Pre Calculus Graphing Trig. Functions Day 1 Draw a graph of the given trigonometric function with the listed amplitude and period. 1. siny x 2. cosy x 3. siny x

Amp: 3 Period: 2π Amp: ½ Period: 4

Amp: 5 Period: 10π

Draw a graph of the given trigonometric function with the listed vertical scale change and period. 4. tany x 5. tany x 6. tany x

v. stretch: 3 period: π v. shrink: 14 period: 6 π v. stretch: –6 period:

5

State the (a) amplitude (if applicable) and (b) period. For tangent, list the vertical stretch or shrink.

7. y = 3 cos x 8. 4

tanx

y

9. y = sin (3x)

10. 4 tan2

xy

11. 5sin 4y x 12. 154 cosy x

13. y = 2 sin x 14. tan 3y x 15. 1 1

sin4 3

y x

16. 1cos 4

5y x 17. 3sin 7y x 18. cosy x

19. Graph at least one period for EACH of the functions in questions 7-18. Attach a separate piece of paper and be sure to label ALL tick marks on your axes. 20. For each tangent function graph in question 19, a) List the equations of two vertical asymptotes of each graph.

b) What relationship do you see between the location of the vertical asymptotes and the period of tangent? Do you think this will always be true?

21. The alternating current I, in amperes, flowing through an ac (alternating current) circuit at time t is

( )120sin 30I tp= , where t > 0.

a) What is the period? b) What is the amplitude? c) Graph this function over two periods. Write two equations of the sine function with each amplitude and period.

22. amplitude = 5, period = 2 23. amplitude = 2

3 , period =

Write two equations of the cosine function with each amplitude and period.

24. amplitude = 1 , period = 2

3

25. amplitude =

1

2, period =

4

In questions 26 –29, list the amplitude (or vertical stretch) and period. Then, write the equation of the given graph. 26. 27. 28. 29. So, what about the reciprocal functions?? The best way to create their graph is based on the original trig. functions. Let’s start with cscy x which is based on _________________________. From last chapter, we know there are certain quadrantal angles which make the reciprocal function

cscy x undefined…Answer the following questions…

When csc x is undefined, sin x ? Where on the unit circle does sin x this value? What angles make us land there on the unit circle? Where are those angles on the sine graph provided? Since cscy x undefined at those x values, what should

we draw on the graph at the x-intercepts of sine? Add them to your graph. Now let’s find some other values…

o What angles make sin x csc x ? o Add these points to your graph.

o Remember that 1

6 2sin

…what is 6

csc

? Add this point to your graph.

o Try some other values…when 1

3sin x , what is csc x ? When

1

4sin x , what is csc x ?

o What pattern do you see with the y-values of csc x when x is between the asymptotes?

We can repeat this process for the other reciprocal functions… Graph secy x below. Graph coty x below.

For 30 – 33, graph the following trigonometric functions: 30. ( )sec 3y x= 31. ( )csc 4y x=

32. ( )2 cot 8y x= 32. ( )3cscy x=

Transformations Review PreCalculus

TRANSFORMATIONS REVIEW Learning Target: Be able to adjust the function when there is a horizontal stretch/shrink AND a left/right movement. A transformation is a stretch/shrink, a reflection, or simple movement of a parent function horizontally or vertically. The general rule … “Inside … x’s … opposite” and “Outside … y’s … same” 1. Suppose you are given the function f (x). If a, b, c, and d are real numbers, our transformed function is

( )( )a f b x c d⋅ + +

Inside Outside

/+ -

( )f x c+ ( )f x c-

( )f x d+ ( )f x d-

/´ ¸

( )f bx ( )f x-

( )a f x⋅ ( )f x-

We are going to focus on the “inside” of the function today because that is where people have the biggest issues.

2. Sketch the function ( ) 2 8f x x= - . The point (0, 0) is the “starting point” of the parent function y x= .

(Use your calculator to verify/help.) Where is this “starting point” after the transformation? Describe the transformation.

x

y

Transformations Review PreCalculus

For each of the following functions, sketch the function and determine where point (0, 0) is after the transformation, then describe the transformation. (Use your calculator to verify/help.)

2. ( ) 12 3g x x= + 3. ( ) 5h x x= -

Using the expression ( )( )a f b x c d⋅ + + from before …

Do you notice that inside the function the b value has been factored out of the expression x + c? 4. Go back to the previous 3 examples and rewrite each of them by factoring out the b value. Compare these “factored forms” to the transformations.

a) ( ) 2 8f x x= -

b) ( ) 12 3g x x= +

c) ( ) 5h x x= -

x

y

x

y

Transformations Review PreCalculus

5. Write the transformed function of y x= after each transformation described below.

a) Horizontal Stretch by 7 and Left 2. b) Horizontal Stretch by 5 and Right 3. c) Horizontal Shrink by 2

3 and Left 4. d) Horizontal Shrink by 38 and Right 9.

The same rules we have been reinforcing using the square root function are used when transforming trig functions. (Horizontal Stretch/Shrinks are simply changes to the period!) 6. Rewrite each inside using factoring, then describe the transformation of each of the functions below.

a) ( ) ( )sin 3f x x p= - b) ( ) ( )2cos 4g x x p= +

c) ( ) ( )13 4sinh x x p= + d) ( )2

3 6tan x p-

7. Write the transformed function of each parent function given below.

a) ( )siny x= with a period of 5p and moved 2p to the right.

b) ( )cosy x= with a period of 23p and moved p to the left.

c) ( )tany x= with a period of 4p and moved right 6

p .

Pre Calculus Graphing Trig. Functions Day 2 State the amplitude (if applicable), period, phase shift and vertical shift of each function. Then, graph at least one period of each function ON GRAPH PAPER…I have plenty!! Be sure to label the scale on your axes.

1. 1sin 2

2y x 2. tan 2 3y x 3. 1

23cosy x

4. tan 24

y x

5. 2cos 1

3

xy

6. 2sin4

y x

One concept that is particularly difficult for students is when the phase shift is not already one of the tick marks on the x axis. This happens when the phase shift is not a factor of the period. We noticed this on question 6, but we simply divided the scale on our x-axis into smaller pieces. It is not always so simple…

7. Given a period of 5 and a phase shift of 3

, create the scale you would use for the x axis. Hint: you may want to

think of a common denominator.

8. Given a period of 2

and a phase shift of

7

, create the scale you would use for the x axis. Hint: you may want to

think of a common denominator. Now you are ready for some more challenging graphing…List the attributes and graph.

9. 1 3

2sin4 2

y x

10. tan 23

y x

11.

1cos

2 3y x

For the next three questions, think about the transformations that we did last class inside the parenthesis!! List the attributes and graph as above.

12. cos 2 33

y x

13. cos 3 5y x 14. 5sin 1

4 8

xy

Factoring out “b” is not easy for some students…Let’s explore another way. Solve each equation for x.

15. 2 03

x

16. 3 0x 17. 04 8

x

18. How do these answers compare to the phase shift on questions 12-14? What’s another way to find “c”?

0

0

Write two equations of the cosine function with:

19. amplitude = 5, period = 4 , phase shift =6

, vertical shift none

20. amplitude = 1, period = 4

, phase shift = , vertical shift = –1

Write two equations of the sine function with:

21. amplitude = 6, period = 2

, phase shift =

8

, vertical shift none.

22. amplitude = 1

2, period = 2 , phase shift = none, vertical shift = 3

In questions 23-25, list the amplitude, period, phase shift, and vertical shift of each graph. Then, write the equation of the graph using the indicated parent function. 23. xy sin 24. xy tan 25. xy cos

p

1

x

y

p

1 x

y

1x

y

p-

For questions 26 and 27, write both a sine AND a cosine function for each function shown. 26. 27. 26. The following data represent the average monthly temperatures for Juneau, Alaska.

Month (x)

1 (Jan)

2 (Feb)

3 (Mar)

4 (Apr)

5 (May)

6 (Jun)

7 (Jul)

8 (Aug)

9 (Sep)

10 (Oct)

11 (Nov)

12 (Dec)

Average Monthly

Temperature (F) 24.2 28.4 32.7 39.7 47.0 53.0 56.0 55.0 49.4 42.2 32.0 27.1

a) Your calculator is capable of finding a Sinusoidal Regression. Find the sine function that fits the data. b) What is the amplitude? c) What is the period? d) What is the phase shift? e) What is the vertical shift?

Pre Calculus Worksheet: Review of Inverses (prerequisites before 4.7) 1. Define Inverse. 2. How do you find the inverse of an equation? The inverse of a graph? 3. Find the inverse of each function. Then, tell the Domain and Range of the Inverse.

a. ( ) 3 xg x

x

+= b. ( ) ( )2 log 3a x x= - -

c. ( ) 4 2 1k x x= + - d. ( ) 27 xm x e +=

4. Determine which, if any, of the inverses in question 3 are NOT functions. Explain why not. 5. Graph the inverse of the given relation. Tell whether each inverse is a function or is not a function. Explain.

a. b. c. 2y x 6. How can you PREDICT whether a function’s inverse will be a function?

7. What did we do first semester to 2y x so that it its inverse was a function? 8. Now, let’s try some inverse trig graphs. Remember to think of the 5 key points from the Parent Functions.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

Think of what we did to 2y x to make the inverse a function first semester... 9. Now highlight a piece of each graph that is a function. The piece you want must include the angles 2(0, ) because

these are the angles in a right triangle. Think about the attributes of the parent function that you should also include. You do NOT have to use the same piece for all three graphs. List an interval for the angles you want to include.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

10. Highlight the part of the unit circle corresponding to the angles you selected in question 9.

a. 1sin ( )y x b. 1cos ( )y x c. 1tan ( )y x

Finally, let’s think of input and outputs…

11. For the original trig. graphs ( sin , cos , tany x y x y x ), x gives an ______________ and y gives a ratio of the

sides of a right triangle. For the inverse trig. graphs ( 1 1 1sin , cos , tany x y x y x ), x gives __________________

and y gives __________________.

1

π

1

π

1

π

Pre Calculus Worksheet 4.7 1. The “input” to a trig function is an _______________, and the “output” is a ______________. 2. Evaluate each of the following trig expressions:

a) 3

cos4

b) 7

sin6

c)

3

5tan

d) sec

2

5

3. The “input” to an inverse trig function is a _____________, and the “output” is an ________________. 4. For each of the following, what quadrant is the answer in? (You don’t have to find the answer.) a) ( )1sin 0.8- b) ( )1cos 0.8- c) ( )1tan 0.8-

5. Based on your answers to question 4, what conclusion can you make when the “input” of an inverse trig function is a positive value? 6. For each of the following, what quadrant is the answer in? (You don’t have to find the answer.) a) ( )1sin 0.8- - b) ( )1cos 0.8- - c) ( )1tan 0.8- -

7. The following represent EVERY inverse sine function that you are expected to know w/o a calculator.

a) 1 1sin

2- æ ö÷ç ÷ç ÷çè ø

b) 1

arcsin2

c) 1 1sin

2

d) 1 1sin

2

e) sin–1 (–1) f) arcsin(1)

g) ( )1sin 0- h) 1 3sin

2-æ ö÷ç ÷ç ÷ç ÷÷çè ø

i) 1 3sin

2-æ ö÷ç ÷ç- ÷ç ÷÷çè ø

8. The following represent EVERY inverse cosine function that you are expected to know w/o a calculator.

a) 1 1cos

2- æ ö÷ç ÷ç ÷çè ø

b) 1

arccos2

c) 1 1cos

2

d) 1 1cos

2

e) cos–1 (–1) f) arccos(1)

g) ( )1cos 0- h) 1 3cos

2-æ ö÷ç ÷ç ÷ç ÷÷çè ø

i) 1 3cos

2-æ ö÷ç ÷ç- ÷ç ÷÷çè ø

9. The following represent EVERY inverse tangent function that you are expected to know w/o a calculator.

a) 1 1tan

3- æ ö÷ç ÷ç ÷÷çè ø

b) 3

arctan3

c) 1tan 3

d) 1tan 3 e) tan–1 (–1) f) arctan(1)

g) ( )1tan 0-

Find the exact value without a calculator.

10. 1sin cos4

11. 1 7cos cos

4

12. 1 1cos 2sin

2

13. 1csc tan 1

Find an algebraic expression equivalent to the given expression.

14. 1sin tan x 15. 1cos tan x

16. tan arcsin x 17. sin arccos 3x

For questions 18 and 19, use a calculator to find the approximate value in degrees.

18. arcsin 0.67 19. 1tan 2.37 Most calculators do not have keys for evaluating the inverse cotangent, cosecant, and secant functions. The easiest way to do this is to first convert the inverse expression into a normal cotangent, cosecant, or secant function, then use the reciprocal properties and their inverses to evaluate.

While ( )( )

1sec

cosx

x= , NOTICE that ( )

( )1

1

1sec

cosx

x-

-¹

For questions 20 – 22, evaluate each using your calculator. Be sure you are in radian mode.

20. ( )1sec 3- 21. 1 1cot

2- æ ö÷ç ÷ç ÷çè ø

22. ( )1csc 4- -

A B

d

14 34

30 km

PreCalculus Worksheet 4.8 1. The length of a shadow of a tree is 130 feet when the angle of elevation to the sun is θ. a) Draw a picture and write the height of the tree as a function of θ.

b) If θ = 25, find the height of the tree.

c) Rewrite your function in part a so that θ is a function of the height of the tree. d) If the height of the tree is 100 feet, what is the angle of elevation? 2. An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to

the ships are 4 and 6.5. How far apart are the ships?

3. Two fire towers are 30 kilometers apart, tower A being due west of tower B. A fire is spotted from the towers, and the

bearings from A and B are E 14 N and W 34 N, respectively.

Find the distance d of the fire from the line segment AB. 4. The hot-air balloon festival held each year in Phoenix, Arizona, is a popular event for photographers. Jo Silver, an award-winning photographer at the event, watches a balloon rising from ground level from a point 500 feet away on level ground. a) Write θ as a function of the height s of the balloon. b) Is the change in θ greater as s changes from 10 ft to 20 ft, or as s changes from 200 ft to 210 ft? Explain.

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9. The Halo: Reach Arena rating system is based on the following formula:

Rating = ( )( )120001000 tanL R Np-+ ⋅ ,

where L = loss penalty, R = raw rating, and N = game normalization. L = 1 if you win, and L = 0.96 if you lose. In other words, losing the game means your rating decreases 4%.

R = kills + assists – deaths

3 – betrayals N =

# of players on each team

score needed to win

a) What is your initial rating? … Let R = 0 & L = 1 b) Your team consisted of 8 players, and you won when you scored 50 points. You also had 15 kills, 4 assists, 12 deaths and 1 betrayal. Find your Rating. c) Losing is simply a rating penalty, so let’s suppose that you win, which means L = 1. Most games have either 4 players on each team with a score of 25 needed to win or 8 players with a score of 50 to win … either way, N = 0.16. Using what you know about the inverse tangent function (try reasoning this out without a calculator if possible), what is the maximum rating possible according to this formula? What about the minimum? Will you ever reach either of these ratings? [Hint … if you are stuck, graph the Rating formula in your calculator and adjust your window … USE x = R] 10. A plane is 160 miles north and 85 miles east of an airport. If the pilot wants to fly directly to the airport, what bearing should be taken? 11. A shoreline runs north-south, and a boat is due east of the shoreline.

The bearings of the boat from two points on the shore are 110 and 100. Assume the points are 550 feet apart. How far is the boat from the shore?

8 feet

25 feet

Simple Harmonic Motion refers to motion that repeats in a regular pattern. The bouncing motion of a mass attached to a spring is a good example of simple harmonic motion. The displacement of the mass as a function of time is a sine or a cosine function. The displacement is the amplitude, and the time it takes to complete one full cycle of motion is the period. 12. The Ferris wheel shown makes one complete turn every 20 seconds. A rider’s height, h, above the ground can be modeled by the equation ( )sinh a t kw= + , where h and k

are given in feet and t is in seconds. a) What is the value of a ? b) What is the value of k ? c) What is the value of w ? d) Using this model, where is the person when they first get on the ride? Is this realistic? e) Write a different equation for h that might be a little more realistic in terms of where a person gets on the ride. 13. A mass on a spring oscillates back and forth and completes one cycle in 0.8 seconds. Its maximum displacement is 4 cm. Write an equation that models this motion. (Assume the mass begins in the middle). Optional … Need More Practice?... try the following from your textbook: Basic Trig & Angles of Elevation & Depression … page 432 - 434 #1 – 13, 19 – 24 Multiple Angles of Elevation & Depression in one problem … p433 – 434 #14 – 16 & p440 #97 Harmonic Motion … p435 #29, 32

0

4 cm

– 4 cm