5, $
10,7
45.4
1
Exercises
Exercises
Mic
hae
l B
lack
stoc
k i
s co
nsi
der
ing
bu
yin
g a
pie
ce o
fin
vest
men
t p
rop
erty
for
$95
,000
.Th
e h
omes
in
th
ear
ea a
re a
pp
reci
atin
g at
an
ave
rage
rat
e of
4%
per
yea
r.F
ind
th
eex
pec
ted
val
ue
of t
he
hom
e in
1 y
ear,
1 ye
ar a
nd
6 m
onth
s,4
year
s,an
d 6
yea
rs a
nd
9 m
onth
s.U
se r
ows
1 an
d 2
to e
nte
r th
e in
itia
l am
oun
t an
d th
e ra
te o
fin
crea
se.T
hen
use
Col
um
n A
to
ente
r th
e am
oun
ts o
f ti
me.
En
ter
the
nu
mbe
rs o
f m
onth
s as
a f
ract
ion
of
a ye
ar s
ince
tis
mea
sure
d in
yea
rs.C
olu
mn
B c
onta
ins
the
form
ula
s fo
rth
e va
lue
of t
he
hom
e.
For
mat
th
e ce
lls
con
tain
ing
the
valu
es a
s cu
rren
cy s
o th
atth
ey a
re d
ispl
ayed
as
doll
ars
and
cen
ts.T
he
expe
cted
val
ue
of t
he
hom
e af
ter
each
am
oun
t of
tim
e is
sh
own
in
th
esp
read
shee
t.
Example
Example
Wh
en a
n a
sset
su
ch a
s a
hou
se i
ncr
ease
s in
val
ue
over
tim
e,it
is
said
to
appr
ecia
te.I
f th
e va
lue
incr
ease
s by
a f
ixed
per
cen
t ea
ch y
ear,
or o
ther
pe
riod
of
tim
e,th
e am
oun
t y
of t
hat
qu
anti
ty a
fter
tye
ars
is g
iven
by
y�
a(1
�r)
t ,
wh
ere
ais
th
e in
itia
l am
oun
t an
d r
is t
he
perc
ent
of i
ncr
ease
exp
ress
ed a
s a
deci
mal
.You
can
use
a s
prea
dsh
eet
to i
nve
stig
ate
futu
re v
alu
es o
f an
ass
et.
© Glencoe/McGraw-Hill 60 Glencoe Algebra 2
Answers (Chapter 6)
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Qu
adra
tic
Ineq
ual
itie
s an
d t
he
Test
Men
u(U
se w
ith L
esso
n 6
-7.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
66
©G
lenc
oe/M
cGra
w-H
ill37
Gle
ncoe
Alg
ebra
2
The
ine
qual
ity
sym
bols
,cal
led
rela
tion
al o
pera
tors
,in
the
TE
ST
men
u ca
n be
used
to
disp
lay
the
solu
tion
of
a qu
adra
tic
ineq
uali
ty.A
noth
er m
etho
d th
at c
anbe
use
d to
fin
d th
e so
luti
on s
et o
f a
quad
rati
c in
equ
alit
y is
to
grap
h e
ach
sid
eof
an
ineq
ualit
y se
para
tely
.Exa
min
e th
e gr
aphs
and
use
the
in
ters
ect
func
tion
to d
eter
min
e th
e ra
nge
of
valu
es f
or w
hic
h t
he
ineq
ual
ity
is t
rue.
Sol
ve e
ach
in
equ
alit
y.
1.�
x2�
10x
�21
�0
2.x2
�9
�0
3.x2
�10
x�
25�
0{x
| x
�7
or
x�
�3}
{x| �
3
x
3}{x
| x�
�5}
4.x2
�3x
�28
5.2x
2�
x�
36.
4x2
�12
x�
9
0{x
| �7
�x
��
4 }
{x| x
��
1.5
or
x�
1}{x
| x
�1.
5 o
r x
��
1.5}
7.23
�
x2�
10x
8.x2
�4x
�13
�0
9.(x
�1)
(x�
3)
0{x
| x�
3.58
or
x�
6.41
}{x
| �2.
12 �
x�
6.12
}{x
| x
�1
or
x�
3}
Exercises
Exercises
Sol
ve x
2�
x�
6.
Pla
ce t
he
calc
ula
tor
in D
ot m
ode.
En
ter
the
ineq
ual
ity
into
Y1.
Th
en t
race
th
e gr
aph
an
d de
scri
be t
he
solu
tion
as
an i
neq
ual
ity.
Key
stro
kes:
[TE
ST] 4
6
4.
Use
TR
AC
Eto
det
erm
ine
the
endp
oin
ts o
f th
e se
gmen
ts.
Th
eses
val
ues
are
use
d to
exp
ress
th
e so
luti
on o
f th
e in
equ
alit
y,{
x|
x�
�3
or x
�2
}.
ZO
OM
2nd
+x
2Y
=
Example
1Example
1
Sol
ve 2
x2�
4x�
5�
3.
Pla
ce t
he
left
sid
e of
th
e in
equ
alit
y in
Y1
and
the
righ
t si
de i
n Y
2.D
eter
min
e th
e po
ints
of
inte
rsec
tion
.Use
th
e in
ters
ecti
on p
oin
ts
to e
xpre
ss t
he
solu
tion
set
of
the
ineq
ual
ity.
Be
sure
to
set
the
calc
ula
tor
to C
onn
ecte
dm
ode.
Key
stro
kes:
2 4
5 3
6.
Pre
ss
[CA
LC
] 5
and
use
th
e ke
y to
mov
e th
e cu
rsor
to
th
e le
ft o
f th
e fi
rst
inte
rsec
tion
poi
nt.
Pre
ss
.Th
en m
ove
the
curs
or t
o th
e ri
ght
of t
he
inte
rsec
tion
poi
nt
and
pres
s .O
ne
of t
he
valu
es u
sed
in t
he
solu
tion
set
is
disp
laye
d.R
epea
t th
e pr
oced
ure
on
th
e ot
her
in
ters
ecti
on p
oin
t.
Th
e so
luti
on i
s {
x|
�3.
24�
x�
1.24
}.
EN
TER
EN
TER
EN
TER
2nd
ZO
OM
EN
TER
EN
TER
—+
x2
Y=
Example
2Example
2
[�4.
7, 4
.7] s
cl:1
by
[�3.
1, 3
.1] s
cl:1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl:1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl:1
©G
lenc
oe/M
cGra
w-H
ill38
Gle
ncoe
Alg
ebra
2
You
hav
e le
arn
ed t
he
Loc
atio
n P
rin
cipl
e,w
hic
h c
an b
e u
sed
to a
ppro
xim
ate
the
real
zer
os o
f a
poly
nom
ial.
In t
he
spre
adsh
eet
abov
e,th
e po
siti
ve r
eal
zero
of
ƒ(x)
�x2
�2
can
be
appr
oxim
ated
in
th
e fo
llow
ing
way
.Set
th
e sp
read
shee
t pr
efer
ence
to
man
ual
calc
ula
tion
.Th
e va
lues
in
A2
and
B2
are
the
endp
oin
ts o
f a
ran
ge o
f va
lues
.T
he
valu
es i
n D
2 th
rou
gh J
2 ar
e va
lues
equ
ally
in
th
e in
terv
al f
rom
A2
toB
2.T
he fo
rmul
as fo
r th
ese
valu
es a
re A
2,A
2�
(B2
�A
2)�6
,A2
� 2
*(B
2�
A2)
/6,
A2
� 3
*(B
2�
A2)
/6,A
2 �
4*(
B2
�A
2)/6
,A2
� 5
*(B
2�
A2)
/6,a
nd
B2,
resp
ecti
vely
.
Row
3 g
ives
th
e fu
nct
ion
val
ues
at
thes
e po
ints
.Th
e fu
nct
ion
ƒ(x
) �
x2�
2 is
ente
red
into
th
e sp
read
shee
t in
Cel
l D
3 as
D2^
2 �
2.T
his
fu
nct
ion
is
then
copi
ed t
o th
e re
mai
nin
g ce
lls
in t
he
row
.
You
can
use
th
is s
prea
dsh
eet
to s
tudy
th
e fu
nct
ion
val
ues
at
the
poin
ts i
nce
lls
D2
thro
ugh
J2.
Th
e va
lue
in c
ell
F3
is p
osit
ive
and
the
valu
e in
cel
l G
3is
neg
ativ
e,so
th
ere
mu
st b
e a
zero
bet
wee
n �
1.66
67 a
nd
0.E
nte
r th
ese
valu
es i
n c
ells
A2
and
B2,
resp
ecti
vely
,an
d re
calc
ula
te t
he
spre
adsh
eet.
(You
wil
l h
ave
to r
ecal
cula
te a
nu
mbe
r of
tim
es.)
Th
e re
sult
is
a n
ew t
able
fro
mw
hic
h y
ou c
an s
ee t
hat
th
ere
is a
zer
o be
twee
n 1
.414
14 a
nd
1.41
4306
.B
ecau
se t
hes
e va
lues
agr
ee t
o th
ree
deci
mal
pla
ces,
the
zero
is
abou
t 1.
414.
Th
is c
an b
e ve
rifi
ed b
y u
sin
g al
gebr
a.
By
solv
ing
x2�
2 �
0,w
e ob
tain
x�
��
2�.T
he
posi
tive
roo
t is
x
��
�2�
�1.
4142
13..
.,w
hic
h v
erif
ies
the
resu
lt.
Spre
adsh
eet
Invest
igati
on
Ap
pro
xim
atin
g t
he
Rea
l Zer
os
of
Po
lyn
om
ials
(Use
wit
h L
esso
n 6
-5.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
66
1.U
se a
spr
eads
hee
t li
ke t
he
one
abov
e to
app
roxi
mat
e th
e ze
ro o
f ƒ(
x) �
3x�
2 to
th
ree
deci
mal
pla
ces.
Th
en v
erif
y yo
ur
answ
er b
y u
sin
g al
gebr
a to
fin
d th
e ex
act
valu
e of
th
ero
ot.
Th
e sp
read
shee
t g
ives
x�
0.66
7. B
y so
lvin
g f
or
xal
geb
raic
ally
, x
��2 3�.
So
, th
e ap
pro
xim
atio
n is
co
rrec
t.
2.U
se a
spr
eads
heet
like
the
one
abo
ve t
o ap
prox
imat
e th
e re
al z
eros
of
f(x)
�x2
�2x
�0.
5.R
oun
d yo
ur
answ
er t
o fo
ur
deci
mal
pla
ces.
Th
en,v
erif
y yo
ur
answ
er b
y u
sin
g th
e qu
adra
tic
form
ula
.T
he
pro
cess
giv
es�
1.70
71 a
nd
�0.
2929
to
th
e n
eare
st
ten
-th
ou
san
dth
. Th
e q
uad
rati
c fo
rmu
la g
ives
x�
�1
��� 22 � �
. �
1�
�� 22 � ��
�1.
7071
an
d �
1�
�� 22 � ��
�0.
2929
.3.
Use
a s
prea
dshe
et li
ke t
he o
ne a
bove
to
appr
oxim
ate
the
real
zer
o of
ƒ(x
) �x3
��3 2�x
2�
6x�
2be
twee
n �
0.4
and
�0.
3.�
0.37
81 t
o t
he
nea
rest
ten
-th
ou
san
dth
Exercises
Exercises
Answers (Chapter 7)
© Glencoe/McGraw-Hill 61 Glencoe Algebra 2
A
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Rat
ion
al R
oo
t T
heo
rem
(Use
wit
h L
esso
n 7
-6.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
77
©G
lenc
oe/M
cGra
w-H
ill39
Gle
ncoe
Alg
ebra
2
Th
e fo
llow
ing
prog
ram
per
form
s sy
nth
etic
div
isio
n a
nd
disp
lays
th
ede
pres
sed
poly
nom
ial
coef
fici
ents
in
rat
ion
al f
orm
.Th
e pr
ogra
m w
ill
allo
wth
e te
stin
g of
pos
sibl
e ra
tion
al z
eros
of
a po
lyn
omia
l fu
nct
ion
.
PR
OG
RA
M:S
YN
TH
DIV
Dis
p "
DE
GR
EE
OF
DIV
IDE
ND
"P
�1→
PQ
→L
2(P
)In
pu
t M
Dis
p "
CO
EF
FIC
IEN
T"
P+1
→P
Dis
p "
CO
EF
FIC
IEN
TS
?"In
pu
t A
If P
�M
+1D
isp
"0�
SA
ME
"A
→L
1(P
)G
oto
3D
isp
"1�
QU
OT
IEN
T"
If P
M
�1
Sto
pD
isp
"2�
NE
W"
Got
o 1
Lb
l 4
Inp
ut
UL
bl
20→
PD
isp
"P
OS
SIB
LE
RO
OT
"1→
PL
bl
5In
pu
t R
0→S
1�P
→P
If U
�0
Lb
l 3
L2(
P)→
L1(
P)
Got
o 2
L1(
P)
→F
If P
M
�1
If U
�1
F�
S→
QG
oto
5G
oto
4D
isp
Q �
Fra
cG
oto
20→
PP
ause
L
bl
1R
Q→
S
Fin
d a
ll o
f th
e ra
tion
al z
eros
of
f(x)
�2x
3�
11x2
�12
x�
9.
Use
th
e pr
ogra
m t
o te
st p
ossi
ble
zero
s.K
eyst
roke
s:[S
YN
TH
DIV
] 3
2 1
2 11
12
9
.P
ress
u
nti
l th
e sc
reen
dis
play
s D
one.
The
col
umn
of n
umbe
rs a
re t
he c
oeff
icie
nts
of t
he d
epre
ssed
pol
ynom
ial.
Sin
ce t
he
last
nu
mbe
r is
not
zer
o,pr
ess
3 .C
hoo
se 0
for
th
e sa
me
coef
fici
ents
.Pre
ss
1 th
en
un
til
fin
ish
ed.R
epea
t th
is u
nti
l a
zero
is
fou
nd.
Th
en p
ress
2
for
the
degr
ee o
f th
e de
pres
sed
poly
nom
ial
and
1 fo
r th
e qu
otie
nt.
Th
e ze
ros
are
3,3,
and
�1 2�.
EN
TER
EN
TER
EN
TER
(–)
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
(–)
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
PR
GM
Fin
d a
ll t
he
zero
s of
eac
h f
un
ctio
n.
1.f(x
)�x3
�8x
2�
23x
�30
1,�
3, 1
02.
f(x) �
x3�
7x2
�2x
�40
�2,
4, 5
3.f(x
) �2x
3�
x2�
32x
�16
4, �
4, �1 2�
4.f(x
) �x4
�x3
�11
x2�
9x�
181,
�2,
3, �
3
5.p(
x) �
3x4
�11
x3�
11x2
�x
�2
�1,
�2,
�1 3�6.
p(x)
�x4
�2x
3�
x2�
8x�
12�
1, 3
,�2i
7.p(
x) �
3x5
�x4
�24
3x�
813,
�3,
��1 3�
8.p(
x) �
3x4
�13
x3�
15x2
�4
�2,
��1
�6�
13 ��
Example
Example
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill40
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Op
erat
ion
s o
n F
un
ctio
ns
(Use
with L
esso
n 7
-7.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
77
Stu
dy
and
use
th
e sp
read
shee
t ab
ove.
1.F
ind
k(x)
�(3
x�
2) �
(x2
�2x
).H
ow d
oes
it c
ompa
re t
o h
(x)?
k(x)
�x
2�
x�
2 �
h(x
)
2.C
hang
e th
e fu
ncti
ons
in t
he s
prea
dshe
et t
o f(
x) �
� 2x �,g(
x) �
1 �
x2,a
nd
h(x)
�1
�� 2x �
�x2
.How
are
th
ese
fun
ctio
ns
rela
ted?
Is
it t
rue
that
f(x)
�g(
x) �
h(x
)?(f
�g
)(x)
�h
(x);
yes
3.M
ake
a co
nje
ctu
re a
bou
t (f
�g)
(x)
for
any
fun
ctio
ns
f(x)
an
d g(
x).
(f�
g)(
x) �
f(x)
�g
(x)
4.M
ake
a co
nje
ctu
re a
bou
t (f
�g)
(x)
for
any
fun
ctio
ns
f(x)
an
d g(
x).U
se t
he
spre
adsh
eet
to t
est
you
r co
nje
ctu
re.D
oes
it a
ppea
r to
be
tru
e? E
xpla
inyo
ur
answ
er.
(f�
g)(
x) �
f(x)
�g
(x);
See
stu
den
ts’w
ork
.
Fin
d (
f�
g)(x
),(f
�g)
(x),
for
each
f(x
) an
d g
(x).
Use
th
e sp
read
shee
tto
fin
d f
un
ctio
n v
alu
es t
o ve
rify
you
r so
luti
ons.
5-7.
See
stu
den
ts’
spre
adsh
eets
.
5.f(
x) �
6x�
86.
f(x)
�x2
�1
7.f(
x) �
10x2
g(x)
�9
�x
g(x)
�3x
�4
g(x)
�6
�x2
7x�
17;
5x�
1x
2�
3x�
3; x
2�
3x�
59x
2�
6; 1
1x2
�6
Exercises
Exercises
It i
s po
ssib
le t
o pe
rfor
m o
pera
tion
s on
fu
nct
ion
s su
ch a
s ad
diti
on,s
ubt
rac-
tion
,mu
ltip
lica
tion
an
d di
visi
on.Y
ou c
an u
se a
spr
eads
hee
t to
in
vest
igat
eth
e re
lati
onsh
ips
amon
g fu
nct
ion
s.
Con
sid
er t
he
fun
ctio
ns
f(x)
�3x
�2,
g(x)
�x2
�2x
,an
d h
(x)
�x2
�x
+ 2.
Fin
d t
he
fun
ctio
n v
alu
es o
f ea
ch f
un
ctio
n f
or s
ever
al v
alu
es o
f x.
Doe
s it
ap
pea
r th
at f
(x)
�g(
x) �
h(x
)?
Use
Col
um
n A
for
th
e ch
osen
val
ues
of
x.C
olu
mn
s B
,C,a
nd
E a
re f
(x),
g(x)
,an
d h
(x)
resp
ecti
vely
.Use
Col
um
n D
for
f(x
)�g(
x).
For
eve
ry v
alu
e of
x,f
(x)�
g(x)
�h
(x).
© Glencoe/McGraw-Hill 62 Glencoe Algebra 2
Answers (Chapter 8)
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Mat
rice
s an
d E
qu
atio
ns
of
Cir
cles
(Use
wit
h L
esso
n 8
-3.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
88
©G
lenc
oe/M
cGra
w-H
ill41
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
wri
te t
he
equ
atio
n o
f a
circ
le i
n t
he
form
x2
�y2
�D
x�
Ey
�F
�0
give
n a
ny
thre
e po
ints
on
th
e ci
rcle
.
Wri
te t
he
equ
atio
n o
f th
e ci
rcle
th
at p
asse
s th
rou
gh t
he
give
n p
oin
ts.I
den
tify
th
e ce
nte
r an
dra
diu
s of
eac
h c
ircl
e.
a.A
(5,3
),B
(�2,
2),a
nd
C(�
1,�
5)
Su
bsti
tute
eac
h o
rder
ed p
air
for
(x,y
) in
x2
�y2
�D
x�
Ey
�F
�0
to f
orm
the
a sy
stem
of
equ
atio
ns.
5D �
3E �
F �
�34
�2D
�2E
�F
��
8�
D �
5E �
F �
�26
Sol
ve t
he
syst
em u
sin
g a
mat
rix
equ
atio
n t
o fi
nd
D,E
,an
d F
.Rep
lace
th
eco
effi
cien
ts in
th
e ex
pan
ded
form
.Th
en,c
ompl
ete
the
squ
are
to w
rite
th
eeq
uat
ion
in s
tan
dard
for
m t
o id
enti
fy t
he
cen
ter
and
radi
us.
Key
stro
kes:
[MA
TR
X]
3 3
5 3
1 2
2 1
1 5
1 [M
AT
RX
] [E
DIT
] 2
3 1
34
8 26
[QU
IT]
[MA
TR
X]
[MA
TR
X] 2
.T
hu
s,D
��
4,E
�2,
and
F�
�20
.T
he
expa
nde
d fo
rm is
x2
�y2
�4x
�2y
�20
�0.
Aft
er c
ompl
etin
g th
e sq
uar
e,th
e st
anda
rd f
orm
is (
x�
2)2
�(y
�1)
2�
25.
Th
e ce
nte
r is
( 2
,�1)
,an
d th
e ra
diu
s is
5.
b.A
(�2,
3),B
(6,�
5),a
nd
C(0
,7)
Fin
d a
syst
em o
f eq
uat
ion
s.T
hen
en
ter
the
equ
atio
ns
into
an
au
gmen
ted
mat
rix.
Red
uce
th
e m
atri
x to
row
red
uce
d e
chel
on fo
rmu
sin
g th
e rr
ef(
com
man
d.T
he
row
red
uce
d ec
hel
on f
orm
of
an a
ugm
ente
d m
atri
x w
ill
disp
lay
the
solu
tion
to
the
syst
em.
�2D
�3E
+ F
��
136D
�5E
�F
��
617E
�F
��
49K
eyst
roke
s:E
nter
the
sys
tem
of e
quat
ions
as
[A],
a 3
�4
augm
ente
d m
atri
x.T
hen
use
th
e re
duce
d ro
w e
chel
on f
orm
by
pres
sin
g [M
AT
RX
] [B
] [M
AT
RX
] .
Th
e so
luti
on is
D=
�10
,E=
�4,
and
F=
�21
.Th
e ex
pan
ded
form
is x
2�
y2�
10x
�4y
�21
= 0
,sta
nda
rd f
orm
is (
x�
5)2
�(y
�2)
2
�50
.Th
e ce
nte
r is
(5,
2) a
nd
the
radi
us
is 5
�2�.
EN
TER
)E
NTE
R2n
dA
LPH
A
2nd
EN
TER
2nd
x
–1E
NTE
R2n
d2n
d
EN
TER
(–)
EN
TER
(–)
EN
TER
(–)
EN
TER
EN
TER
EN
TER
2nd
EN
TER
EN
TER
(–)
EN
TER
(–)
EN
TER
EN
TER
EN
TER
(–)
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
2nd
Example
Example
Wri
te t
he
equ
atio
n o
f th
e ci
rcle
th
at p
asse
s th
rou
gh t
he
give
np
oin
ts.I
den
tify
th
e ce
nte
r an
d r
adiu
s of
eac
h c
ircl
e.
1.(0
,�1)
,(�
3,�
2),a
nd (�
6,�
1)2.
(7,�
1),(
11,�
5),a
nd (3
,�5)
3.(�
2,7)
,(�
9,0)
,and
(�10
,�5)
x2�
y2�
6x�
6y�
7�
0;x2
�y2
�14
x�
10y
�58
�0;
x2�
y2�
6x�
10y
�13
5�
0;C
(�3,
3),
R�
5C
(7, �
5), R
�4
C(3
, �5)
, R�
13
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill42
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Par
abo
las
(Use
wit
h L
esso
n 8
-2.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
88
Th
e sp
read
shee
t b
elow
use
s th
e eq
uat
ion
of
a p
arab
ola
in t
he
form
y
�a
(x�
h)2
�k
or x
�a
(y�
k)2
�h
to f
ind
in
form
atio
n a
bou
t th
ep
arab
ola.
xor
yis
en
tere
d i
n C
olu
mn
D a
nd
th
e va
lues
of
a,h
,an
d k
are
ente
red
in
to C
olu
mn
s A
,B,a
nd
C r
esp
ecti
vely
.
1.W
hic
h r
ow r
epre
sen
ts t
he
equ
atio
n y
�3x
2�
24x
�50
?ro
w 3
2.W
rite
th
e st
anda
rd f
orm
of
the
equ
atio
n r
epre
sen
ted
by r
ow 2
.
x�
�1 4�(y
�1)
2�
33.
Wh
at f
orm
ula
sh
ould
be
use
d in
cel
l F
2?1/
AB
S(A
2)
4.F
ind
the
vert
ex,l
engt
h o
f la
tus
rect
um
,axi
s of
sym
met
ry,f
ocu
s,di
rect
rix,
and
dire
ctio
n o
f op
enin
g of
a p
arab
ola
wit
h e
quat
ion
(y
�8)
2�
�4(
x�
4).
(8, 4
); 4
; y
�4;
(7,
4);
x�
9; le
ft
Exercises
Exercises
You
hav
e le
arn
ed m
any
of t
he
char
acte
rist
ics
of p
arab
olas
wit
h v
erti
cal
and
hor
izon
tal
axes
of
sym
met
ry.T
he
info
rmat
ion
is
sum
mar
ized
in
th
e ta
ble
at t
he
righ
t.Yo
uca
n u
se w
hat
you
kn
ow t
o cr
eate
a s
prea
dsh
eet
to
anal
yze
give
n e
quat
ion
s of
para
bola
s.
form
of e
quat
ion
y�
a(x
�h)
2�
kx
�a(
y�
k)2
�h
verte
x(h
, k)
(h, k
)ax
is o
f sym
met
ryx
�h
y�
kfo
cus
(h, k
� � 41 a�
)(h
� � 41 a�
, k )
dire
ctrix
y�
k�
� 41 a�x
�h
�� 41 a�
dire
ctio
n of
ope
ning
upw
ard
if a
0,
right
if a
0,
left
dow
nwar
d if
a�
0if
a�
0
leng
th o
f lat
us
|�1 a� |un
its|�1 a� |
units
rect
um
Answers (Chapter 9)
© Glencoe/McGraw-Hill 63 Glencoe Algebra 2
A
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Ho
rizo
nta
l Asy
mp
tote
s an
d T
able
s(U
se w
ith L
esso
n 9
-3.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
99
©G
lenc
oe/M
cGra
w-H
ill43
Gle
ncoe
Alg
ebra
2
Th
e li
ne
y�
bis
a h
oriz
onta
l as
ympt
ote
for
the
rati
onal
fu
nct
ion
f(x
) if
f(
x)→
bas
x→
�or
as
x→
��
.Th
e h
oriz
onta
l as
ympt
ote
can
be
fou
nd
byu
sin
g th
e T
AB
LE
feat
ure
of
the
grap
hin
g ca
lcu
lato
r.
Exercises
Exercises
Fin
d t
he
hor
izon
tal
asym
pto
te f
or e
ach
fu
nct
ion
.
a.f(
x)�� x2
�41 x
�5
�
En
ter
the
fun
ctio
n in
to Y
1.P
lace
[T
blS
et]
in t
he
Ask
mod
e.E
nte
r th
en
um
bers
10,
000,
100,
000,
1,00
0,00
0,an
d 5,
000,
000
and
thei
r op
posi
tes
inth
e x-
list
.K
eyst
roke
s:1
4 5
[TB
LS
ET
] [T
AB
LE
].T
hen
en
ter
the
valu
es f
or x
.
Not
ice
that
as
xin
crea
ses,
yap
proa
ches
0.T
hu
s,w
hen
y�
0 is
th
eh
oriz
onta
l asy
mpt
ote.
b.f
(x)
�� 2x
2�3x
52 x�
6�
En
ter
the
equ
atio
n in
to Y
1.E
nte
r th
e n
um
bers
10,
000,
100,
000,
1,00
0,00
0,an
d 5,
000,
000
and
thei
r op
posi
tes
in t
he
x-li
st.N
ote
the
patt
ern
.As
xin
crea
ses,
yap
proa
ches
1.5
.Th
us,
y�
1.5
is t
he
hor
izon
tal a
sym
ptot
e.
2nd
EN
TER
2nd
)—
+x
2(
�Y
=
Example
Example
Fin
d t
he
hor
izon
tal
asym
pto
te f
or e
ach
fu
nct
ion
.
1.f(
x)�
� x2 �x
1�
y�
22.
f(x)
�� 2x
2x �2
7� x1 �
12�
y�
�1 2�3.
f(x)
�� 2x
3�
6 2x x3
2�
2�
y�
3
4.f(
x)�� 3x
2�
2 5x x�
1�
y�
05.
f(x)
��15
x2� x33x
�7
�y
�0
6.f(
x)�
y�
0
7.f(
x)�
�5 xx2 ��23
�n
on
e8.
f(x)
�� 2x
2�6x
33 x�
6�
no
ne
9.f(
x)�
�2x2�
4�
no
ne
x3�
8x2
�4x
�11
��
�x4
�3x
3 �4x
�6
You
have
lear
ned
to s
olve
pro
blem
s in
volv
ing
dire
ct,i
nver
se,a
nd jo
int
vari
atio
n.M
any
phys
ical
sit
uat
ion
s in
volv
e at
lea
st o
ne
of t
hes
e ty
pes
of v
aria
tion
.For
exam
ple,
acco
rdin
g to
New
ton
’s l
aw o
f u
niv
ersa
l gr
avit
atio
n,t
he
wei
ght
of a
mas
s n
ear
Ear
th d
epen
ds o
n t
he
dist
ance
bet
wee
n t
he
mas
s an
d th
e ce
nte
rof
Ear
th.S
tudy
th
e sp
read
shee
t be
low
to
dete
rmin
e th
e ty
pe o
f va
riat
ion
that
exi
sts
betw
een
th
e qu
anti
ty o
f an
ast
ron
aut’s
wei
ght
and
the
dist
ance
of
th
e as
tron
aut
from
th
e ce
nte
r of
Ear
th.
In t
he
spre
adsh
eet,
the
valu
es f
or t
he
astr
onau
t’s w
eigh
t in
new
ton
s ar
een
tere
d in
th
e ce
lls
in c
olu
mn
A,a
nd
the
valu
es f
or t
he
astr
onau
t’s d
ista
nce
in m
eter
s fr
om t
he c
ente
r of
Ear
th a
re e
nter
ed in
cel
ls in
col
umn
B.C
olum
n C
con
tain
s th
e as
tron
aut’s
dis
tan
ce f
rom
Ear
th’s
su
rfac
e.
©G
lenc
oe/M
cGra
w-H
ill44
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Var
iati
on
(Use
wit
h L
esso
n 9
-4.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
99
1.U
se t
he
valu
es i
n t
he
spre
adsh
eet
to m
ake
a gr
aph
of
the
astr
onau
t’s w
eigh
t pl
otte
d ag
ain
st t
he
astr
onau
t’s
dist
ance
fro
m E
arth
’s c
ente
r.
2.B
ased
on
you
r gr
aph
,is
this
an
in
vers
e or
dir
ect
vari
atio
n?
inve
rse
3.W
rite
an
equ
atio
n t
hat
rep
rese
nts
th
is s
itua
tion
.Let
Wre
pres
ent
the
astr
onau
t’sw
eigh
t,k
the
con
stan
t of
vari
atio
n,a
nd
R t
he
dist
ance
fro
m E
arth
’s c
ente
r.
W�
� RK2�
4.U
se t
he
equ
atio
n t
o fi
nd
the
wei
ght
of t
he
astr
onau
t at
th
ese
dist
ance
s fr
om E
arth
’s s
urf
ace.
(Hin
t:R
emem
ber
to a
dd t
hes
e va
lues
to
the
valu
e in
cel
l B
2 to
fin
d th
e di
stan
ce f
rom
Ear
th’s
cen
ter.
)a.
145,
300,
000
mb
.65
mc.
25,6
00 m
1.29
9615
N73
4.54
94 N
728.
7047
N
d.3
00,8
00,7
00 m
e.65
80 m
f.18
0,56
0 m
0.31
6872
N73
3.05
15 N
694.
6873
N
Exercises
Exercises
Weight (N)
200
300
100 0
400
500
600
700
800
Dis
tan
ce (
mill
ion
s o
f m
eter
s)20
4060
100
© Glencoe/McGraw-Hill 64 Glencoe Algebra 2
Answers (Chapter 10)
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Reg
ress
ion
Eq
uat
ion
L
ab(U
se w
ith L
esso
n 1
0-1
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1010
©G
lenc
oe/M
cGra
w-H
ill45
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
det
erm
ine
a re
gres
sion
equ
atio
n t
hat
best
fit
s a
set
of d
ata.
Th
is a
ctiv
ity
requ
ires
til
es l
abel
ed o
n o
ne
side
,an
d a
con
tain
er.
Col
lect
th
e D
ata
Ste
p1
Pla
ce t
he
tile
s on
th
e de
skto
p an
d co
un
t th
e to
tal
nu
mbe
r.R
ecor
dth
e to
tal
nu
mbe
r.T
hen
pla
ce t
he
tile
s in
th
e co
nta
iner
an
d ge
ntl
ysh
ake.
Ste
p2
Pou
r th
e ti
les
onto
th
e de
skto
p,re
mov
e al
l th
e ti
les
wit
h a
lab
elsh
owin
g,an
d se
t th
ese
asid
e.C
oun
t th
e re
mai
nin
g ti
les
wit
hou
t th
ela
bels
sh
owin
g an
d re
turn
th
em t
o th
e co
nta
iner
.S
tep
3R
ecor
d th
e da
ta i
n a
tabl
e li
ke t
his
on
e.
Ste
p4
Rep
eat
step
2 a
nd
3 u
nti
l th
e n
um
ber
of t
iles
wit
hou
t la
bels
is
zero
or t
he
nu
mbe
r re
mai
ns
con
stan
t.S
tep
5T
ake
the
tile
s th
at w
ere
set
asid
e in
Ste
p 2
and
pou
r th
em o
ut
ofth
e co
nta
iner
on
to t
he
desk
top.
Rem
ove
the
tile
s w
ith
out
the
labe
lsh
owin
g an
d co
un
t th
e ti
les
wit
h t
he
labe
l sh
owin
g.R
epea
t th
ispr
oces
s u
nti
l al
l th
e ti
les
hav
e be
en r
emov
ed.
Ste
p6
Rec
ord
the
data
in
ata
ble
like
th
is o
ne.
An
alyz
e th
e D
ata
1-6
. An
swer
s w
ill v
ary.
1.E
nte
r tr
ials
in
L1
and
nu
mbe
r of
til
es w
ith
out
labe
l sh
owin
g in
L2.
En
ter
tria
ls i
n L
3an
d n
um
ber
of t
iles
wit
h t
he
labe
l sh
owin
g in
L4.
2.U
se [
ST
AT
PL
OT
] to
mak
e a
scat
ter
plot
.Mak
e a
grap
h o
n p
aper
for
eac
hpl
ot.R
ecor
d th
e w
indo
w u
sed.
Des
crib
e th
e pa
tter
n o
f th
e po
ints
.
3.F
rom
th
e [C
AL
C]
men
u f
ind
the
regr
essi
on e
quat
ion
th
at b
est
fits
the
data
.Rec
ord
the
two
clos
est
equ
atio
ns,
rou
ndi
ng
valu
es t
o th
e n
eare
sth
un
dred
ths.
Lis
t an
d di
scu
ss t
he
ran
d/or
r2
valu
es.A
lso
incl
ude
th
egr
aph
s in
det
erm
inin
g th
e be
st-f
itti
ng
regr
essi
on e
quat
ion
.
4.S
ketc
h yo
ur b
est-
fit
regr
essi
on e
quat
ion
choi
ce f
or e
ach
scat
ter-
plot
on
pape
r.
5.D
escr
ibe
any
prob
lem
s w
ith
th
e da
ta o
r th
e re
gres
sion
equ
atio
ns.
6.In
sert
(0,
tota
l n
um
ber
of t
iles
) in
th
e ta
bles
an
d th
e li
sts.
Des
crib
e th
eef
fect
on
th
e gr
aph
s.W
hat
hap
pen
s w
ith
[P
wrR
eg]
and
[Exp
Reg
]w
hen
this
ord
ered
pai
r is
in
sert
ed?
Exp
lain
wh
y th
is o
ccu
rs?
ST
AT
Tria
lsN
umbe
r of
tile
s w
ithou
t lab
el s
how
ing
xy
1 2
Tria
lsN
umbe
r of
tile
s w
ith th
e la
bel s
how
ing
xy
1 2
©G
lenc
oe/M
cGra
w-H
ill46
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Net
Pre
sen
t V
alu
e(U
se a
fter
Les
son 1
0-6
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1010
1.If
th
e N
PV
is
grea
ter
than
th
e co
st,t
he
inve
stm
ent
wil
l pa
y fo
r it
self
.B
ased
on
th
e sp
read
shee
t sh
own
abo
ve,w
ould
it
be c
ost-
effe
ctiv
e fo
r th
eco
mpa
ny
to b
uy
the
van
? E
xpla
in.
Th
e co
st is
act
ual
ly a
bo
ut
$75
gre
ater
th
an t
he
NP
V, s
o it
wo
uld
no
t b
e co
st-e
ffec
tive
to
bu
yth
e va
n.
2.F
our
tim
es a
yea
r,Jo
sey
and
Dre
w p
ubl
ish
a m
agaz
ine.
Th
ey w
ant
to b
uy
aco
lor
prin
ter
that
cos
ts $
1750
.Th
e co
st o
f ca
pita
l fo
r th
is p
urc
has
e w
ould
be 6
%.T
hey
are
pla
nn
ing
to r
aise
th
e pr
ice
of t
hei
r m
agaz
ine
from
$1
to$2
.Cre
ate
a sp
read
shee
t to
det
erm
ine
the
NP
V f
or t
his
pu
rch
ase.
a.T
he la
st is
sue
of t
he m
agaz
ine
sold
500
cop
ies.
If e
ach
issu
e of
the
mag
azin
epr
inte
d in
col
or s
ells
100
cop
ies
mor
e th
an t
he
prev
iou
s is
sue,
is t
he
prin
ter
a go
od i
nve
stm
ent
afte
r on
e ye
ar?
Exp
lain
.N
o, a
fter
on
eye
ar t
he
NP
V is
on
ly a
bo
ut
$168
2.14
.b
.If
the
sale
s of
th
e m
agaz
ine
con
tin
ue
to r
ise
at t
he
sam
e ra
te,i
s th
epr
inte
r a
good
in
vest
men
t af
ter
two
year
s?Y
es, a
fter
tw
o y
ears
th
eN
PV
is a
bo
ut
$521
0.28
. Th
e N
PV
is a
bo
ut
$346
0.28
gre
ater
than
th
e co
st.
3.a.
Cal
cula
te t
he
NP
V f
or a
n i
nve
stm
ent
over
a p
erio
d of
six
yea
rs i
f th
eco
st o
f ca
pita
l is
4.5
% a
nd
the
inve
stm
ent
wil
l br
ing
a ca
sh f
low
of
$750
ever
y ye
ar.
Th
e N
PV
wo
uld
be
abo
ut
$386
8.40
.
b.W
ould
th
is b
e a
good
in
vest
men
t of
$30
00?
Exp
lain
?Y
es, t
he
NP
V is
$113
1.60
gre
ater
th
an t
he
cost
.
Exercises
Exercises
You
have
lear
ned
how
to
use
expo
nent
ial a
nd lo
gari
thm
ic f
unct
ions
to
perf
orm
a nu
mbe
r of
fin
anci
al a
naly
ses.
Spr
eads
heet
s ca
n be
use
d to
per
form
man
yty
pes
of a
naly
ses,
such
as
calc
ulat
ing
the
Net
Pre
sent
Val
ue o
f ex
pend
itur
esor
in
vest
men
ts.F
or e
xam
ple,
wh
en a
bu
sin
ess
own
er i
s co
nsi
deri
ng
a m
ajor
pu
rch
ase,
it i
s a
good
ide
a to
fin
d ou
t w
het
her
th
e in
vest
men
tw
ill
be p
rofi
tabl
e in
th
e fu
ture
.Con
side
r th
eex
ampl
e of
a l
ocal
res
tau
ran
t-de
live
ry s
ervi
ceth
at i
s de
bati
ng
wh
eth
er t
o bu
y a
van
for
$800
0.T
he
own
ers
of t
he
com
pan
y es
tim
ate
that
th
e va
n w
ill
brin
g in
$25
00 p
er y
ear
over
fou
r ye
ars.
Th
ey c
an u
se t
he
foll
owin
g fo
rmu
lato
fin
d th
e pr
esen
t va
lue
of t
he
futu
re c
ash
flo
wto
fin
d th
e N
et P
rese
nt
Val
ue
(NP
V),
that
is,
how
mu
ch t
he
prof
its
wou
ld b
e w
orth
in
tod
ay’s
doll
ars.
NP
V�
� (1C �F
n r)n
�,w
her
e C
Fn
�th
e ca
sh
flow
in
per
iod
nan
d r
�th
e co
st o
f ca
pita
l,w
hic
h i
s ei
ther
the
inte
rest
tha
t w
ill
be p
aid
on a
loan
or
the
inte
rest
tha
tth
e m
oney
wou
ld e
arn
if it
wer
e in
vest
ed.
Answers (Chapter 11)
© Glencoe/McGraw-Hill 65 Glencoe Algebra 2
A
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Rec
urs
ion
an
d It
erat
ion
(Use
wit
h L
esso
n 1
1-6
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1111
©G
lenc
oe/M
cGra
w-H
ill47
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
per
form
ite
rati
ons
and
recu
rsio
ns.
Fin
d t
he
firs
t 3
iter
ates
of
f(x)
= 4
x+1
5 if
x0
= 5.
Sto
re x
0in
X.T
hen
en
ter
the
expr
essi
on o
n t
he
hom
e sc
reen
.Sto
reth
e re
sult
to
X.R
epea
t th
e ca
lcu
lati
on f
or e
ach
ite
rate
.K
eyst
roke
s:5
4 15
.
x 1�
35,x
2�
155,
and
x 3�
635
EN
TER
EN
TER
EN
TER
ST
O+
EN
TER
ST
O
Fin
d t
he
firs
t th
ree
iter
ates
of
each
fu
nct
ion
.
1.f(
x)�
6x�
12 i
f x 0
�5
2.f(
x)�
2x2
�3
if
x 0�
�1
x 1�
42, x
2�
264,
x3
�15
96x 1
��
1,x 2
��
1,x 3
��
1
3.f(
x)�
x2�
4x�
5 if
x0
�1
4.f(
x)�
2x2
�2x
�1
if x
0�
�1 2�
x 1�
2, x
2�
1, x
3�
2x 1
��5 2�,
x2
��3 27 �
, x3
��14
245 �
A b
ank
acc
oun
t h
as a
n i
nit
ial
bal
ance
of
$11,
250.
00.I
nte
rest
is
pai
dat
th
e en
d o
f ea
ch y
ear.
Fin
d t
he
acco
un
t b
alan
ce u
nd
er t
he
give
nin
tere
st r
ate
afte
r th
e st
ated
tim
e p
erio
d.
5.3.
8%,2
yea
rs6.
4.75
%,5
yea
rs7.
6.05
%,1
0 ye
ars
8.7.
44%
,15
year
s$1
2,12
1.25
$14,
188.
05$2
0,24
2.27
$33,
009.
77
Example
1Example
1
A s
avin
gs a
ccou
nt
has
an
in
itia
l b
alan
ce o
f $3
000.
00.
At
the
end
of
each
yea
r,th
e b
ank
pay
s 6%
in
tere
st a
nd
char
ges
a $2
0 an
nu
al f
ee.F
ind
th
e ac
cou
nt
bal
ance
aft
er 6
yea
rs.
Sto
re t
he
init
ial
valu
e an
d en
ter
an e
xpre
ssio
n t
o ca
lcu
late
th
e ba
lan
ce a
t th
e en
d of
a y
ear.
Key
stro
kes:
3000
1.
06
20
.
At
the
end
of s
ix y
ears
,th
e ac
cou
nt
has
a b
alan
ce o
f $4
116.
05.
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
ST
O—
EN
TER
ST
O
Example
2Example
2
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill48
Gle
ncoe
Alg
ebra
2
You
hav
e le
arn
ed a
bou
t th
e ch
arac
teri
stic
s of
nu
mbe
rs i
n a
seq
uen
ce.A
spre
adsh
eet
can
cal
cula
te a
seq
uen
ce a
nd
enab
le y
ou t
o fi
nd
the
sum
of
term
s in
th
e se
ries
.
Spre
adsh
eet
Invest
igati
on
Seq
uen
ces
and
Ser
ies
(Use
aft
er L
esso
n 1
1-2
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1111
1.C
reat
e a
spre
adsh
eet
like
th
e on
e in
th
e ex
ampl
e ab
ove.
Rec
ord
the
init
ial
sequ
ence
as
�4,
�1,
and
2.R
epea
t th
e pr
oces
s yo
u fo
llow
ed in
the
exa
mpl
e.W
hat
are
th
e n
ext
six
nu
mbe
rs i
n t
he
sequ
ence
?5,
8, 1
1, 1
4, 1
7, a
nd
20
2.D
escr
ibe
the
step
s th
e sp
read
shee
t pr
ogra
m c
ompl
etes
to
fin
d th
e n
ext
term
in
th
e se
quen
ce.
Fir
st, t
he
pro
gra
m c
alcu
late
s th
e co
mm
on
dif
fere
nce
by
sub
trac
tin
g a
ny
term
fro
m it
s su
ccee
din
g t
erm
.T
hen
, it
add
s th
e co
mm
on
dif
fere
nce
to
th
e la
st t
erm
to
fin
dth
e n
ext
term
in t
he
seq
uen
ce.
3.U
se t
he
spre
adsh
eet
to f
ind
the
valu
e fo
r th
e 16
th t
erm
in
th
e se
quen
ce.
41
4.F
ind
the
sum
of
the
3rd
thro
ugh
13t
h t
erm
s in
th
e se
quen
ce.
187
Exercises
Exercises
Cre
ate
a sp
read
shee
t li
ke
the
one
bel
ow a
nd
en
ter
the
firs
t th
ree
term
s of
a s
equ
ence
.Fin
d t
he
firs
t te
nte
rms
of t
he
seq
uen
ce.T
hen
fin
d t
he
sum
of
the
firs
t te
n t
erm
s of
th
ese
ries
.
Hig
hli
ght
cell
s B
2 th
rou
gh D
2 an
d m
ove
you
r cu
rsor
to
any
corn
er o
f th
eh
igh
ligh
ted
cell
s u
nti
l a
blac
k cr
oss
appe
ars.
Dra
g ac
ross
th
e ro
w a
nd
rele
ase
it a
t ce
ll K
2.T
he
nex
t va
lues
in
th
e se
quen
ce w
ill
appe
ar i
n t
he
cell
s.
To
fin
d th
e su
m o
f th
e fi
rst
10 t
erm
s in
th
e se
ries
,hig
hli
ght
the
cell
s co
nta
inin
g th
e te
rms,
then
cli
ck t
he
�sy
mbo
l on
th
e to
olba
r.T
he
sum
wil
lap
pear
in
th
e n
ext
cell
.Not
e th
at t
his
wil
l w
ork
for
arit
hm
etic
ser
ies
only
.T
he
sum
of
the
firs
t te
n t
erm
s of
th
is s
erie
s is
7.5
Example
Example
© Glencoe/McGraw-Hill 66 Glencoe Algebra 2
Answers (Chapter 12)
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Pro
bab
iliti
es(U
se w
ith L
esso
n 1
2-5
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1212
©G
lenc
oe/M
cGra
w-H
ill49
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
per
form
cal
cula
tion
s in
volv
ing
perm
u-
tati
ons,
com
bin
atio
ns,
and
prob
abil
ity.
Th
ere
are
5 gi
rls
and
3 b
oys
on a
cla
ss c
omm
itte
e.A
sub
com
mit
tee
of 3
peo
ple
is
bei
ng
chos
en a
t ra
nd
om.
Wh
at i
s th
e p
rob
abil
ity
that
th
e su
bco
mm
itte
e w
ill
hav
e at
lea
st 2
gir
ls?
P(a
t le
ast
2 gi
rls)
�P
(2 g
irls
)�P
(3 g
irls
).E
ach
pro
babi
lity
is
the
prod
uct
of
the
com
bin
atio
ns
of g
irls
an
d bo
ys d
ivid
ed b
y th
e co
mbi
nat
ion
s of
all
th
e st
ude
nts
tak
en 3
at
a ti
me.
Key
stro
kes:
5 3
2 3
3 1
5
3 3
3 3
0 8
3 3
.
Th
e pr
obab
ilit
y th
at t
he
subc
omm
itte
e h
as a
t le
ast
2 gi
rls
is �5 7�.
EN
TER
EN
TER
MA
TH
MA
TH
�)
MA
TH
MA
TH
+M
AT
H
MA
TH
(
Fin
d e
ach
pro
bab
ilit
y.
1.T
her
e ar
e 5
girl
s an
d 4
boys
on
th
e sc
hoo
l pu
blic
atio
ns
com
mit
tee.
A g
rou
p of
5 m
embe
rsis
bei
ng c
hose
n at
ran
dom
to
atte
nd a
wor
ksho
p on
sch
ool n
ewsp
aper
s.F
ind
each
pro
babi
lity.
a.at
leas
t 3
girl
sb
.4 g
irls
or
4 bo
ysc.
at le
ast
2 bo
ys
�1 20 1�� 12 25 6�
�1 20 1�
2.T
wo
card
s ar
e dr
awn
fro
m a
sta
nda
rd d
eck
of c
ards
.Fin
d ea
ch p
roba
bili
ty.
a.bo
th q
uee
ns
or b
oth
bla
ckb
.bot
h k
ings
or
both
ace
sc.
both
fac
e ca
rds
or b
oth
bla
ck
� 25 25 1�� 22 21�
�1 68 68 3�
3.F
ind
the
prob
abil
ity
that
a c
omm
itte
e of
6 U
.S.R
epre
sen
tati
ves
sele
cted
at
ran
dom
fro
m
7 D
emoc
rats
an
d 7
Rep
ubl
ican
s w
ill
hav
e at
lea
st 3
Rep
ubl
ican
s on
th
e co
mm
itte
e.�3 40 02 9�
4.T
hre
e C
Ds
are
ran
dom
ly s
elec
ted
from
a c
olle
ctio
n o
f 6
rock
an
d 5
rap
CD
s.F
ind
the
prob
abil
ity
that
at
leas
t 2
are
rock
.�1 39 3�
Example
1Example
1
Tw
o ca
rds
are
ran
dom
ly s
elec
ted
fro
m a
sta
nd
ard
dec
k o
f ca
rds.
Fin
d t
he
pro
bab
ilit
y th
at b
oth
car
ds
are
kin
gs o
r th
at b
oth
car
ds
are
red
.S
ince
th
ese
even
ts a
re m
utu
ally
in
clu
sive
fin
d th
e co
mbi
nat
ion
s of
4ki
ngs
tak
en 2
at
a ti
me
plu
s 26
red
car
ds t
aken
2 a
t a
tim
e m
inu
s 2
red
kin
gs t
aken
2 a
t a
tim
e di
vide
d by
th
e co
mbi
nat
ion
s of
52
card
sta
ken
2 a
t a
tim
e.K
eyst
roke
s:4
3 2
26
3 2
2 3
2 52
3
2 .
Th
e pr
obab
ilit
y of
ch
oosi
ng
2 ki
ngs
or
two
red
card
s is
� 25 25 1�.
EN
TER
EN
TER
MA
TH
MA
TH
�)
MA
TH
—M
AT
H+
MA
TH
(
Example
2Example
2
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill50
Gle
ncoe
Alg
ebra
2
You
hav
e le
arn
ed t
he
form
ula
s fo
r th
e n
um
ber
of p
erm
uta
tion
s of
nob
ject
sta
ken
rat
a t
ime,
P(n
,r),
and
the
nu
mbe
r of
com
bin
atio
ns
of n
obje
cts
take
nr
at a
tim
e,C
(n,r
).Yo
u a
re g
oin
g to
set
up
a sp
read
shee
t li
ke t
he
one
show
nbe
low
to
perf
orm
an
alys
es o
f th
ese
fun
ctio
ns.
In t
he
spre
adsh
eet,
the
valu
es i
n r
ow 1
rep
rese
nt
n,t
he
valu
es i
n r
ow 2
re
pres
ent
r,an
d th
e fo
rmu
las
for
P(n
,r)
and
C(n
,r)
are
in r
ows
3 an
d 4,
resp
ecti
vely
.
Th
e fo
rmu
la t
o ca
lcu
late
P(n
,r)
is�
FA
CT
(B1)
/FA
CT
(B1-
B2)
.
FA
CT
is a
spe
cial
fu
nct
ion
fro
m t
he
fun
ctio
n l
ist
and
shou
ld n
ot b
e en
tere
dfr
om t
he
lett
ers
on t
he
keyb
oard
.En
ter
the
form
ula
in
B3.
Th
en d
rag
the
curs
or a
cros
s th
e ro
w t
o co
py t
he
form
ula
in
to c
ells
C3
thro
ugh
G3.
Th
e fo
rmu
la f
or C
(n,r
) is
�F
AC
T(B
1)/(
FA
CT
(B1�
B2)
*FA
CT
(B2)
) an
d sh
ould
be e
nte
red
in c
ell
B4.
Cop
y th
e fo
rmu
la i
nto
cel
ls C
4 th
rou
gh G
4.
Spre
adsh
eet
Invest
igati
on
Per
mu
tati
on
s an
d C
om
bin
atio
ns
(Use
aft
er L
esso
n 1
2-2
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1212
1.C
ompa
re t
he
valu
es o
f P
(n,r
) an
d C
(n,r
) fo
r n
�5
and
r�
0 th
rou
gh 5
,as
wel
l as
for
tw
o ot
her
ch
oice
s of
nan
d r.
Mo
st o
f th
e va
lues
of
P(n
, r)
are
mu
ch la
rger
th
an t
he
corr
esp
on
din
g v
alu
es o
f C
(n, r
). T
he
valu
es o
f P
(n, r
) te
nd
to
incr
ease
, wh
ile t
he
valu
es o
f C
(n, r
)te
nd
to
incr
ease
an
d t
hen
dec
reas
e.
2.S
ever
al i
den
titi
es h
old
for
P(n
,r)
and
C(n
,r).
Use
th
e sp
read
shee
t to
ver
ify
the
foll
owin
g id
enti
ties
by
fin
din
g th
ree
exam
ples
of
each
.2a
-2c.
See
stu
den
ts’w
ork
.a.
P(n
,n)�
P(n
,n�
1)
b.C
(n�
1,r)
�C
(n,r
�1)
�C
(n,r
)
c.C
(n,0
)�C
(n,1
)�C
(n,2
)�..
.�C
(n,n
)�2n
Exercises
Exercises
Answers (Chapter 13)
© Glencoe/McGraw-Hill 67 Glencoe Algebra 2
A
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Law
of
Sin
es:
Am
big
uo
us
Cas
e(U
se w
ith L
esso
n 1
3-4
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1313
©G
lenc
oe/M
cGra
w-H
ill51
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
ill
ust
rate
th
e A
mbi
guou
s C
ase
for
the
Law
of
Sin
es.T
his
pro
gram
con
stru
cts
a vi
sual
rep
rese
nta
tion
of
give
n
info
rmat
ion
.Fro
m t
he
draw
ing,
the
nu
mbe
r of
sol
uti
ons
can
be
dete
rmin
ed.
LA
WS
INE
S:
Lb
l 1
bco
s(A
→D
:bsi
n(A
→E
Dis
p "
A�
" m
ax(1
,in
t((D
�a�
.999
))→
Xm
axIn
pu
t A
min
(�1,
D-i
nt(
(D�
a�.9
99))
→X
min
Dis
p "
a�"
int(
(E�
a�.9
99)→
Ym
axIn
pu
t M
min
(�1,
Ym
ax�
2(X
max
�X
min
)/3)
→Y
min
Dis
p "
b�
"Z
squ
are
Inp
ut
BL
ine(
0,0,
D,E
){0
,1,2
}→L
1:M
L1�
B→
L2
Lin
e(0,
0,X
max
,0)
Lin
Reg
(ax�
b)
L1,
L2
Cir
cle(
D,E
,a)
Axe
sOff
:Clr
Dra
w In �
AB
C,A
�35
°,a
�34
,an
d b
�45
.Det
erm
ine
wh
eth
er �
AB
Ch
as o
ne,
two,
or n
o so
luti
ons.
Ru
n t
he
prog
ram
an
d en
ter
the
give
n i
nfo
rmat
ion
.Exa
min
e th
ere
sult
ing
figu
re f
or i
nte
rsec
tion
poi
nts
.K
eyst
roke
s:to
hig
hli
ght
the
LA
WS
INE
Sth
en p
ress
.F
ollo
w t
he
prom
pts.
A�
35
a�
34
b�
45
.Not
ice
that
th
e ci
rcle
wh
ose
radi
us
is a
un
its
inte
rsec
tsth
e h
oriz
onta
l se
gmen
t tw
ice.
Th
is i
ndi
cate
s th
ere
are
two
solu
tion
sor
tw
o tr
ian
gles
are
pos
sibl
e.
EN
TER
EN
TER
EN
TER
EN
TER
EN
TER
PR
GM
Det
erm
ine
wh
eth
er e
ach
tri
angl
e h
as o
ne,
two,
or n
o p
ossi
ble
sol
uti
ons.
1.A
�44
.3°,
a�
22,a
nd
b�
20.1
12.
A�
126°
,a�
12,a
nd
b�
72
3.A
�21
°,a
�2,
and
b�
32
4.A
�55
°,a
�11
,an
d b
�15
0
5.A
�11
2°,a
�5,
and
b�
70
6.B
�38
.6°,
b�
22.9
,an
d c
�33
.72
7.C
�30
°,c
�20
.2,a
nd
b�
40.4
18.
B�
50°,
b�
13,a
nd
c�
152
Example
Example
Exercises
Exercises
©G
lenc
oe/M
cGra
w-H
ill52
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Co
fun
ctio
ns
(Use
wit
h L
esso
n 1
3-1
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1313
1.U
se t
he
spre
adsh
eet
to m
ake
a gr
aph
of
the
sin
e va
lues
for
th
e an
gles
fro
m 0
°to
360
°.T
hen
mak
e a
grap
h o
f th
e co
sin
e va
lues
.
2.If
f(x
) an
d g(
x) a
re c
ofu
nct
ion
s,th
en f
(x)�
g(90
°�
x).C
ompa
re t
he
shap
esof
th
e gr
aph
s.H
ow c
an y
ou t
ell
that
sin
e an
d co
sin
e ar
e co
fun
ctio
ns
by t
hei
r sh
apes
?T
he
gra
ph
s h
ave
a si
mila
r sh
ape,
bu
tth
e co
sin
e g
rap
h is
sh
ifte
d 9
0°co
mp
ared
wit
h t
he
sin
e g
rap
hb
ecau
se t
hey
are
co
fun
ctio
ns
wit
h s
in �
�co
s (9
0 �
�).
Exercises
Exercises
Th
e fu
nct
ion
s of
sin
e an
d co
sin
e ar
e co
fun
ctio
ns.
Set
up
a sp
read
shee
t li
keth
e on
e sh
own
bel
ow t
o in
vest
igat
e th
e re
lati
onsh
ips
betw
een
cof
un
ctio
ns.
In t
he
spre
adsh
eet,
the
valu
es i
n r
ow 1
are
th
e an
gle
valu
es i
n d
egre
es.T
he
valu
es i
n r
ows
2 an
d 3
are
the
calc
ula
ted
valu
es f
or t
he
sin
e an
d co
sin
e fo
rea
ch a
ngl
e,re
spec
tive
ly.T
o u
se t
he
spre
adsh
eet
to f
ind
the
fun
ctio
ns
for
any
angl
e,fi
rst
ente
r ea
ch f
un
ctio
n i
nto
th
e sp
read
shee
t in
th
e fo
rm s
how
n b
elow
.
�S
IN(B
1*P
I()/
180)
�C
OS
(B1*
PI(
)/18
0)
Th
is f
orm
of
the
form
ula
con
tain
s ad
diti
onal
in
form
atio
n t
hat
is
nec
essa
ryfo
r th
e sp
read
shee
t to
use
deg
rees
to
calc
ula
te t
he
answ
er.W
ith
out
it,t
he
spre
adsh
eet
can
not
rec
ogn
ize
that
th
e an
gle
is m
easu
red
in d
egre
es a
nd
wil
lre
turn
th
e w
ron
g an
swer
.
Th
en,c
ompl
ete
the
spre
adsh
eet
by e
nte
rin
g th
e an
gles
up
to 3
60°
wh
ose
mea
sure
s ar
e m
uli
tple
s of
30°
and
45°.
0
1.50
1.00
0.50
0.00
�0.
50
�1.
00
�1.
50
6012
018
024
030
036
060
120
180
240
300
360
0
1.50
1.00
0.50
0.00
�0.
50
�1.
00
�1.
50
© Glencoe/McGraw-Hill 68 Glencoe Algebra 2
Gra
ph
ing C
alc
ula
tor
Invest
igati
on
Sin
uso
idal
Eq
uat
ion
s(U
se w
ith L
esso
n 1
4-2
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1414
©G
lenc
oe/M
cGra
w-H
ill53
Gle
ncoe
Alg
ebra
2
A g
raph
ing
calc
ula
tor
can
be
use
d to
ver
ify
a si
nu
soid
al r
egre
ssio
n e
quat
ion
in t
he
form
y�
asi
n (
bx�
c)�
dgi
ven
fou
r da
ta p
oin
ts.T
he
sin
uso
idal
regr
essi
on i
s fo
un
d u
nde
r [C
AL
C]
[C].
ST
AT
As
a pe
rson
rid
es a
Fer
ris
wh
eel,
the
pers
on’s
dis
tan
ce f
rom
the
grou
nd
vari
es s
inu
soid
ally
wit
h t
ime.
Let
tbe
th
e n
um
ber
of s
econ
ds t
hat
hav
e el
apse
d si
nce
th
e F
erri
s w
hee
l st
arte
d.T
he
ride
r’s
posi
tion
wh
en t
he
last
sea
t is
fil
led
and
the
Fer
ris
wh
eel
star
ts i
sw
hen
t�
0.S
upp
ose
it t
akes
3 s
econ
ds t
o re
ach
th
e to
p of
th
e F
erri
s w
hee
l,43
fee
t ab
ove
the
grou
nd.
Th
e di
amet
er o
f th
e w
hee
l is
40
feet
,an
d it
mak
esa
revo
luti
on e
very
8 s
econ
ds.C
reat
e a
tabl
e of
val
ues
and
wri
te t
he s
inus
oida
leq
uat
ion
.
Key
stro
kes:
En
ter
the
data
in
L1
and
L2.
Ch
oose
an
app
ropr
iate
win
dow
.Use
[S
TA
TP
LO
T]
to g
raph
th
e po
ints
.[C
] [L
1]
[L2]
.
a�
20,b
�� 4π �,
c�
1,an
d d
�23
h(t
)�20
sin
� 4π �(t�
1) �
23
EN
TER
EN
TER
EN
TER
VA
RS
,2n
d,
2nd
ALP
HA
ST
AT
As
the
pad
dle
wh
eel
of a
ste
amb
oat
turn
s,a
poi
nt
on t
he
pad
dle
bla
de
mov
es s
o th
at i
ts d
ista
nce
,h,f
rom
th
e w
ater
’s s
urf
ace
is a
sin
uso
idal
fu
nct
ion
of
tim
e.T
he
wh
eel’s
dia
met
er i
s 18
fee
t,an
d i
tco
mp
lete
s a
revo
luti
on e
very
10
seco
nd
s.T
he
hei
ght
of t
he
poi
nt
atva
riou
s ti
mes
is
show
n i
n t
he
tab
le.
1.W
hy
is t
he
hei
ght
the
sam
e af
ter
14 s
econ
ds a
s it
is
afte
r 4
seco
nds
? T
he
wh
eel c
om
ple
tes
a re
volu
tio
n e
very
10
seco
nd
s.
2.W
hat
are
th
e va
lues
of
a,b,
c,an
d d
?
a�
9, b
�� 5π �,
c�
�3 2�, a
nd
d�
7
3.W
rite
a r
egre
ssio
n e
quat
ion
.
h(t
) �
9 si
n [� 5π � (t�
�3 2� )]�7
Example
Example
Exercises
Exercises
t sec
.1
35
79
11h(
t) ft.
2343
233
2343
t 1.
54
6.5
911
.514
(sec
onds
)h
(t)
716
7�
27
16(f
eet)
Answers (Chapter 14)
©G
lenc
oe/M
cGra
w-H
ill54
Gle
ncoe
Alg
ebra
2
Spre
adsh
eet
Invest
igati
on
Trig
on
om
etri
c Id
enti
ties
(Use
aft
er L
esso
n 1
4-3
.)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
1414
1.S
tudy
th
e va
lues
in
Col
um
ns
D a
nd
E.W
hat
ide
nti
ty s
eem
s to
be
poss
ible
from
th
is p
atte
rn?
tan
(�
B)
��
tan
B
2.E
nte
r th
e fo
rmu
la S
IN(B
) in
Col
um
n F
.Th
en e
nte
r th
e fo
rmu
laC
OS
(B)*
TA
N(B
) in
Col
um
n G
.Wh
at i
den
tity
do
thes
e tw
o n
ew c
olu
mn
ssu
gges
t?si
n B
� c
os
Bta
n B
3.M
ake
a co
lum
n w
ith
th
e fo
rmu
la S
IN(P
I()-
B).
Wh
at i
den
tity
do
you
di
scov
er?
sin
(π
�B
)�si
n B
or
sin
(π
�B
)�co
s B
tan
B
Exercises
Exercises
A t
rigo
nom
etri
c id
enti
ty h
olds
for
all
val
ues
of
w
her
e ea
ch e
xpre
ssio
n i
sde
fin
ed.F
or e
xam
ple,
sin
�
cos
�
tan
.Y
ou h
ave
lear
ned
to
prov
eal
gbra
ical
ly t
hat
an
equ
atio
n i
s an
ide
nti
ty.Y
ou c
an u
se a
spr
eads
hee
t to
tes
teq
uat
ion
s fo
r sp
ecif
ic v
alu
es t
o se
e if
an
equ
atio
n m
igh
t be
an
ide
nti
ty.
To
use
th
e sp
read
shee
t to
tes
t th
e va
lues
of
expr
essi
ons
for
diff
eren
t an
gles
,en
ter
the
angl
e m
easu
res
in t
he
cell
s in
Col
um
n A
,an
d en
ter
the
expr
essi
ons
from
th
e eq
uat
ion
s yo
u w
ant
to t
est
in t
he
colu
mn
s to
th
e ri
ght.
Fir
st,e
nte
rth
e fo
rmu
la�
RA
DIA
NS
(A)
in C
olu
mn
B t
o co
nve
rt d
egre
es t
o ra
dian
s.(R
ecal
l th
at y
ou c
an d
o th
is b
y en
teri
ng
the
form
ula
RA
DIA
NS
(A2)
in
cel
lB
2,co
pyin
g ce
ll B
2,an
d pa
stin
g to
fil
l th
e re
st o
f C
olu
mn
B.)
In
th
e sp
read
shee
t sh
own
,th
e fo
rmu
laS
IN(B
)/C
OS
(B)
is i
n t
he
cell
s in
Col
um
n C
.T
he
cell
s in
Col
um
n D
con
tain
th
e fo
rmu
laT
AN
(B).
Col
um
n E
con
tain
s th
efo
rmu
laT
AN
(�B
).N
otic
e th
at t
he
valu
es i
n C
olu
mn
s C
an
d D
agr
ee w
ith
the
iden
tity
sta
ted
abov
e.