PreCalculus Resources eManual

Lesson 1.1.1 Resource Page

76 Pre-Calculus with Trigonometry

Tubular Vision In this experiment, you will measure the diameter of the field of view as a function of your distance from the object (a wall). Materials: A tube, meter stick or a tape measure, and a ruler. Collecting the Data:

1. You will need the following roles for your group:

• A viewer (the one that will look through the tube) • A spotter (the one who will mark and measure on the board or wall the field of view) • A measurer (the one who measures the distance from the wall to the end of the tube) • A recorder

2. Measure a distance away from the wall to the end of the tube (the end away from the viewer eye) and record the distance. Stand facing the wall.

3. 4.

Instruct the spotter to make chalk (or pen marks if you are using paper) marks at the left and right edges of your field of view. Look straight ahead through the tube; do not roll your eyes. Use your peripheral vision to judge. Measure the distance (in cm) between the chalk marks. Then erase them to prepare for the next measurement.

><

Field of view looking through the tube.

5. Repeat steps 2 - 4 for different distances. Be sure to obtain eight data points. Record your results below:

Distance from Wall Field of View

Lesson 1.1.4 Resource Page Page 1 of 2

Chapter 1: Tools for Your Journey 77

Transformations of Graphs ONE STEP TRANSFORMATIONS Given the graph to the left, sketch the transformed graph to the right.

1. )2( !xf f(x)

2. )(2 xg!

g(x)

3.

�

h(x !1) + 3

)(xh



MULTI-STEP TRANSFORMATIONS Given the graph to the left, sketch the transformed graphs to the right. Show each step of the two-step transformation. 4.

�

k(x) = 3 f (x) +1 5.

�

m(x) = !2 f (x !1) 6.

�

p(x) =12f (x) + 3

7.

�

q(x) =12f (x ! 2)

f(x)

f(x)

f(x)

f(x)

Lesson 1.2.2A Resource Page


Quadratic Formula—TI-83/84

STEPS DISPLAY Select , then select NEW EXEC EDIT

Create New

NEW

1: Press Í

Type QUADFORM and press Í

PROGRAM: QUADFORM :

Press then select I/O Select 8 then press Í This command will clear the screen at the start of the program.

PROGRAM: QUADFORM :ClrHome

Input

Select and arrow over to I/O. Select 1: (Input). To get the quote, push ƒ then . Press ƒ will give the A. Press y

[TEST] and choose = . Press ƒ then again to end the quote. Press ¢ (found above ¬). Use ƒ again and select A. Repeat the same steps so B and C can be inputted.

PROGRAM: QUADFORM : ClrHome :Input "A=", A

Process

These variables are obtained by using the ƒ key. We want to rename B2 – 4AC as D. To store this new quantity into memory, we use the ¿ key followed byƒ—[D].

:B2 – 4AC→D

We need to do two more calculations, one for each root. Enter the line (-B+√(D))/(2A)→R into the program. Write another line that will calculate the second root and store the result into S.

Output

The Disp is found as before. Note that the R and S do not have quotes around them.

:Disp R :Disp S

Select yz 5to exit the program.

+

+

Lesson 1.2.2B Resource Page


Sierpinski’s Triangle 1. Choose any point inside the triangle. 2. Roll a die to randomly pick a vertex, A, B, or C. If the result is a 1 or 2, find the

midpoint that goes toward A. If the result is a 3 or 4, find the midpoint that goes toward B. If the result is a 5 or 6, find the midpoint that goes toward C.

3. Plot the midpoint between the point from part 1 and the vertex from part 2. 4. Using the point you created in part 3, repeat steps 2 and 3 at least 6 times.

C

B A



Angles in a Unit Circle

0

Chapter 1 Closure Resource Page: Key Ideas


Function Definition

Domain and Range

Parent Graphs

Transformations: Shifting and Stretching Functions

Working with Exponents (Including Negative and Fractional)

Inverse of a Function

Operations with Functions

Transformations of Non-Parent Functions

Point-Slope Form of a Line

Law of Sines and Law of Cosines

Special Triangles

Radians: Measuring Angles in the Unit Circle

Key Ideas Ideas Chapter 1



!"#

=___________

___________)(xh

B.

D.

SHIFTING GRAPHS

�

k(x) =___________

___________

!

"

#

A.

C.


Chapter 2: Finding the Area Under a Curve 175

Periodic Functions

10 8 6 4 2 -4 -2

(1, 4) (6, 4)

(5, 2)


Sum Formula—TI-83/84

STEPS DISPLAY Select , then select NEW EXEC EDIT

Create NewNEW

1: Press Í

Type SUM and press Í

PROGRAM: SUM :

Press then select I/O Select 8 then press Í This command will clear the screen at the start of the program.

PROGRAM: SUM :ClrHome

Input

Press ¿ ƒ µ Í. Press ¿ „ Í.

PROGRAM: SUM : ClrHome : 0 → S : 1 → X

Process

Select and scroll down to 9: Lbl Press Í and then Â Í

Press ƒ µ To get the Y1, press , arrow over to Y-VARS, select 1:FUNCTION and press Í Press „ ¿ „ Í

Select and then select 1: If (press Í) Press „ y ¸, then type · Í

Select and scroll down to 0: Goto Press Í and then Â Í

: Lbl 3 : S + Y1 → S : X + 1 → X : If X ≤ 5 : Goto 3

Output

Select and arrow over to I/O. Select 3: Disp Press ƒ µ

:Disp S

Select y z 5to exit the program. Before running the program, be sure that you have a function entered in Y1.

¥

¥



Area Under a Curve – Part I 2-64. 2-65.


Chapter 2: Finding the Area Under a Curve 177

Piecewise Functions

Intuitive Notion of Continuity

Horizontal and Vertical Shifts of Piecewise Functions

Periodic Functions

Sigma Notation

Estimating Area Under a Curve with:

– Left-Endpoint Rectangles

– Right-Endpoint Rectangles

– Trapezoids

– Midpoint Rectangles

Shifting Areas

Area as a Function



Chapter 3: Exponentials and Logs 247

3-5. a. 2 f (x) b. f (2x) c. ! f (x) d. f (!x)

f (x)

f (x)

f (x)

f (x)



kf(x) and f(kx) Transformations

Applications of Exponential Functions

Equivalent Transformations

Inverse Functions

– Vertical Line Test

– “Switch and Solve” method

Definition of Logarithm

Log Graphs

Laws of Logarithms

Solving Exponential and Logarithmic Equations

LN vs. LOG




UNIT CIRCLE AND SPECIAL TRIANGLES

!


Chapter 4: Circular Functions 323

�

32

�

2

2

�

!3

�

!4

�

!6

�

!2

�

1

2

�

1

2

�

2

2

�

32

�

!"6

�

!"4

�

!"3

�

!"2

-1

-1

.5

-.5

Lesson 4.1.1C Resource Page


Unit Circle

!6

3

2, 1

2( )π/6

!4

2

2,

2

2( )

!3

1

2,

3

2( )

!2

0,1( ) !

1

2,

3

2( ) 2"3

!2

2,

2

2( ) 3"4

!3

2, 1

2( ) 5"6

!1, 0( ) "

!3

2, ! 1

2( ) 7"6

!2

2, !

2

2( ) 5"4

!1

2, !

3

2( ) 4"3 3!2

0, "1( )

5!3

12

, "3

2( )

7!4

2

2, "

2

2( )

0 1, 0( )

11!6

3

2, " 1

2( )

(+ , +) (+ , –)

(– , –)

(– , +)



Building a Sine Curve

π/6

π/4

π/3 π/2

2π/3

3π/4

5π/6

π

7π/6

5π/4 4π/3

3π/2 5π/3

7π/4

0, 2π

11π/6

Cut

Off



y =

sinθ

y =

cosθ



Trig Graphs



sin x

cos x

y = sin x + cos x



y = x + sin(x)

Chapter 4 Closure Resource Page


GRAPHICAL REPRESENTATION OF TRIG FUNCTIONS

P

A

C

B

θ D

1



Angles and Coordinates in the Unit Circle

Sine and Cosine in the Unit Circle

The Fundamental Pythagorean Identity

Using Right Triangles to Find Trigonometric Ratios

Graphs and Transformations of Sine and Cosine

The Five-Point Method for Graphing Sine and Cosine

The Reciprocal Trig Functions

Other Trigonometric Functions in the Unit Circle

Simplifying Complex Fractions

Angular Frequency and Period

The General Sine Function

The Three Pythagorean Identities

Verifying Identities

Modeling with Periodic Functions

Graphical Addition




View Tube Revisited

Name:_______________

Your group’s distance from the wall:__________

Tube Tube length (cm) View diameter (cm) A B C D E F G H

Sketch your stat plot here: Observations and conclusions:


Chapter 5: Introduction to Limits 407

Graphs of Secant and Cosecant

The graph of f (x) = cos x is shown below. Use this graph to sketch 1

f (x)= sec x .

The graph of f (x) = sin x is shown below. Use this graph to sketch 1

f (x)= csc x .

2! !"

2 !"

!3"

2 !2"

1

2

3

4

–1

–2

–3

–4

!

2 ! 3!

2

y

x

2! !"

2 !"

!3"

2 !2"

1

2

3

4

–1

–2

–3

–4

!

2 ! 3!

2

y

x



Direct and Inverse Variations

Transforming Rational Functions

Simplifying Algebraic Fractions

Graphing

�

1

f (x)

Definition of a Limit

One-Sided Limits

Continuous Function

Piecewise Functions and Limits




Inverse Sine Inverse Cosine

π 2π –π –2π

π

2π

–π

–2π

x

y

π 2π –π –2π

π

2π

–π

–2π

x

y


Chapter 6: Extending Periodic Functions 483

Tangent Graph (cosine provided for reference)

Inverse Tangent



Trig Modeling Resource Page

PROBLEM RELEVANT INFORMATION EQUATION and SKETCH 1 Period

Number of cycles in 2π Amplitude Vertical Shift Horizontal Shift

2 Period Number of cycles in 2π Amplitude Vertical Shift Horizontal Shift










The Spring Problem Materials Each group will need the following: • One Slinky Jr. • Clay (or play dough) to act as a weight • A meter stick or other measuring device ( butcher paper marked every 2 cm) • A stop watch Tasks for the team: Holder: The person who holds the slinky in place. Timer: The person who times the period of an oscillation. Low Spotter: The person who spots the lowest position. High Spotter: The person who spots the highest position. 1. Attach the clay to the end of the slinky. 2. The holder should hold the non-clay end of the slinky

and extend his/her arm out so that it can be held steady for a period of time. Holding the top against a door or window jamb will make this easier. You will need to hold about

�

1

4 of the coils from the top.

Separate this section by using a ruler or other thin flat object, as shown in the picture to the right.

3.

Allow the slinky to hang loose so that you get an idea of the middle position during the oscillations. Adjust the number of coils above the ruler if the spring hangs too low. The upper spotter should raise the clay between 20 and 30 cm above the center position. The measurements will go as follows: DROP LOW POINT HIGHT POINT LOW POINT HIGH POINT -- Mark here LOW POINT -- Mark here HIGH POINT -- Mark again

Practice a couple of test drops to get an idea of the high and low positions. Once you have established your standards you are ready to begin the experiment. 4. Draw a line on the paper or use the floor as a reference point to measure from.



Team Task:

Holder: To keep your arm and hand steady to avoid any secondary motion.

Spotters: The high spotter is in charge of dropping the clay. Make a mark at the starting position so that each trial will start from the same point. Once the motion has started, mark the low and high points of the clay on the paper as specified previously. Use different colors or numbers for each trial.

Timer: Start the stop watch when the weight hits its the high marking point. Stop the watch when it reaches its next high point.

PART 1: Run the experiment three times. Record your data on the chart below. All heights are measured from the reference line or floor.

TRIAL 1 TRIAL 2 TRIAL 3 MEAN

FIRST HIGH POINT

LOW POINT

SECOND HIGH POINT

TIME

PART 2: In this part we wish to concentrate on the period of an oscillation. Measure the time it takes for the spring to make the specified number of oscillations. If you have more than one stop watch you can speed up the process. Choose a dropping height similar to the one you used in Part 1. An approximation is fine here since we are only concerned about the timing. Start timing after one or two oscillations to minimize the effects of secondary motion. To calculate the period, divide the time by the number of oscillations. We will sketch a graph of the motion during our analysis. To help develop the graph, the spotters should pay close attention to the motion of the slinky and record their observations. Complete the chart below:

Oscillations 2 4 6 8 10 20

Time

Period



Solving Trig Equations

Inverse Sine and Inverse Cosine

Ambiguous Case for the Law of Sines

Tangent and Inverse Tangent

Graphing Trig Functions of the Form

�

y = asin(b(x ! h)) + k

Angle Sum and Difference Formulas

Modeling With Trig Functions

Double and Half Angle Formulas

Solving Complex Trig Equations




ODD AND EVEN FUNCTIONS

Fill in the table below for the following power functions. Why are they called power functions? How are they different from exponential functions? In the column labeled “symmetry,” record whether they are symmetric about the x-axis, the y-axis, the origin, or none of these.

Equation Sketch f (!x) f (!x) Simplified Symmetry

�

f (x) = x4

�

f (x) = x2

�

f (x) = x0

�

f (x) = x!2

�

f (x) = x!4

a. Why might these functions be called “even?”

b. Write an identity describing the relationship between

�

f (!x) and

�

f (x) in all cases. f (!x) = ?

c. Describe the symmetry of the graphs of these “even” functions.

A function

�

f (x) is called an EVEN FUNCTION if, for all x,

�

f (!x) =

�

f (x) .

d. In your study teams, come up with at least one other function that is even other than using even integer exponents. Think symmetry!


Chapter 7: Algebra for College Mathematics 555

Fill in the table below for the following power functions.

Equation Sketch f (!x) f (!x) Simplified Symmetry

�

f (x) = x5

�

f (x) = x3

�

f (x) = x1

�

f (x) = x!1

�

f (x) = x!3

a. Why might these functions be called “odd?”

b. Write an identity describing the relationship between

�

f (!x) and

�

f (x) in all cases. f (!x) = ?

c. Describe the symmetry of the graphs of these “odd” functions.

A function

�

f (x) is called an ODD FUNCTION if, for all x,

�

f (!x) = –

�

f (x) .

d. In your group come up with at least one other function that is odd, other than using odd exponents. Think symmetry!



Polynomial Division

Fill in the generic rectangles to complete the multiplication of the terms. State the resulting product. a. (x ! 3)(2x3 + x2 ! 2x +1) b. (2x +1)(x3 + 2x2 ! 3)

�

___ x4_____ x

3______ x

2_____ x ______

�

___ x4_____ x

3______ x

2_____ x ______

Use the generic rectangles to find the following quotients: c. (x4 ! x3 ! 4x2 + 8x + 8) ÷ (x + 2) d. (4x3 + 4x2 ! 7x ! 6) ÷ (2x + 3) x4

! x3

! 4x2

+ 8x + 8 4x3

+ 4x2

! 7x ! 6 What if it does not divide completely? The examples below show two methods for dividing, both have remainders that we write as fractions.

x4! 6x

3+18x !1

x ! 2

Using Long division:

x ! 2 x4! 6x

3+ 0x

2+18x !1

x4! 2x

3

! 4x3+ 0x

2

! 4x3+ 8x

2

! 8x2+18x

! 8x2+16x

2x !1

2x ! 4

3

x3! 4x

2! 8x + 2

Final Answer: x3 ! 4x2 ! 8x + 2 + 3

x!2

Using Generic Rectangles: x4! 6x

3+ 0x

2+18x !1

Final Answer: x3 ! 4x2 ! 8x + 2 + 3

x!2

x3 !4x

2 !8x +2

x x4 !4x

3 !8x2 + 2x 3

–2 !2x3 +8x

2 + 16x –4

Remainder

2x3 +x

2 !2x +1

x

–3

x3 +2x

2 +0x !3

2x

+1

x x4

+2

2x 4x3

+3

Remainder


Chapter 7: Algebra for College Mathematics 557

Pascal’s Triangle Fill in the appropriate values below.

Row 0

Row 1

Row 2

Row 3

Row 4

Row 5

Row 6



Properties of Functions

– Increasing and Decreasing Functions

– Concavity

– Even and Odd Functions

Setting up Word Problems

Simplifying Algebraic Expressions

Using Substitution

Completing the Square

Polynomial Division

Addition of Series

– Arithmetic Series

– Geometric Series

Pascal’s Triangle

– Binomial Expressions

– Binomial Probabilities




RACE TO INFINITY

Contestants:

�

a(x) = 50 x

�

e(x) = 2x!1000

�

b(x) = x10

�

f (x) = x

�

c(x) =100x2

�

g(x) =1.1x

�

d(x) = 800log x

�

h(x) = 20x3

Listed above are several functions which all go to infinity as x gets large. Your team’s task is to determine the order of the finish. You will need to figure out which one moves to infinity the quickest as x gets large. In other words, you will determine which function “dominates” the others. Use the chart below to help determine the order of the functions for various values of x. You will need to use scientific notation for several of the entries.

Function x = 1 x = 10 x = 100 x = ??

�

a(x) = 50 x

�

b(x) = x10

�

c(x) =100x2

�

d(x) = 800log x

�

e(x) = 2x!1000

�

f (x) = x

�

g(x) =1.1x

�

h(x) = 20x3

a. What is the order of the functions when x = 1?

b. What is the order of the functions when x = 10?

c. What is the order of the functions when x = 100?

d. What is the final order of finish?

e. Challenge: At what point (value for x) will the order no longer change? Which functions change position at this point?


Chapter 8: More on Limits 629

Finding the Area of a Circle 8-48. First the area of the inscribed dodecagon.

a. What is the central angle of each of the isosceles triangle shown?

b. Given the radius is one unit, what is the height of one of the isosceles triangles. Do not find the decimal approximation.

c. What is the base of each isosceles triangle? Do not find the decimal approximation.

d. Find the area of the inscribed polygon. 8-49. Now find the area of the circumscribed

dodecagon.

a. What is the base of each isosceles triangle? Do not find the decimal approximation?

b. Given the height of each triangle is one unit. Find the area of the polygon.

Use the diagrams below to assist you with problems 8-50 and 8-52. 8-50. 8-52.

1

1



Dominant Terms

Limits to Infinity

Holes and Asymptotes

Squeeze Method

The Number e

�

ex and

�

ln x

Pert and Applications of e

Infinite Geometric Series

The Harmonic Series

The Fibonacci Series




�

g(x) = 3x +1

14)( 2++!= xxxf


Chapter 9: Rates of Change 709

Resource Page: Scenario 1

1. Walk slowly at a constant speed from “start” for 10 seconds.

2. Stop for 5 seconds.

3. Walk slowly again for 5 seconds.


5. Run towards “finish” for 5 seconds. -------------------------------------------------------------------------------------------------------------------------------------- Lesson 9.3.2 Resource Page

Resource page: Scenario 2

1. Walk slowly from “start” for 10 seconds.


3. Walk slowly back towards “start” for 5 seconds.


5. Run towards “finish” for 5 seconds.



Rates of Change

Slope and Rates of Change

Average Rate of Change: AROC

Instantaneous Rate of Change: IROC

Secant Line vs. Tangent Line

Velocity and Position Graphs

Definition of a Derivative

Slope of a Function and Area Under a Curve



Chapter 10: Vectors and Parametric Equations 767

Vector Line Dance

Step Move Vector Angle and Magnitude

Sketch

1 Right 2 2, 0 0°, 2

2 Up 1 0, 1

3 Back 2 0, ! 2

4 Up 1, Left 1 !1, 1 135°,

�

2

5 Left 2

6 Up 1, Right 1 1, 1

7 Down 1, Right 1

Chapter 10 Closure Resource Page: Merge Problem


Math Magic Land – Spinning Cups One of the most popular rides at Math Magic Land is the Spinning Cups. During the ride, the large outer disk rotates counter-clockwise and makes a complete revolution every 30 seconds. The medium size disk rotate clockwise every 4 seconds. The passenger can rotate the smallest disk (the cup) counter-clockwise as fast as they want. Janelle (labeled as J) loves making her friends dizzy on the ride. When she rides, she spins the cup one full turn every two seconds. The main disc has a radius of 100 feet. The medium disc has a radius of 40 feet and the cups each have a radius of 4 feet.

100 ft 40 ft

4 ft

30 ft

50 ft

J

M


Chapter 10: Vectors and Parametric Equations 769

Vector Addition

Magnitude and Standard Angle of a Vector

Component Form of a Vector

Unit Vectors

Applications of Vectors

Dot Product

Parametric Equations

Vector Equations

Inverses and Parametric Equations

Applications of Parametric Equations


Lesson 11.1.1 Resource Page: Polar Graph Paper

Chapter 11: Polar Coordinates and Complex Numbers 811

Polar Graph Paper

Ch 10 Closure Resource Page: Key Ideas


Polar Coordinates

Conversions Between Polar and Rectangular Forms

Polar Graphs: Common Forms, Rotations

Complex numbers: Simplifying, Graphing

Polar Form of Complex numbers

Multiplying and Dividing Complex Numbers

Powers and Roots of Complex Numbers

DeMoivre’s Theorem



Chapter 12: Linear Transformations 851

Matrices

Matrix Operations (adding, multiplying)

Identity Matrix

Inverse Matrix

Linear Transformation

Rotation Matrix

Vectors and Matrices

Composition of Matrices

Eigenvalues


PreCalculus Resources eManual

Education

Transcript of PreCalculus Resources eManual