Lesson 17-2

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/HVVRQ�� Writing Trigonometric Functions

Learning Targets: • Find the amplitude and period of a trigonometric function. • Write a trigonometric function given its graph. • 0RGHO�VLWXDWLRQV�ZLWK�WULJRQRPHWULF�IXQFWLRQV�

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Predict and Confirm, Self Revision/Peer Revision, Quickwrite

1. Sine and cosine functions can be used to model real-world phenomena, including electric currents, radio waves, and tides. Based on your observations from Item 8 in Lesson 17-1, state the period of each function given below, and then sketch the graph of the function over one period. Be sure to label the axes carefully. After you have completed your answers, use a graphing calculator to verify results. a. f(x) = cos (4x)

b. f(x) = sin (!x)

230 SpringBoard® Mathematics Precalculus, Unit 3 • Trigonometric Functions

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Writing Trigonometric Functions

2. Consider a function of the form y = A sin[B(x ! C)] + D or y = A cos[B(x ! C)] + D. a. What is the amplitude?

b. What is the period?

c. Which parameter can cause a vertical stretch or shrink or a reflection over the x-axis?

d. Which parameter causes a horizontal shift?

e. Which parameter causes a vertical shift?

3. State the period and amplitude of each function and describe any horizontal or vertical shifts. Sketch the graph of each function over one period. Carefully label the scale on each axis. a. y = 2 cos x

b. y = sin (3x)

c. y = 3 sin x ! 1

d. y x= ! +( )" # $$

% & ''

cos 2 2 !

e. y x= ( )cos !2

f. y x= !" #$

% &' +3 1

4 2sin ( )!

g. y = 4 cos (3x ! !) + 1

4. Make sense of problems. Write a set of ordered steps that explains how to graph functions of the form y = A sin [B(x ! C)] + D and y = A cos [B(x ! C)] + D.

Activity 17 • Graphs of the Form y = A sin [B(x ! C )] + D 231

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Lesson 17-2

Writing Trigonometric Functions

When modeling real-world situations, it may be necessary to determine the equation of the sine or cosine function from a graph or a set of data.

Example A Write an equation for the graph below in terms of sine.

–1

–2

–3

–4

–5

P 2P 3P 4P 5P 6P 7P 8P 9P 10P 11P

y = A sin B(x ! C) + D

|A| = !maximum minimum 2

|A| = ! !

= 1 5

2 3 ( )

D = +maximum minimum 2

D = + !

= ! 1 5

2 2 ( )

The period may be determined by finding the distance between two consecutive maximum values or two consecutive minimum values.

Period = 11! ! 3! = 8!

B = 2! period

B = =2 8

1 4

! !

A horizontal shift C can be determined for sine by finding the x-coordinate of a point of intersection of the graph and the line y = D.

The graph intersects y = !2 at (!, !2), so a possible value of C is !.

Determine whether A is positive or negative by determining whether or not there is a vertical reflection.

The first extrema for x > ! is a maximum. Therefore, there is no vertical reflection, and so A = 3.

y x= !" #$

% &' !3 14 2sin ( )!

For cosine, a horizontal shift C can be determined by finding the distance a maximum (or minimum) point has been shifted from the y-axis.

MATH TIP

232 SpringBoard® Mathematics Precalculus, Unit 3 t Trigonometric Functions

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. Lesson 17-2

Writing Trigonometric Functions

Try These A a. Write an equation for Example A in terms of cosine.

b. Write a second equation for Example A in terms of sine.

5. Write two equations for the graph shown below, one in terms of sine and the other in terms of cosine.

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–2

P 4

P 2

3P 4

P 2P 7P 4

5P 4

3P 2

–

6. Write two equations for the graph shown below, one in terms of sine and the other in terms of cosine.

–1

–2

–3

1 2 3 4 5–1

7. Suppose that the depth of the water at a popular surfing spot varies from 3 ft to 11 ft, depending on the time. Suppose that on Monday, high tide occurred at 6:00 a.m. and the next high tide occurred at 7:00 p.m. a. Draw a graph to model the depth of water as a function of time t in

hours since midnight on Monday morning. b. Write an equation for the graph. c. Use the equation in Item 6(b) to predict the depth of the water at

2 p.m., correct to three decimal places. d. Reason quantitatively. Describe how the parameters of the

equation that models this situation change if the depth of the water is measured in meters instead of feet.

Check Your Understanding

The depth of water at high and low tides follows a periodic pattern that can be modeled with a sine or cosine function.

OCEANOGRAPHYCONNECT TO

Activity 17 t�Graphs of the Form y = A sin [B(x ! C )] + D 233

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Writing Trigonometric Functions

8. Consider the graph of f(x).

42–2–4

–6

a. Write an equation for f(x) in terms of sine. State the period and amplitude of f(x), and describe any horizontal or vertical shifts relative to the parent graph.

b. Write an equation for f(x) in terms of cosine. State the period and amplitude of f(x), and describe any horizontal or vertical shifts relative to the parent graph.

9. Consider the graph of f(x).

4 62–2–4–6 –2

–4

Write an equation for f(x) in terms of sine. State the period and amplitude of f(x), and describe any horizontal or vertical shifts relative to the parent graph.

A graphing calculator can be used to check your answers. Use the TRACE or TABLE feature to see ordered pairs which satisfy the equation.

TECHNOLOGYCONNECT TO

234 SpringBoard® Mathematics Precalculus, Unit 3 • Trigonometric Functions

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ACTIVITY 17 PRACTICE

Lesson 17-1 Write your answers on notebook paper. Show your work. 1. Identify the interval(s) on [0, 2!] for which the

sine function is increasing. 2. Identify the interval(s) on [0, 2!] for which the

cosine function is increasing. 3. a. Sketch the graphs of f(x) = sin x and

g(x) = sin x ! 1 over at least one period, labeling each axis.

b. State the period and amplitude of g(x) = sin x ! 1, and describe any horizontal or vertical shifts.

4. a. Sketch the graphs of f(x) = cos x and g(x) = 4 cos x over at least one period, labeling each axis.

b. State the period and amplitude of g(x) = 4 cos x, and describe any horizontal or vertical shifts.

5. a. Sketch the graphs of f(x) = sin x and g(x) = sin (x ! !) over at least one period, labeling each axis.

b. State the period and amplitude of g(x) = sin (x ! !), and describe any horizontal or vertical shifts.

6. a. Sketch the graphs of f(x) = cos x and g(x) = 2 cos x + 1 over at least one period, labeling each axis.

b. State the period and amplitude of g(x) = 2 cos x + 1, and describe any horizontal or vertical shifts.

7. Which pair of equations generates the same graph? A. f(x) = sin x ! 3 and g(x) = sin (x ! 3) B. f(x) = sin !( )!x 2 and g(x) = sin x +( )!2 C. f(x) = !cos x and g(x) = cos (x + !) D. f(x) = 2 cos x and g x x( ) cos= ( )12

8. Find a cosine equation which will generate the same graph as f(x) = 2 sin x ! 2.

Lesson 17-2 9. Write a sine equation that models the graph.

4 62–2–4–6 –1

10. Write a cosine equation that models the graph.

42–2–4 –2

11. Write two equations, one in terms of sine and the other in terms of cosine, that model the graph.

42–2–4 –2

Graphs of the form y = A sin[B(x ! C )] + D Trigonometric Graphs

Activity 17 t�Graphs of the Form y = A sin [B(x ! C )] + D 235

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12. Write two equations, one in terms of sine and the other in terms of cosine, that model the graph.

42–2–4 –2

13. Write two equations, one in terms of sine and the other in terms of cosine, that model the graph.

42–2–4 –2

14. Which of the following functions has a period greater than the period of f(x) = sin x? A. f x x( ) sin ( )= 1

5 B. f x x( ) sin= 1

5 5

C. f(x) = sin 5x D. f(x) = 5 sin x

15. The graph of a sine function shows the function has maximum values at x = !3, x = 1, x = 5, and x = 9. Explain what this tells you about the equation of the function.

16. The graph of a cosine function shows the function has maximum values at x = ! 9

5 !,

x = ! 4 5 !, x = !

5 , and x = 6

5 !. Explain what this

tells you about the equation of the function. 17. The graph of a sine function has a maximum

value of y = 4 and a minimum value of y = 2. Explain what this tells you about the equation of the function.

MATHEMATICAL PRACTICES Make Sense of Problems and Persevere in Solving Them

18. A weight suspended from a spring vibrates vertically in a periodic pattern. The height of the weight relative to its rest position is f(t) centimeters t seconds after the weight is at its lowest point.

Spring

Rest Position

Time

Periodic Motion–Weight on a Spring

Explain why f(t) could be a sine or cosine function, and identify any additional information needed to determine each of the parameters, A, B, C, and D.

Graphs of the form y = A sin[B(x ! C )] + D Trigonometric Graphs

236 SpringBoard® Mathematics Precalculus, Unit 3 t Trigonometric Functions

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