14471351 Worksheets
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Handouts
Marjorie Fernandez Karwowski
Valencia Community College
East Campus
Mathematics Department
MFK/Fall 08
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Calculator Help and Basic Function Guide
Calculator Check: Before using your calculator first do the following
Choose the mode key, make sure everything on the left hand side is highlighted
Choose the y= key, make sure all the plots are not highlighted. If they are then use
your up arrow key, place your cursor on the ones that are highlighted then hit enter.
Under the y= key use your down arrow key to access any equations and press clear
to erase them. On the TI-83 you are able to enter 10 equations.
Choose zoom, then choose #6, that is z-standard. This will give you a window that is
-10 to 10 on the x and y axis with a scale of 1, or that is [-10,10,1] by [-10,10,1].
Basic Functions
1.) Difference between a negative sign and a subtraction symbol For example, if you want to compute - 9 10.8 you must first choose the ( - ) key
(left of the enter key) then choose 10.8. *Please note that the negative sign will
appear smaller than the subtraction symbol. Remember the negative sign must
always be used when an equation, number or computation begins with a negative.
2.) The n th root The ^ raises a number to a power or that is exponent.
For example if you want to find the 6.78 , you want to use the key or typein 78.6^(1 2 ) or 78.6^0.5.
If you want to find the 3 6.78 its easiest to type in 78.6^( 31 ). You can also use
the math key option #4.
3.) Changing decimals to fractions and vica versa For example, to convert .236 to a reduced fraction, type in .236 then Math,
choose Frac then hit enter. Do the same to change a fraction to a decimal by
choosing Dec under the Math key.
4.) Basic viewing windows under the zoom key There are 2 basic built in windows that we use in the class, they are found under
the zoom key. The standard window, as stated, above gives you a [-10,10,1] by
[-10,10,1] window. The z-decimal window will give you a close up view of the graph
in comparison to the z-standard window. The z-decimal window gives you a
[-4.7,4.7,1] by [-3.1,3.1,1] window. You should always hit the window
key to see the window that you are currently using.
5.) Graphing using the y= key
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After entering your equation(s) under the y= key, always hit trace afterwardsto explore the coordinates. You may enter up to 10 equations at a time.
If you want to save an equation under the y= key but not display its graph, useyour left arrow key, go on top of the = sign then hit enter. When the equal sign is
not highlighted the graph is so called turned off. To turn the graph back on just
simply go over the equal sign and hit enter.
The trace key allows you to see the points on the graph by displaying thecoordinates on the bottom of your screen.
6.) Evaluating at a particular x-value You may evaluate a graph (remember hit trace first) at x-values by simply typing
in the x value on the keypad then hit enter. *Please note that your x-values must be
no smaller than your x-min and no larger than your x-max *.
For example lets evaluate2
2
+
=x
y at x = 6. First type in )2(2 + x under the
y= key, hit trace, type in 6 then hit enter. The calculator will give you the y-value
and the cursor will be located at the point (6, -.25). Note: Notice that we usedparenthesis for (x + 2). If you type 2 x + 2, the calculator will only divide 2 byx and not by the whole expression (x + 2).
You will need to be in the z-standard window for the example above because usingz-decimal will result in an error due to 6 being larger than x-max which is 3.1.
7.) Using tables, always set up the table first*There are 2 choices, you may have the calculator complete the table for you or you
may enter various x-values then have the calculator give you the corresponding y-
values*
Having the calculator complete the table for you. To have the calculator completethe table for you first go to Tblset. Enter the x-value that you want to start with,
then enter the increment value or that is the Tbl value. Choose auto for both
the Indpnt and Depend options (the Indpnt represents the x, and the Depend
represents the y). Now hit 2nd then Graph and your table is now complete.
You may use your up and down arrow keys to look at other ordered pairs.
Entering various x-values on your own. To complete a table by entering various x-values once again go to Tblset first. Once you are there just simply choose the
Ask option for the Indpnt , now hit 2
nd
then Graph and your table shouldappear blank. You may now enter your x-values then hit enter to obtain the
corresponding y-value.
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Note: Fractions are most important in this class, they will be used throughout theentire course. You must be able to add, subtract, multiply and divide fractions byhand without using a calculator. We will also be evaluating expressions throughoutthe course that involves fractions.
Fraction review
Adding and subtracting: A common denominator must be found. (The leastcommon denominator, L.C.D.) Multiplying: Always simplify before you multiply. Dividing: Keep change and flip, then proceed per instructions with
multiplication.
Squaring a fraction: Square the numerator and denominator. Remember that anegative square is always positive.
Add or Subtract
1.)5
7
3
2 2.) 2
2
5 3.)
12
10
12
7 4.) -1 +
4
9
Multiply or divide
5.)24
5
25
4 6.) 6
36
5 7.)
14
7
8
49 8.) - 24
5
14
Simplify (remember PEMDAS)
9.) ( )322
10.) ( )612 11.) - 9
53 12.)
34 ( )35
2
13.) 19 0 14.) 0 10
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1105 Chapter 1
Functions and Their GraphsSec 1.1 Linear Models
The general (or standard) form of a liner equation is denoted by ax + by = c, where a and bcannot both be = 0
To graph: find the x-intercept (set y = 0), find the y-intercept (set x = 0) then a line connecting the
points.
To find the slope: (shortcut for slope m =int
int
x
ythen change the sign)
1. Graph the following by hand and state the intercepts and slope.
a. 3x 9y = 9 y-int_____, x-int______ b. - 20x 10y = 40 y-int_____, x-int____
m = ___________ m = _______
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
c. - 2x +3y = 4 y-int_____, x-int______ d. 1
43= yx y-int______, x-int________
m = ___________ m = _______
-5 -4 -3 -2 -1 1 2 3 4 5
-20
-16
-12
-8
-4
4
8
12
16
20
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
5
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2. Given: 2x 4y = - 48 a. x int. _______ b. y-int ________c. Solve for y _____________
d. State a window in which both intercepts are visible [ , ] by [ , ] then graph.
x
y
3. Given: -2x +3
y= 10 a. x int. _______ b. y-int ________c. Solve for y _____________
d. State a window in which both intercepts are visible [ , ] by [ , ] then graph.
x
y
Slope is discussed in section 1.4 but we will discuss here.
The slope intercept form is denoted by: y = mx + b, where m is the slope and b is the y intercept.
4. Write an equation in slope-intercept form for the lines below.
-18 -16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 1 2 14
-550
-500
-450
-400
-350
-300
-250
-200
-150
-100
-50
50
100
150
x
y
1 2 3 4 5 6 7 8 9 10 1110
20
30
40
50
60
70
80
90
100
110
x
y
Slope: In the context of applications is the rate of change. For example someones salary
changing over time, the rate at which tree is growing over time, the rate at which a cars value
depreciates over time, the rate at which a drug leaves the body etc. Were assuming the rate is
constant.
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See page 14 #1 3 for applications using the slope-intercept form.
See page 17 # 25 28 for applications using the standard/general form.
5. The temperature in the Kent, Ohio at 5 a.m. was 52 0 F. The temperature rose 4 degrees every hour
until is reached its maximum temperature at about 5 p.m. Complete the table of values for thetemperature, T, at h hours after 5 a.m.
h 0 3 6 8 10
T
a. Write an equation (in slope intercept form) for the temperature, T, in terms of h.
b. Graph the equation.
c. What is the temperature at 10 a.m.? At noon?
d. When was the temperature 88 0 ?
1 2 3 4 5 6 7 8 9 10 11 12
10
20
30
40
50
60
70
80
90
100
x
y
6. Tony has $450 in his savings account for school expenses. He withdraws $50 every month formiscellaneous items per month. Complete the table of values for the amount of money, A, left in hisaccount after m months.
m 0 4 6 8 9
A
a. Write an equation that expresses the amount of money, A, in his account in terms of the number ofmonths, m.
b. Graph the equation.
c. How much did his money decrease between the third month and the sixth month? Show this on thegraph.
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d. When will his account contain more than $200? Show this on the graph.
1 2 3 4 5 6 7 8 9
50
100
150
200
250
300
350
400
450
x
y
7. Juanita plans to diversify her retirement savings in 2 different accounts. The IRA CD account pays3.5% interest per year, the Traditional CD account pays 4.0% interest per year. Juanita plans to earn$600 per year on her retirement savings.
a. How much interest will she earn in the IRA account? Use the variable x.
b. How much interest will she earn in the Traditional account? Use the variable y.
c. Write an equation in general form that describes her retirement savings account if she earns $600 peryear.
d. Find the intercepts and describe their meaning. Round to the nearest dollar.
Sec. 1.1 starting on page 14Homework: 1, 3, 14 17 all, 25a-d, 29a-c, 33, 45, 46, 49
Sec. 1.2 Functions
A function is a relationship between 2 variables such that for each x value (input) there exists a
unique y value (output).
1. The following table denotes the Daytona 500 winners average speed (in miles per hour)where x = 0 corresponds to the year 2000.
x 0 1 2 3 4
r 155.669 161.783 142.971 133.870 156.345
a.) Is r a function of x? _______
b.) From the table find r(3). _____________
c.) Find x such that r(x) = 161.783. ___________________
d.) Interpret your result from part b. _________________________________________
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2. The graph below describes the population of a town where P is a function of t. P denotes thepopulation (in thousands) and t denotes the time in years since 1995.
a. In what year was the population 10,000?
b. Approximate the population in 2000.
c. How long did it take the population in part b to double?
d. During what years was the population between 60,000 and 110,000?
1 2 3 4 5 6 7 8 9
10
20
30
40
50
60
70
80
90
100
110
t
P
3. Evaluate the following, simplify if possible. Do by hand then verify with your calculator.
a. f(x) = - 722 xx , find f(-2) b. f(x) = 422 + xx find f( - 2/3)
c. g(x) = 8x , g(12) then find g(4) d. h(x) =42
3
xx
, h( 5) then find h(2)
Sec. 1.2 starting on page 30 homework: 15, 17, 19, 27, 31a-b, 35a-b, 37a-c, 41a-d, 43a-d
Sec. 1.3 Graphs of Functions
Vertical Line Test A graph represents a function if and only if when you draw a line through the graphit crosses the graph atmost once.
1. See page 50 #9 of your textbook.
2. Complete the table for g(x) = 722 xx
3. Complete the table for f(x) = 3.5x x22 + x 3
x - 5 -3 -1 1
y
x3
7
12
27
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4. Complete the table for f(x) = 23.5x x22 4 5x
Note: Its often helpful to rewrite
function notation, that is set f(x) = y, P(r) = y etc.
6. Given the graph find the following: a. Find f( - 1) b. For what value(s) of x is f(x) = 1?
c. the value(s) of x where f(x) = - 2 d. Find where f(x) = 0
e. Find the minimum value of f(x). f. Find the intervals of increasing and decreasing.
g. Find where f(x) < 1 h. Find where f(x) > - 2
- ^ -
-3 -2 -1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
x
y
7. Use the graph to find: a. Estimate R(- 1) b. Estimate the value(s) where R(s) = 0.6
c. Find the maximum and minimum value(s) of R(s).
d. For what value(s) of s does R have a maximum and minimum values?
e. Intervals of increasing and decreasing.
-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
s
R(s)
x4
7
10
13
10
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8. The graph below is y = 3/2x 1, use the graph to find the following:
a. y = - 2 b. y > - 2 c. y < - 1 d. solve 3/2x 1 = 5
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
Sec. 1.3 starting on page 48 homework: 1a-f, 2a-f, 5a-d, 9, 13, 17a-c19a-c, 21a-c, 29a-b, 33a, 37
Sec. 1.4 Slope and Rate of Change
1. Find the slope for each linea. b.
Recall that given an x and y intercept the slope can be found using the shortcut
m =int
int
x
ythen change the sign
2. Graph the following using the intercept (cover method) method and find the slope for each.
a. 2x + 4y = - 8 m = _______ b. 8y + 6x = 48 m = _______
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
x
f(x)
The slope between two points ( 11,yx ) and ( 22 ,yx ) is m =12
12
xx
yy
The slope m or that is the constant rate of change is also described by
run
riseorfall
changehorizontal
angeverticalch =
3. Find the slope between the points ( 3, - 4) and ( - 5, 7)
4. Find the slope from the graph.
-30 -20 -10 10 20 30 40 50 60 70
-25
-20
-15
-10
-5
5
10
15
20
25
30
35
40
x
f(x)
Again, the slope can be described as a rate of change when working with applications.
5. The graph below denotes cost, C (in dollars) of renting a laptop computer in terms of h
hours.
a. Find the slope
b. Explain the meaning of the slope with respect to the context of the problem.
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1 2 3 4 5 6 7 8 9
10
20
30
40
50
60
70
80
90
h
C(h)
Sec. 1.4 starting on page 67 homework: 5-7 all, 9, 11, 13, 21a-d, 23, 33, 35
Sec. 1.5 Linear Functions
Recall that the slope-intercept form of a linear equation is given by y = mx + b,
Where m is the slope and b is the y-intercept. To graph using our calculator it is
necessary to write the equation in this form in order to enter it under the y = key.
1. Write the equations in the slope intercept form, then state the slope and y-intercept.
a. - 3x + 7y = 9 b. 2x 3y = 1
2 . i. Sketch by hand the graph of the line with given slope and y-intercept.ii. Write the equation of the line in slope intercept form.iii. Find the x-intercept.
a. m = - 1/2 and b = - 2 b. m = 3/4 and b = 3
equation _________________ equation _________________
x-int _______ x-int _______
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
Note that you can start at any pt. that lies on the graph not just the y-intercept when
sketching a graph. When given a point (other than b) and a slope we can use the
point slope form, then from here we can write it in slope intercept form.
The point slope form of a linear equation that passes through the pt. ( 11,yx ) with slope m
is y = m(x x1) + y1 , distribute the m and simplify to express in slope intercept form.
3. i. Sketch by hand the graph.
ii. Write equation in point slope form
iii. Write equation in slope intercept form.
a. ( -1, 2); m = -2/3 b. (3, - 2); m = 2
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
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Interpreting the slope: Often we are interested as to how a situation changes over time.
How much does our car depreciate each year, how does our salary change each year, is
the price of gas increasing or decreasing each day?
4. The prenatal growth of a fetus more than 12 weeks old can be approximated by theformula L = 1.53t 6.7, where L is the length in cm and t is the age in weeks. Prenatallength can be determined by X rays.
a. What is the slope? _________
b. Interpret the slope as a rate of change.
c. Estimate the age of a fetus that is 12 cm long.
5. The value in dollars of a copy machine is denoted by the functionf(x) = -370x + 5100, where x is years after 1998.
a. Interpret the slope as a rate of change.
b. Interpret the y-intercept. (vertical intercept)
c. What was the value of the copy machine in 2001?
6. The graph below denotes the value, V, (in thousands) of a tool making machine after t years.
a. Find the slope from the graph and explain its meaning with context of the problem.
b. Find a formula for the function that describes the graph. (in slope intercept form)
c. Explain the meaning of the vertical intercept. (the y-intercept)
3 6 9 12 15 18 21 24 27 30 33 36
10
20
30
40
50
60
70
80
90
t
V(t)
7. Match the equation with its graph. The scales on the graphs are all the same.
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a. y = 23
2x b. y = 2
2
3x c. y = - 2x + 3 d. y = - 4x + 3
x
y
L1
L2
L3
L4
Sec. 1.5 starting on page 85 homework: 1, 2, 11, 13,23, 33, 35,53a-d
Chapter 8 Linear Systems
Sec. 8.1 Systems of Linear Equations in 2 variables
The solution(s) given any 2 graphs is at the point of intersection.
1. Find the solution from the graph below.
= -
=-
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
Solving a system of linear equations using your calculator:
Solve for y for both equations and place under the y1 and y2 key. Hit graph, then trace.
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2nd trace, choose #5 then hit enter 3 times.
2. Solve the following using your grapher.
a. y =.68x 2.1 and y = 4.98 4.23x b. 3.1x + 2.3y = -1.1 and y = -2.3x 5.6
We will be discussing 3 different ways in which to solve a system of 2 equations; graphing by hand
or with our calculator, using linear combinations, that is using substitution or elimination. Before
doing so lets discuss the different types of systems.
In this section we discuss consistent, inconsistent, independent and dependent systems.
A consistent system will have at least one solution. An inconsistent system will have no solutions. (parallel lines) An independent system for 2 linear equations will have 0 or 1 solutions. A dependent system will have infinite solutions (coinciding lines).
3 Case Scenarios when solving a system of linear equations.
From a graph, the solution(s) if they exist, are at the point(s) of intersection
I. Consistent and independent: One solutionexists, one example would beperpendicular lines.
the solution is at (2.5, -0.5)
**When solving by hand you will get a value for x and a value for y.For example, x = 2.5, y = -0.5
II. Consistent and dependent: Infinite solutions, the lines are the same.
For example: The system of equations x y = 2 and 2x 2y = 4 are the same
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equations, one is just a multiple of the other, the graphs are the same.
**When solving by hand you will get a true result 0 = 0, or 5 = -5 etc.
III. Inconsistent and independent: No solutions the lines are paralell.
For example, the system y = 2x + 3 and y = 2x -1, the slopes are equal thereforeparallel lines.
**When solving by hand you will get a false result 7 =3, 0 = -9 etc.3. Solve by graphing, identify as consistent or inconsistent, then dependent or independent.
a. y = 3x 4 b. 3y 6x = 12
x + 2y = 6 y = 2x + 1
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
Solving using linear combinations, that is using substitution or elimination
For the following problems, solve by hand (symbolically) using substitution then
elimination. State whether the system is consistent or inconsistent and whether
the system is dependent or independent.
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4. x + y = 23x + y = 4
5. 16x 6y = - 4
- 8x + 3y = 2
6. x + y = 7
x + y = 8
7. 3x y = 6 - 4x + 2y = - 8
Note that Revenue = price quantity and cost = fixed cost(s) + variable cost(s)
Write equations for the following, do not solve.
8. Ron and Javier are ticket sellers for their band to raise money for charity, Ron handlingstudent tickets that sell for $2.00 each and Javier selling adult tickets for $4.50 each. Iftheir total income for 364 tickets was $1,175.50, how many did Ron sell and how many didJavier sell? Use R for Ron and J for Javier.
a. Write an equation for the number of tickets sold.
b. Write an equation for the revenue from the tickets sold.
9. Lauri Annes beaded jewelry company sells there bracelets for $30 each. When she first startedthe company her initial fixed cost was $350 and $6.00 per bracelet.
a. Express Lauri Annes cost C in terms of the numberx of bracelets produced.
b. Express her revenue R in terms of the numberx of bracelets sold.
Sec. 8.1starting on page 651 homework: 3, 5, 7, 10, 11, 17, 18, 25a-b, 27b-c
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Sec. 8.4 Linear Inequalities
1. Graph each inequalitya. - 2x + 3y > -6 b. y > 3
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
c. 3 < x 1 d. y + 2x > 4, y 1 x
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
e. - x 2y - 4 , 4x 6y < 12 f. y < - 22+
x, x 0, y 0
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
Sec. 8.4starting on page 689 homework: 1, 3, 9, 10, 15, 21, 23, 28
Chapter 6 Quadratic Functions
Sec. 6.1 Factors and x-intercepts
Solving quadratics using factoring:20
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Move all terms to the left, right hand side = 0 If the left is not factored and there are parenthesis, remove the parenthesis then factor Once the left is factored then set each quantity = 0 and solve for the variable.
1. Solve by factoring. a. (x 2)(x + 9) = 0 b. 9y 2 + 12y 5 = 0
c. 5x 2 + 27x = - 10 d. 3(2x 2 - 1) = 7x e. (x + 1)(x = 3) = 5
Recall 1a from above, notice how if x 2 is a factor then x = 2 is a solution? Therefore thereverse is true, if x = 2 is a solution then (x - 2) is a factor, if x = -9 is a solution then (x + 9)is a factor etc.
2. Write a quadratic equation (in standard form with integer coefficients) whose solutions aregiven.
a. x = 8, x = - 2 b. x = 1/3, x = - 7
3. A model rocket is launched upward at an initial velocity of 128 ft. per second. Its heighth above ground can be modeled by h(t) = - 16t 2 + 128t.
a.) Complete the table starting at x = 2 and incrementing by 1
b.) Determine when the rocket hits the ground
c.) Determine when the rocket is 240 ft. above the ground on the way up
d.) Determine the maximum height of the rocket
4. Nancy stands at the top of an 282 ft cliff and throws her College Algebra bookdirectly upward with an initial velocity of 25 feet per second. The height of her book
above the ground t seconds later is given by h(t) = - 16t 2 + 25t + 282, where h is in feet.
a. Use your grapher to make a table of values for the height equation starting at 0 andincrementing by 0.5 seconds.
tH(t)
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b. Graph the equation and use the table of values to help you choose the window.
c. Whats the highest altitude Nancys book reaches? Use the table and the trace key.
d. How long does it take for her book to reach the maximum height? Use the table and thetrace key.
e. When does Nancys book pass her on its way down? Remember shes standing on a282 ft cliff.
f. How long will it take her book to reach the ground?
Sec. 6.1starting on page 485 homework: 1a-e, 3, 7, 11, 13, 15, 17, 2329, 33
Sec. 6.2 Solving Quadratic Equations
In this section we will solve quadratic equations using the quadratic formula. The quadratic
formula can be used when factoring is not possible.
When using the quadratic formula make sure your equation is in standard form with all
terms on the left hand side and the right hand side equal to zero.
That is, ax 2 + bx + c = 0 where the quadratic formula is defined by
x =a
acbb
2
42
1. Solve the following using the quadratic formula, write answers in exact form and thenround to 3 decimal places where possible.
a. 3x 2 + 9x = -5 b. 2x 2 = 4x + 3
t
H(t)
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2. An object is propelled upward from a height of 80 feet at an initial velocity of 80 ft
per second, then its height after t seconds is given by the equation h = -16t 2 + 80t +80
where h is in feet.
a. Construct a table of values showing the objects height in 0.5 second intervals after it ispropelled.
b. What is the height of the object in 2 seconds on the way up?
c. How long will it take the object to reach the ground? Solve algebraically.
Sec. 6.2 starting on page 499 homework: 27, 31, 39a-c
Sec. 6.3 Graphing Parabolas
A quadratic equation is nothing more than a polynomial that has degree 2. We will
begin discussing quadratic equations that are in vertex form. The nice thing about themwritten in vertex form is that they are much easier to graph by hand, and we can easilyidentify many of its characteristics like the vertex, line of symmetry and the x and yintercepts.
*Note that when we talk about characteristics we are referring to a comparison to the
standard parabola which is defined by f(x) = x 2 . This graph opens upward with the vertex
at the origin and is symmetric about the y-axis.
-5 -4 -3 -2 -1 1 2 3 4 5
-2
-1
1
2
3
4
5
6
7
x
y
t
h
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Given f(x) = a x 2 What does the a value describe?
a describes the concavity, that is how wide or narrow it opens,
if a is positive then the parabola opens upward (concave up)
if a is negative then it opens downward (concave down)
Concavity: If a is greater then 1 the graph opens narrower
If a is between 0 and 1 then the graph opens wider
1. State whether the graph opens downward or upward.
State whether the graph opens narrower or wider.
a. y = 2.5 2x b. f(x) = - 2.5 2x c. f(x) = 0.24 2x d. y = - 0.24 2x
The vertex form of a quadratic equation is defined by
f(x) = a(x h) 2 + k
What does the (h,k) describe?
The (h,k) is called the vertex, it is where the minimum or maximum occurs on the
parabola. If you draw a vertical line through this point then you will see that the
parabola is symmetric about this line. To find the line of symmetry which is denoted
by x = h, simply set x h = 0 than solve for x.
The vertex also describes the shifting of the parabola with respect to y = 2x . For
example if the vertex is (- 2, 3) then we know that the graph was shifted left 2 then
up 3. Sometimes its easier to first determine the vertex then describe the shifting.
2. i. Determine if the parabola is concave up or down. ii. State the vertex.iii. State the line of symmetry. iv. State the shifting v. opens wider or narrower
a. f(x) = (x 9) 2 b. y = x 2 - 12
c. y = (x 1) 2 + 2 d. (x + 3) 2 - 1
e. f(x) = 4(x 8) 2 + 9 f. y =5
2(x + 9) 2 - 11
Finding the vertex from standard form y = ax 2 + bx + c
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The h value or that is the x-coordinate is x =a
b
2
To find the k value or that is the y-coordinate, simply plug the x-value that you found from step 1 into the
equation to find the y. (Remember when we evaluated equations earlier in the semester.)
For example, lets say y = 3x 2 - 6x + 1 the x-coordinate will be)3(2
)6(= 1 Now when we plug
1 into the equation we get y = 3(1) 2 - 6(1) + 1 = - 2. Therefore the vertex for the equation is (1, - 2)
3. Do the following by hand, show all work, do not use decimals.
o Find the vertex using the formula above, do by hand. (Remember to write the vertex in
ordered pair form)
o State whether there is a minimum or a maximum point on the graph.
o Find the coordinates of the intercepts.
a. y = - 3x 2 - 12x + 4 b. f(x) = 2x 2 + 4x 2
Determining the nature of solutions from the discriminant
Recall the quadratic formula, x = a
acbb
2
42 , acb 4
2
is called the discriminant
There are 3 case scenarios that determine the type of solutions
I. Disc. is negative , that is acb 42 < 0, there are 0 real solutions(Quantity under the radical is negative, there are 0 real solutions)
When the graph does not cross the x-axis, then there will be no real solutions
II. Disc. = 0 that is acb 42 = 0, there is 1 real solution (Quantity under theradical is 0, theres 1 real solution)
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When the graph just touches the x-axis there will be 1 real solution at the x-intercept
III. Disc. is positive, that is acb 42 > 0, there are 2 real solutions(Quantity under the radical is positive, theres 2 real solutions)
When the graph crosses the x-axis twice, then there will be 2 real solutions at those x-
intercepts
*Remember solutions exist at the x-intercept(s) Disc. # of real solutions Positive 2
Zero 1
Negative 0
4. Use the discriminat to determine the nature of the solutions.
a. x 2 - 2x + 3 = 0 b. 3x 2 + 5x = 9 c. 2x 2 - 4x + 2 = 0
Sec. 6.3 starting on page 510 homework: 1a-d, 3a-d, 15a, 15c, 17, 21, 29, 30, 43a-c, 44c
Sec. 6.4 Problem Solving
* Remember earlier that we discussed how minimums and maximums occur at the vertex
with quadratic equations. In business, we are most concerned with minimizing costs and
maximizing revenue and profit.
1. A projectile is thrown upward where its distance above the ground after t seconds is
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given by h(t) = - 16t 2 + 532t. Use your calculator.
Find its maximum height and the time it takes to reach its maximum height.
2. Bob owns a watch repair shop and has found that the cost of operating his shop is given
by C(x) = 4x
2
- 344x + 61 where x is the number of watches repaired.
How many watches must he repair to have the lowest cost and what will the total cost be?
Break-Even: Break-even is when a company has zero loss or zero profit. The profit equation is:
profit = revenue cost. Therefore break-even is when, profit = 0 or revenue cost = 0
Remember to write the break-even points in ordered pair form.
Remember when asked to find a minimum or maximum value use x = -b/2a!
There are 2 different ways you can solve these word problems. You may choose either way for thetest.
i. Find the profit equation, that is revenue cost = 0, solve for x by using the quadratic formula.Plug the x-value into either the revenue equation or the cost equation to get the y value. Plugginginto the revenue equation is always easier.
ii. Put the profit equation under y1, the revenue equation under y2, find the point of intersectionusing your grapher. You will be given the window on your test. This method is the easiest.
3. Juans Ladder company produces ladders and it costs his company C = .065x 2 + x + 1500 to
make x number of ladders. His company receives R = 60x dollars in revenue from the sale of eachladder. Round all answers to the nearest whole number.
a. Write a profit equation for his company.
b. Use the window [- 100, 1000, 40] by [- 100, 80000, 1000] to find the break-even points on yourgrapher.
c. Now use the quadratic formula to find the break-even points.
d. Find the value of x for which profit is a maximum.
Sec. 6.4 starting on page 524 homework: 5, 7, 17a-c, 21a-b, 22a-b,
49 use the window [ - 10, 2000] by [ - 10, 22000], 51use the window [ - 10, 500] by [ - 10, 270000]
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Sec. 6.5 Graphing Inequalities
Recall in chapter 1 when we solve equations graphically then also solve where f(x) > 0 etc.Remember solutions are at the x-intercepts, for inequalities you want to start off by finding the x-intercepts.
1. Using the graph below find where: a. (x + 5)(x 2) = 0
b. (x + 5)(x 2) > 0 c. (x + 5)(x 2) < 0
- +
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
x
y
2. Use your grapher to solve x 2 + 4.9x 10.56 > 6, with the window [ - 9.4, 9.4] by [ - 50, 50]
Recall the zero-product property in which if ab = 0 then a = 0 or b = 0
For example, (x 3)(x + 2) = 0, set x 3 = 0 solve for x then set x + 2 = 0 and solve for x.3. Solve (x 3)(x + 2) > 0, first solve for x then use test points to find the intervals in
which the solutions lie.
4. Solve, write answer in interval notation 2x 2 - 7x < 4
5. A rocket is fired from the ground. Its height in feet aftertseconds is denoted by
h = - 16t 2 + 297t.
a. Write an inequality for the following. In what time interval will the rockets height behigher than 1331 feet?
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b. Use your grapher to find the time interval in which the rocket is higher than 1331 feet.Use the window [- 10, 20, 1] by [1200, 1500, 50]
6. A company sells personalized pen at 480 20p pen each month in which they chargep dollarsPer pen. The company wants their revenue from the pens to be over $2160 per month.
a. Write an inequality that denotes their revenue. (Remember revenue =price quantity)
b. In what range should they keep the price of their personalized pens?Use [ - 2, 30 ] by [ - 10, 3000]
7. The cost in dollars of producingx number of items is given by C = - 0.2x 2 + 13x + 1500
where 0 x 680. How many items can be produced if the total cost is to be under $2500?Write answer in interval notation. Use [ - 100, 900, 100 ] by [- 10, 5000, 200 ]
Sec. 6.5 starting on page 535 homework: 5a-b, 7a-b, 19, 21, 31, 35, 51, 53, 55
Chapter 2 Modeling with Functions
Sec. 2.1 Non-linear models
In this section we will be solving by extracting roots. Its important to remember that theopposite of squaring is taking the square root and vica versa.
Extracting roots: ax 2 + c = 0, isolate the x 2 term, then take the of the other side.
Example 1, solve x 2 - 11 = 0, first put the 11 on the right, x 2 = 11
now take ndsidetherightha , x = 11
Example 2, solve -3(x + 2) 2 + 15 = 0, first put the 15 on the right, - 3(x + 2) 2 = - 15
now divide both sides by 3, (x + 2) 2 = 5
now take the x + 2 and set equal to ndsidetherightha , x + 2 = 5now solve for x, x = - 2 5
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Example 3, solve 2(x 9) 2 32 = 0, 2(x 9) 2 = 32, (x 9) 2 = 16, x 9 = 16
x 9 = 4
x 9 = 4 x 9 = - 4
x = 4 + 9 x = - 4 + 9
x = 13 and x = 5
Solve the following, give exact answers then round to 3 decimal places where needed.
1. a. 5x 2 = 30 b.3
4 2x= 12
c. (2x 5) 2 = 4 d. (5t 8) 2 = 17 e. 7(x + 8) 2 21 = 0
2. Solve for the specified variable. D = yx2
3
1, solve for x
Compound Interest: B = P(1 + r) n , A is the amount of the money in an account
compounded annually for n years at an interest rate r and P is the principal.
3. Alba wants to invest $5500 in a savings account that pays interest compounded annually.
a. Write a formula for the balance B in her account after 3 years as a function of the interestrate r.
b. If Alba wants to have $6750 in her account after 3 years, what would the interest rate r have to be?
Again recall that when solving an x 2 equation we take the square root of both sides.
Therefore when solve a square root equation we square both sides.
3. Solve the following, dont forget to check your answers. Isolate the term then solve.
a. 6x = 3 b. 62 x + 2 = 13 c. 2 13 +x = 6
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Sec. 2.1 starting on page 133 homework: 1, 3, 5, 7, 9, 13, 31, 33, 35, 36, 51, 52, 55a-b
Sec. 2.2 Some Basic Functions
This section goes over the 8 basic functions that are often used. There will be matchingof these basic functions and shifting of them on the final exam. In this section you will be
required to graph by hand shifting and or transformations of only y = x and y = x
Turn to page 145 to see the 8 basic graphs that are shown below. Note that the y = x graph is
incorrect in the book, the correct graph below.
1. y = x 2. y = x 2 3. y = x
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
4. y = x 5. y =3
x 6. y = x3
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
7. y = x
18. y = 2
1
x
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
First lets go over some basic absolute value problems. A few of these will be on the test.
Simplify
1. a. - 9 b. 2 - 84 c. 3 - 7 104
In section 2.3 shifting/transformations is discussed, well go ahead and discuss it now.
Given y = f(x) and y = f(x) + k, the graph will be shifted up k units.Given y = f(x) and y = f(x) k, the graph will be shifted down k units.
For example, y = x and y = x - 2 is shown below, state the shifting.
Given y = f(x + k) is the graph of f(x) shifted left k units.
Given y = f(x k) is the graph of f(x) shifted right k units.
The graph of y = x and y = 2x is shown below, state the shifting.
Recall in the earlier chapters how we solved equalities and inequalities from a graphicalperspective. Assume there are arrows on the graphs.
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2. The graph of f(x) = x 3 - 6 is given below. a. Decsribe the shifting
b. Solve x 3 - 6 = 21 c. Solve x 3 - 6 < 21 d. Solve x 3 - 6 > - 6
-4 -3 -2 -1 1 2 3 4
-24
-21
-18
-15
-12
-9
-6
-3
3
69
12
15
18
21
24
x
f(x)
3. The graph of f(x) = 23 +x is given below. a. Decsribe the shifting
b. Solve 23 +x = 0 c. Solve 23 +x < 5 d. Solve 23 +x > 3
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
Piecewise-defined function: Piecewise-defined functions are functions represented by more thanone equation. Sometimes an application can be modeled by a linear equation then it changes to aquadratic model.
4. Graph the piecewise-defined functions by hand
a. y = { x if x 1 and y = { x + 2 if x > 1
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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
b. y = { - x + 3 if x 2 and y = {2
3x - 2 if x < 2
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
Sec. 2.2 starting on page 148 homework: 1a-b, 5, 19a-c, 23a-c, 25, 26, 27a, 27c, 28a, 28c, 41, 43
Sec. 2.3 Transformations of Graphs
Recall from sec. 2.2
Given y = f(x) and y = f(x) + k, the graph will be shifted up k units.
Given y = f(x) and y = f(x) k, the graph will be shifted down k units.
Given y = f(x + k) is the graph of f(x) shifted left k units.
Given y = f(x k) is the graph of f(x) shifted right k units.
Identify each graph as a translation of a basic function, and write the formula for the graph.
1. 2. 3.
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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
4. a. State the basic graph. b. Describe the shifting
c. Sketch the basic graph and the function that is given, label 3 points on each for each function.
i. g(x) = 9+x ii. y = x + 2
-15-12 -9 -6 -3 3 6 9 12 15 18 21 24 27 30 33 36 39
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
g(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
iii. f(x) =3
1
x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-1.25
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
1.25
x
f(x)
Scale Factors and Reflections
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Recall from chapter 6 the effect the a value had on the standard parabola y = 2x , recall the
difference between y = 2x , y = 3 2x and y =3
12x . Recall y = - a 2x will result in the same
graph but reflected across the x-axis. In this section we call this a value the scale factor,
if a > 1 then the graph will be stretched vertically (narrower), if 0 < a < 1 the graph is
compressed vertically, it will open wider .
5. a. Identify the scale factor and describe how it affects the graph of its basic functionb. Sketch the basic graph and then the given function. Label 3 points on the graphs.
i. y = x2
1ii. y = 2 x and iii. y = -2 x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3-2
-1
1
2
3
4
5
6
x
f(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3-2
-1
1
2
3
4
5
6
x
f(x)
6. Given the graphs below find the following:a. State the graph as a transformation of the basic. b. State the equation for the graph.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
(- 2, 3)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
(3, -4)
Sec. 2.3 starting on page 163 homework: 1, 2, 5, 6, 7, 8, 10, 12, 14, 15, 21, 22, 23, 24, 26, 33, 51a, 53a, 65,
68, 69
Sec. 2.5 The Absolute Value Function
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There are 3 types of absolute value functions that we will be solving.
The rules are as follows:
I. Forx = a and a > 0, set x = a then x = -a and solve for x. (Theres no solution to x = -
For example: Solve the following: 53 x = 7
II. Forx < a set a < x < a, then solve for x. Do the same forx a
For example: Solve the following: 35 +x < 8
III. Forx > a set x > a solve for x, then set x < - a solve for x. Do the same for Forx a.
For example: Solve the following: 16 x 3
Solve the following: (first isolate the absolute value expression then use one of the rules above)
1. a. 35 +x - 2 < 8 b. 52 x + 10 15
c. 4 13 x = 8 d. 2 24 +x + 1 11 e. 13 x = - 8
Sec. 2.5 starting on page 201 homework: 13, 18, 21, 23, 25
Sec. 2.6 Domain and Range
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Finding the domain from a graph . Because the domain is related to the x-values, find theleftmost or the smallest x-value, then find the rightmost or the largest x-value then write
answer in interval notation. We will assume all the graphs have endpoints, that is there are no
arrows.
Finding the range from a graph. Because the range is related to the y-values, find the bottom
most or the smallest y-value, then find the top most or the largest y-value then write answer in
interval notation. Again, we will assume all the graphs have endpoints, that is there are no
arrows
1. From the graphs below, find the domain and range using interval notation.
a.) Domain ______________ b.) Domain _____________
Range _______________ Range ______________
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
f(x)
c.) Domain ______________ d.) Domain _____________
Range _______________ Range ______________
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
45
x
f(x)
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
56
x
f(x)
Finding the domain and range given a function:
When given an equation that contains a denominator, simply set the denominator 0, then solve forx to find the domain. Graph the function to assist in finding the range.
2. State the domain only for the following functions.
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a. f(x) =1
3
xx
b. f(x) =53
3
+xx
When given an equation that contains a , set the radicand 0 then solve for x. Graph thefunction to assist in finding the range.
3. State the domain and then graph with your grapher to find the range.
a. f(x) = 3x b. f(x) = 4+x - 3
Sec. 2.6 starting on page 211 homework: 1, 5, 26a-b, 29a-b
Sec. 3.1 Variation
An example of variation is the area of a circle. A = r 2 states that the area of a circle with
radius r is directly proportional to the square of its radius, the constant of proportionality
is . Notice how the dependent variable A increases as the radius r increases.
Direct Variation with a Power: Ify varies directly with x to the power n, then y = kx n , k and n are
positive exponents with k being the constant of proportionality or the constant of variation.
1. Write an equation for the following then find the constant of proportionality.
a. L varies directly to V. L = 10 and V = 2
b. G varies directly to the square of h. G = 26 and h = 3
Ify varies inversely with x to the power n, then y =nx
k.
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2. Write an equation for the following then find the constant of proportionality.
a. y is inversely proportional to x squared. y = 12 and x = 2
b. t varies inversely to m. t = 9 and m = 5
3. a. Use the table of values to find the constant of variationb. Write y as a function of x. That is find the equation of variation.c. Fill in the rest of the table with the correct values.
y varies directly with x y varies directly with the square of x
y varies inversely with x y varies inversely with the cube of x
See page 250 # 15, 17, 19
Solving applications: i. Write the equation of variation. ii. Use 2 data points to find k.
iii. Plug k into your equation. iv. Use this equation containing to answer any followingquestions.
Sec. 3.1 starting on page 249 homework: 13a-c, 15a-c, 17a-c, 19a-c, 21, 23, 25
This is the formula sheet that you will be able to use on the Chapter 4 and
Chapter 5 tests. Please have it with you in class also.
x y
4
7 17.5
11
32.5
x y
10.8
- 1
1.2
5 30
x y
- 1
- .7
14 -.25
20
x y
- 10 -.002
2
.016
11
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Formula Sheet for Chapter 4 and 5
P(t) = P 0 b t P = P 0 (1 + r) t P = P 0 (1 - r) t A(t) = P(1 + nr ) nt
y = log b x then x = by
y = ln(x) then x = e y
1. log b b = 1 2. log b 1 = 0
3. log b b x = x (basically log b bsomething = something)
4. b xblog = x b somethingblog = something
Base 10 and base e logarithmic properties
5. log10 = 1 ln e = 1
6. log 1 = 0 ln 1 = 0
7. log10 x = x ln e x = x log10something = something
and the ln e something = something
8. 10 xlog = x e xln = x e )ln(something = something
9. product property: log b (xy) = log b x + log b y
10. quotient property: log b
y
x= log b x - log b y
11. power property: log b x k = k log b x
12. log a x =a
x
log
logor log a x =
a
x
ln
ln.
13. i. x m x n = x nm+ ii. (x m ) n = x nm iii. n mx =nm
x / iv. .nm
n
m
xx
x =
Chapter 4 Exponential Functions
Sec. 4.1 Exponential Growth and Decay
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Compare the graphs. One is denoting exponential growth, the other linear growth.f(x) = 3 x f(x)= 3x both graphs
y1 = 3x y2 = 3 x
From the table above notice how for each unit increase in x 3 is added to the value of y1. However, foreach unit increase in x, 3 is multiplied to the value of y2. This is one example of exponential growth.
Now lets compare y = 3 x to y = 2 3 x (this is the darker graph)
y = 3 x y1 = 3 x and y2 = 2 3 x
1. From the table above determine the initial value for when x = 0 for both graphs.
2. From the table and graphs, state the growth for each.
The Exponential Growth function is given by P(t) = 0Pbt for b > 1, 0P denotes the initial value,
b denotes the growth factor and t is for time.
3. A rancher started with 65 alpacas and finds that the number of alpacas increases by a factor of 2 everyyear.
a. Write the exponential growth function.
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b. Complete the table starting at 0 and incrementing by 1.
t0 1 2
3
4
5
P(t)
c. Observing your table of values give an appropriate graphing window[ , ] by [ , ]
d. Graph the function using your calculator.
e. How many alpacas does the rancher have after 9 years?
Note:
If the problem states that there is an increase by a factor of b every month, weekyear etc. then the equation is simply P(t) = 0Pb t .
If it states every 2 weeks, 6 months, 3 years etc. then the equation will beP(t) = 0Pb
2/t , P(t) = 0Pb6/t , P(t) = 0Pb
3/t etc.
***Hence the words every n amount of time t will be t/n.
4. A less experienced rancher starts off with 65 alpacas and finds that his number increases by a factorof 2 every 2 years.
a. Write the exponential growth function.
b. Complete the table starting at 0 and incrementing by 1.
t0 1 2
3
4
5
P(t)
c. How many alpacas does this rancher have after 9 years?
d. Comment on the differences in the tables above.
5. A colony of bacteria starts with 450 organisms and triples every 3 weeks.
a. Write a formula for the population of the bacteria colony after t weeks.
b. How many bacteria will there be in 2 weeks?
c. How many bacteria will there be in 7 weeks?
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Recall the compounded annually interest formula from chapter 2 A = P(1 + r) t, where P is the
amount invested, r is the interest rate, t is time in years and A is the amount in the account after t
years.
In this chapter we call the (1 + r) the growth factor where r is the percent increase.
6. Since the opening of a manufacturing facility, the population of a certain city has grown according
to the model f(t) = 210,000(1.015)
t
, where t is the number of years since 1999.
a. State the initial value. b. What does 210,000 represent? c. What is the growth factor?
d. What does t = 0 represent? e. What is the percent increase? f. What was the population in2005?
7. Reggie invests $6,500 into a retirement certificate of deposit account that pays 4.5%, interest
compounded annually.
a. Write an equation that denotes exponential growth.
b. How much will he have in his account after 5 years?
8. Lakisha bought a house in Alameda California for $400,500. Since 1995 the prices of homes inthat area rose on average 7% per year.
a. State the initial value of the home.
b. What does t = 0 represent?
c. What is the percent increase?
d. Write an equation that denotes exponential growth.
e. How much was her house worth in 2000?
The Exponential Decay function is given by P(t) = 0Pbt for 0 < b < 1, 0P denotes the initial value,
b denotes the decay factor and t is for time.
y1 = 3 x y2 =
3
1x y1 = 3 x and y2 =
3
1x
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For percent decrease word problems, b = 1 r is the decay factor and r represents the percent
decrease.
9. Vics boat cost $14,000 in 2000 and has depreciated by 8% every 2 years.
a. State the decay factor and the percent decrease.
b. Write a function that describes the exponential decay.
c. How much was his boat worth in 2005?
10. The number of bass in Lake Campbell has declined to one-third of its previous number due topollution every 4 years since 1990 when the number of bass was estimated at 5,000.
a. State the 0P. b. Write a function that describes the exponential decay.
c. How many bass were there in 1995?
Note: Recall that when solving an nx equation we take the n the root of the other side.
11. Solve for the variable, give exact answer then round estimate to 3 decimal places, assume all positiveanswers.
a. 1875 = 3b 4 b. 151,875 = 20,000b 5 c. 10.56 = 12.4(1 r) 20
12. A new Toyota Corolla cost $18,000 in 2005. The value of course decpreciates exponentiallyover time. A 2 year old Toyota Corolla cost $14,600.
a. Write an equation that denotes the value of the Toyota Corolla.
b. Find the decay factor and the percent rate of depreciation.
c. How much would a 5 year old Toyota Corolla?
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13. Given the table below find the following.
a. Determine whether the tablerepresents exponential growth or decay.
b. State the initial value.
c. Write an equation that denotes exponential growth or decay.
d. Find the growth or decay factor.
e. Find the percent rate or percent decrease.
f. What is the q-intercept?
g. Write a formula that represents the data in the table.
h. Complete the table.
14. Given the graph below find the following.
a. Find the initial value.
b. Find the growth factor b. (no decimals)
c. Write a formula for the function.
1 2 3 4 5 6 7 8 9 10 11 12
2
4
6
8
10
12
14
x
y
p 01
`2
3 4
q 84
30.24 18.144
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For #15 and 16, assume that each population grows exponentially with constant annual percent
increase, r.
15. a. The population of Ohio was 11,353,140 in 2000. Write a formula in terms of r for the populationof Ohio t years later.
b. In 2006 the population of Ohio was 11,478,006. Write an equation and solve for r.
c. What was the annual percent increase to the nearest hundredth of a percent?
16. The population of Tiffin was 2,500 in 1995 and tripled in 4 years.
a. Write an equation that denotes Tiffins population growth.
b. Find the growth rate r.
c. Find the annual percent of increase to the nearest hundredth of a percent?
Sec. 4.1 starting on page 334 homework: 5a-b, 7a-b, 9, 11, 13, 15, 17, 33, 37, 41a-c, 45, 49, 55, 57,
63a-b, 65a
Sec. 4.2 Exponential Functions
Recall in sec. 4.1 that an exponential growth function was denoted by P(t) = 0Pbt for b > 1,
and the exponential decay function was denoted by P(t) = 0Pbt for 0 < b < 1. We were working with all
applications in sec. 4.1. Both functions are exponential, its just that with applications we use the wordsgrowth and decay and also we use the above mathematical notation.
In section 4.2 we also work with exponential functions but not within applications, therefore we willbe using slightly different notation. Instead of using the words growth and decay we will be indentifyingwhether or not the functions are increasing or decreasing.
Also in this section we will be working with shifting, reflection, domain, range, x and y intercepts as wedid in previous chapters.
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An exponential function that is increasing is denoted by: f(x) = a(b x ) For y = a(b x ) , b > 1 and a > 0
no x-intercept
y-int (0,a) (recall b0= 1)
domain: all real numbers or that is ( - , )
range (0, )
horizontal asymptote: y = 0 increases from left to right
The graphs will have the basic shape below. The graphs are increasing from left to right. The larger the baseb and the larger the initial value a the faster the graph will increase. Notice also how the graphs do notcross the x-axis, therefore there exists a horizontal asymptote at y = 0.
y = 10 x y = 2 x y = 3(2 x )
Note: You will need to memorize the basic shape and their properties.
Note the same type of shifting applies with exponential functions as they did with quadratics. With
f(x) k there is a vertical shift up or down and with f(x h) there is a horizontal left or right.
y = 3 x y = 3 x + 4 y = 3 )4( +x
An exponential function that is decreasing is denoted by: f(x) = a(b x )(the properties below are the same for the increasing exponential)
For y = a(bx
) where 0 < b < 1 and a > 0 no x-intercept
y-int (0,a) (recall b0= 1)
domain: all real numbers that is ( - , )
range (0, ) horizontal asymptote: y = 0
decreases from left to right
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** Be Careful: Recall that x n = nx
1, so x 2 = 2
1
xetc., therefore 2 x = x2
1=
x
2
1
Therefore exponential functions that are decreasing can also be denoted by
y = a(b x ) where b > 1 and a > 0 because y = a(b x ) can be re-written as y = ax
b
1
I would advise for any negative exponents, rewrite with positive exponents then decide if the
function is increasing or decreasing.
y = 2 x y = 2 x that is y =
x
2
1graphed together
1. Find the y-intercept (x = 0) and determine whether the functions are increasing or decreasing.
a. f(x) = 21(1.015) x b. y = 1.3(3) x c. y =
x
2
3d. g(x) =
x
5
2
When we replace x with x that is given y = f(x) the y = f(-x) will be reflected across the y-axis.
The y 2 graphs are darkened. That is
y1 = 2 x and y 2 = 6 x y1 = 2 x and y 2 = 6 x
Recall earlier the difference between y = x and y = - x , the y = - x is the same except it is reflected
across the x-axis.
2. Observing the graphs below comment on their differences.
y = 2 x y = 2 x that is y =
x
2
1y = - 2 x
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3. Compare the following with y = 4 x . State whether the function is increasing or decreasing and describe
any shifting and reflection.
a. y = 4 x - 2 b. y = 4 1+x c. h(x) = - 4 x d. y = 4 x e. y = 4 5x
4. See page 348 # 3a-b. Make a table of values and graph each function.
x -
2-
10
1
2
3
4
y1 = 2x
y 2 =
x
2
1
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
2
4
6
8
10
12
14
16
x
y
Lastly in this section we will start to solve exponential functions. Here in this section, the key is torewrite one or both of the bases the b so they are the same, then set the exponents equal to eachotherthen solve for the variable.
For example, if 8 x = 8 3 , what would x have to be in order to make the statement true?
What if instead we had 2 x = 8, what would x have to be? Couldnt we rewrite 8 as 2 3 ,
therefore we now have 2 x = 2 3 , now its more obvious that x = 3.
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Recall the following exponent rules to assist you in rewriting one or both of the bases in order to makethem
i. x m x n = x nm+ ii. (x m ) n = x nm iii. n mx =nm
x / remember that 27 = 3 3 , 9 = 3 2 , 3 8 = 2 etc.
5. Solve each by hand, no calculator.
a. 5 x = 5 43 x b. 3 52 +x = 9 c. 9 3+x = 81 x2
d. 8 43 x = 64 e. 4 x = 2 f. 49 32 +x = 7
g. 27 1+x = 9 h. 2 4 3x = 8 x
Sec. 4.2starting on page 348 homework: 1a-d, 3a-b, 5a-b, 7a-d, 27, 31, 33, 35, 36
Sec. 4.3 Logarithms
Recall in section 4.1 the following word problem:
Since the opening of a manufacturing facility, the population of a certain city has grown according
to the model f(t) = 210,000(1.015) t, where t is the number of years since 1999. We were then asked to
find the population in 2005. We simply plugged 6 into t.
What if we wanted to know the year in which the population reached 250,000? We would have to solvefor the t. In order to do so we have to learn the properties oflogarithms.
Lets say we have y = 2 x and we want to solve for x, we can do so using the property of logarithms.
We will learn that if y = 2 x then x = log 2 y.
The logarithmic function is denoted by y = log b x where b > 0, x > 0 and b # 1.
and can be written as x = b y
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The logarithm y = log(x) is called the common logarithm, the base is 10 though its not
written.
Recall the exponential function from sec. 4.2 then observe the graph of y = log(x).
y = 10 x y = log(x) both graphs along with y = x
Notice on your calculator how the 10 x key and the log(x) are right behind eachother?Likewise with the x 2 and the x . The opposite of squaring is taking the square root, hence the opposite
of y = 10 x is y = log(x), we say they are inverses of eachother. If 2 functions are inverses of eachother
that means that when you graph them they are reflections on one another about the line y = x. This isshown in the last graph above. We will be working with inverses in chapter 5.
In this section we will mainly be rewriting from one form to another, evaluating without a calculator andwill be solving just a few word problems.
1. Find each logarithm without using a calculator. (Look at the rule/formula sheet. You may need to
rewrite using the fact that ify = log b x then x = by )
a. log 5 (25) b. log 8 ( 8 ) c. log 2 (8)
d. log 49 (7) e. log 6 (6) f. log 4 (1)
g. log(10) 4 h. log 3
3
1i. log(1)
2. Evaluate with a calculator to 3 decimal places. What happens as the x value increases? Comment on
the amount of increase.
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a. log(12) b. log(120) c. log (1200)
3. Rewrite each equation in logarithmic form. (Remember that ifthen x = b y y = log b x )
a. 8 = 2 3 b. x = 7 y c. 5 p = q
d. 4 x2 = y e. 5 2 = 25 f. m = 4.5 2.3
Solving exponential equations
Isolate the term that contains the exponent
Rewrite the exponential equation to logarithmic form, remember if x = by
then y = log b x.
Solve for the variable.
Note that 2(10 x ) # 20 x . Also2
14log# log7
For example: Solve and round to 3 decimal places.
2(10 x3 ) = 50 Isolate the term with the exponent by dividing both sides by 2
10 x3 = 25 Rewrite to log form: 3x = log(25) Solve for x: x =3
)25log((exact solution)
Note that you cannot divide the 25 by the 3, first find the log of 25 then divide by 3
x = .466 is the approximate solution.
4. Solve the following, give exact answers where possible then round to 3 decimal places if needed.
a. 36 = 10 x b. 1120 = 40(10 x ) c. 5(10 x7 ) = 365
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d. 2 + 4(10 x2 ) = 10 e. 24(10 x2.3 ) 23.5 = 364
5. The population of a certain city increased during the years 1990 to 2000 according to the
formula P(t) = 2,456,000(10)x
035. . Where t is the number of years since 1990.
a. What was the population in 1998?
b. When will the population reach 5,350,000? Round to the nearest year.
Sec. 4.3 starting on page 363 homework: 1 7 odd, 11, 15, 17, 23a, 31a-d, 35-41 odd,
45, 49a, 49c
Sec. 4.4 Properties of Logarithms
In this section we will mainly be simplifying, rewriting and solving. There will be of course someapplications.
On page 46 from the formula/rule sheet we will be using the following properties and rules:
product property: log b (xy) = log b x + log b y
quotient property: log b
y
x= log b x - log b y
power property: log b x k = k log b x
i. x m x n = x nm+ ii. (x m ) n = x nm iii. n mx =nmx / iv.
nm
n
m
xx
x =
1. Using the properties above, combine into one logarithm and simplify.
a. log x + log (8) b. 3log(x) + log(y) c. log x -2
1log y
d. log(3a) + log(b) e. log (2x 7) log b 2
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f. 7 log (x) + 8 log(y) g. log(x 2 - 25) - log(x + 5)
2. Use properties to expand in terms of simpler logs.
a. log(7x) b. log7
2
xx
c. log[x 4 (y 5 )]
d. log [x 2 (4x + 3) 2 ] e. log 3 42
x
x
3. If log b 2 = 0.43 and log b 3 = 0.68, evaluate the following.
a. log b 6 b. log b 9 c. log b
2
3
2 methods in solving exponential equations that contain a base other than 10.
Method 1: i. Isolate the term that contains the exponent. ii. Take the log of both sides.
iii. Use the property log(b) x = xlog(b) then solve for x.
For example: Solve 2 x = 6, log(2) x = log(6), xlog(2) = log(6),
x =)3log(
)6log(is the exact answer, 2.585 is the estimate. Note:
)3log(
)6log( log(2)!
Method 2: i. Isolate the term that contains the exponent. ii. Rewrite using the fact thatif y = b x then x = log b y. iii. Use the change of base formula to solve for x.
Change of base formula: log a x =a
x
log
log
4. Solve , give exact answer then round to 3 decimal places.
a. 4 x = 9 b. 4 2x = 7 c. 2.13 x = 8.1
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5. In 2003 the Soccer for Charity organization had 2,575 members with an annual growth rate of4.5%.
a. Write a formula for the membership in the Soccer for Charity as a function of time, assumingthat the organization continued to grow at the same rate.
b. How many members did they have in 2006?
c. When will the organization have 4,500 members? Round to the nearest year.
The formula for compound interest is as follows: A(t) = P(1 + nr ) nt
A(t) is the amount in the account after P dollars is invested at an interest rate r
after t years and n is the number of times compounded per year.
If the money is compounded monthly n = 12, quarterly then n = 4, semi-annually n = 2.
6. What rate of interest is required so that $750 will yield $975 after 3 years if the interest is
a. Compounded quarterly?
b. Compounded monthly?
Sec. 4.4 starting on page 373 homework:
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Chapter 5 Logarithmic Functions
Sec. 5.1 Inverse Functions
If 2 functions are inverses. then their graphs are reflections of eachother about the line y = x.
Recall from sec. 4.3 the following.
y1 = 10 x y 2 = log(x) both graphs along with y = x
Observe the table of values:
Note: If 2 functions are inverses of eachother then their domain and range values are interchanged.
The inverse of f(x) is denoted by f 1 (x). Note that f 1 (x) )(
1
xf
1. Let f(4) = 7, f(5) = 10, f(6) = 13 and f(7) = 16.
a. Make a table of values for f(x) and another table for its inverse f 1 (x) .
x y10 1
1 10
2 100
3 1000
x y 21 0
10 1
100 2
1
000
3
x f(x) x f 1 (x)
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Steps to finding the inverse given a function
I. Replace f(x) with y
II. Solve for x.
III. Replace x with f 1 (x) and y with x.
For example: f(x) = -3x 1, y = -3x 1, 3x = -y 1, x =3
)1( y, therefore f 1 (x) =
3
)1( x
2. Find a formula for the inverse then graph using your calculator to verify.
a. y = 2x + 4 b. f(x) =2
14 x
3. Find the inverse then make a table of values for f(x) = 4x - 8
4. Find the function for the following then graph its inverse.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
5. Use the graph below to draw its inverse beside it. Label all the ticks and label 2 points on the graph.
x f(x) x f 1 (x)
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-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
10
20
30
40
50
x
y
x
y
One-to-One Function and Horizontal Line Test
A function is one-to-one if for each output there exists only one input.
Note that example below is not one-to-one. This is a quadratic function; if we draw a horizontal linethe line would intersect the graph in more than one place. This is called the horizontal line test.
The graph represents a function but not a one-to-one function. Hence the function does not have aninverse.
6. Determine whether the graphs and functions below have inverses.
a. b. c.
d. y = - 4.9x + 7 e. f(x) = 3 - 2x f. y = 4x 3
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Sec. 5.1 starting on page 415 homework:1a-c, 23, 27, 28, 43a
Sec. 5.2 Logarithmic Functions
Recall from sec. 5.1 how to find an inverse, recall that if y = log b x then x = b y
1. Given y = 3 x
a. Find its inverse b. Make a table of values c. graph both functions
-9-8-7-6 -5-4-3-2-1 1 2 3 4 5 6 7 8 9
-9-8-7-6-5-4-3-2-1
12345678
9
x
y
2. Compare the following with y = log(x) to describe the shifting or the reflection.
a. y = -log(x) b. f(x) = log(x 3) c. y = log(x) 4 d. f(x) = log(-x)
x
f 1 (x) -1
0
1
2
x -1
0
1
2
f(x)
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Note the 2 properties: log10 x = x and 10 xlog = x
3. Simplify: a. log1016 b. 10 )16log( x c. 10 xlog2
4. Given f(x) = log10 x Solve for x a. f(x) = 3.45 b. f(x) = -1.76
5. Convert each logarithm equation to exponential form.
a. y = log 6 x b. log12
(4) = w c. log(5) = p d. log c (g) = d
Note: When asked to solve a logarithm convert to exponential, when asked to solve an exponentialconvert to a logarithm.
Note: Evaluate log(-10) Recall that for log b (x), x > 0, therefore check your answers when
solving to make sure the x-values are not where the log is undefined.
6. Solve for the unknown value. Leave answer in exact form where needed. Remember to combinelogarithms into one by using the previously learned properties.
a. log b 25 = 5 b. log b 49 =2
1c. log 2 (x) = 12
d. 3log 6 ( x) = 4 e. log 4 x = 2 f. log(x) = 2
g. 4log 2 (x) = 12 h. log 2 (4x - 1) = 2 i. log 3 x = -2
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j. 5 + 2logx = 13 k. log x + log(x 3) = 1
l. log 3 (5x 1) - log 3 (x) = 2 m. log 4 (x + 8) + log 4 (x + 2) = 2
Sec. 5.2 starting on page 430 homework: 1, 3, 19a-c, 20a-c, 25a-d, 29a-c, 31, 33, 37-53 odd
Sec. 5.3 The Natural Base
The number e
The natural exponential function is the function y = e x .Many real world applications within the biology and finance involves the irrational number e. This
specific exponential function is defined by y = e x . Because e is approximately equal to 2.718,
its graph lies between y = 2 x and y = 3 x .
Graph all 3 functions under the z-decimal window.
As with the other exponential functions, this specific function can also describe growth or decay.
Notice how they are symmetric about the y-axis, therefore they are even functions.
y = e x . y = e x .
The natural logarithm
The logarithm function with base e , that is y = log e x which can be rewritten as x = ey is
called the natural logarithm. The natural logarithm function is written as y = ln(x). Recall
That y = log10 (x) is written y = log(x). The natural logarithm y = log e x is written as y = ln(x).
Notice how the graphs of y = ln(x) and y = e x are inverses of each other.
Just as y = 10 x and y = log(x) were.
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y = lnx y = e x graphed together
Keep in mind that the natural and common logarithmic functions are merely log functions withunique bases that are often used in science and business applications.
Converting log functions to exponential and vica versa
To solve a natural exponential function we convert it to the natural logarithm, to solve a
natural logarithm, we convert it to the natural exponential function. When solving we do the
same as with the other exponential and logarithmic functions.
y = ln(x) is rewritten as x = ey
y = ex
is rewritten as x = ln(y)
1. Solve for x. Give exact answers where possible, then round to 3 decimal places.
a. 5.9 = e x b. 192 = 16e x4 c. 2 + 4(10 x2 ) = 10
d. 4 - e x3 = 2 e. ln(x) = 3.5 f. 4.5 = 4e x1.2 + 3.3
Recall the following properties given in chapter 4.
i. ln e x = x that is ln e something = something
ii. e xln = x that is e )ln(something = something
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iii. ln e = 1 iv. ln 1 = 0
2. Use the properties above to simplify.
a. e 14ln b. ln e )4( x c. e xln5 d. ln( e ) e. ln
xe
1
3. The growth of a colony of bacteria is given by Q(t) = 450e t195.0 . There are initially 450
bacteria present and t is given in hours.
a. Does this model indicate that the population is increasing or decreasing? Explain.
b. How many bacteria are there after 11 hours?
c. Graph using [0, 10] by [450, 3160]. Determine the bacteria present after 6 hours.
d. Using the table function on your grapher, determine the number of hours that will pass inwhich the amount of bacteria reach approximately 2600.
4. The number of milligrams of a drug in a persons system after t hours is given by the function
D = 30e t4. .
a. Does the model indicate that the amount of drug will increase or decrease as time goes by?Explain.
b. Find the amount of the drug after 3 hours.
c. When will the amount of drug be 4.959 milligrams? Which method will you use to determine
this?
d. Use the table function to interpret D(2).
e. When will the amount of the drug be 0.09 milligrams, that is almost completely gone from thesystem?
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5. The price of an airline ticket to Paris was $850 in 1998. In 2005 the prices rose to $1900.
a. What is P o if t = 0 represents 1998?
b. Use the 2005 price find the growth factor e k .
c. Find a growth law of the form P(t) = P o ekt for the price of an airline ticket to Paris.
Sec. 5.3 starting on page 446 homework: 1, 3, 5a-d, 7, 9, 11a,b,d, 13b,c, 23-29 odd, 47a-d,48a-c, 49a-c
Chapter 7 Polynomial and Rational Functions
Sec. 7.1 Polynomial Functions
In this section we go over polynomial functions and the basic characteristics such as the degreeand leading coefficient. We will also multiply polynomials and factor the sum or difference of
cubes.
1. Multiply: a. (x 3)(x + 3)(2x 5) b, (4x 2)(3x 2 + 7x 5)
2. Given: 3x 2 + 7x 5 2x 4 Find: The degree and the leading coefficient
3. Without performing multiplication, give the degree of the product.
a. (3x 2 + 7x)( 2x 5) b. (7x 5)(4x + 6)( 3x 2 5)
Factoring the sum or difference of cubes:
x 3 - y 3 = (x y)(x 2 +xy + y 2 ) x 3 + y 3 = (x + y)(x 2 - xy + y 2 )
4. Factor: a. x 3 + 8 b. 27 x 3 - 8y 3
Sec. 7.1 starting on page 572 homework: 1, 3, 17a-b, 35, 37, 41
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Sec. 7.2 Graphing Polynomial Functions
In chapter 1 we worked with linear functions (degree 1) described by y = mx + b. In chapter 2 we
worked with quadratic functions of degree 2 described by y = ax 2 + bx + c. A linear function has
atmost 1 zero or solution, a quadratic function has atmost 2 zeros or solutions.
Therefore a polynomial of degree n will have atmost n solutions/zeros/x-intercepts. A polynomial of degree 3 is called a cubic, and a polynomial of degree 4 is called a quartic.
Recall that a quadratic had a minimum or a maximum. At this minimum point the graph changes
from decreasing to increasing. The graph with a maximum changes from increasing to decreasing.
We called the point a vertex. With higher degree polynomials we call these turning points.
The graph below has 2 turning points. The window is [-5, 5] by [-10,10]
There is a turning point at (-2.5, 10) and another turning point at (2.5, -10)
Characteristics of functions/polynomials
function degree # of turning points end behavior
linear 1 none positive or negative slope
y = 2x 1 y = - 2x -1
quadratic degree 2 1 turning pt.
increases left and right if a >0 decreases left and right if a
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(positive leading coefficient) (negative leading coefficient)
y = 2x 2 + 3x y = - 2x 2 + 3x
cubic degree 3 0 or 2 turning pts.
decreases left, increase right if a >0 increases left, decreases right if a 0 decreases left and right if a
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2. Use your calculator to graph to find the following for y = x 3 + 4x 2 - 11x - 30
a. The x-intercepts. Use xmin = -10 and xmax = 10 and adjust the ymin and ymax to get a
good graph.b. Write the polynomial in factored form.
3. Find the zeros of each by factoring, where needed write answer(s) in exact form
a. y = 4x 2 - 28x + 49 b. P(x) = 4x 4 - 28x 2 + 49
c. f(x) = 8x - 4x 4 d. r(x) = x 3 - 11x 2 e. p(x) = x 3 - 25x
4. Find a possible equation for the polynomial whose graph is shown below. Write answer infactored form.
Sec. 7.2 starting on page 584 homework: 1, 3, 11, 13, 21a-b, 23a-b, 39a, 41a, 43a, 44a, 45a, 47, 51
Sec. 7.4 Graphing Rational Functions
Definition: A rational function is nothing more than a polynomial divided by a polynomial, denoted by
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f(x) =)(
)(
xQ
xPwhere Q(x) # 0.
We learned earlier about vertical asymptotes, that is imaginary vertical lines in which the graph
approaches but does not cross. These vertical asymptotes occur at the x-values that make the
denominator = 0, or that is where the function is undefined.
For example, the graph below shows the equation y =
3
1
x, note that when x = 3 the
denominator = 0.
The graphs below show the function under 3 different window, the standard window and thez-decimal.
The standard window shows the equation of the vertical asymptote whereas the z-decimal
does not.
Notice how the graph using the z-decimal approaches x = 3 but does not cross the graph at
that particular value.
Also note that at x = 3 the function gives you an error using the table, which implies that
when x = 3 the function is undefined.
standard window(asymptote shown at x = 3) z-decimal window f(3) is undefined
To find a vertical asymptote set the denominator = 0 and solve for x.
1.) Find any vertical asymptote(s) for the following, graph to verify your answers.
a. f(x) =
4
2
+xb. y =
32 x
x
c. f(x) =4
12
x
xd. f(x) =
xx
x
3
142
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2.) Does f(x) =4
12 +
x
xhave a vertical asymptote? Why or why not?
Determining horizontal asymptotes
For the rational function f(x) =)(
)(
xQ
xP=
0
1
1
01
...........
............
bxbxb
axaxam
m
n
n
++++++
n is the degree of the numerator and na is the leading coefficient
m is the degree of the denominator and mb is the leading coefficient
3 case scenarios
I. If n < m, theres a horizontal asymptote at y = 0. (the x-axis)
II. If n = m, theres a horizontal asymptote at y =m
n
b
a.
III. If n > m, there is no horizontal asymptote.
3.) Find any horizontal asymptotes (if they exist) for the following.
a. f(x) =4
2
+xb. y =
32 xx
c. f(x) =4
12
x
xd. y =
9
22
3
xx
4.) Find any horizontal or vertical asymp
