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# 6.1.1E: Sinusoidal Graphs (Exercises)


Section 6.1 Exercises

1. Sketch a graph of $$f\left(x\right)=-3\sin \left(x\right)$$.

2. Sketch a graph of $$f\left(x\right)=4\sin \left(x\right)$$.

3. Sketch a graph of $$f\left(x\right)=2\cos \left(x\right)$$.

4. Sketch a graph of $$f\left(x\right)=-4\cos \left(x\right)$$.

For the graphs below, determine the amplitude, midline, and period, then find a formula for the function.

5. 6.

7. 8.

9. 10.

For each of the following equations, find the amplitude, period, horizontal shift, and midline.

11. $$y=3\sin (8(x+4))+5$$

12. $$y=4\sin \left(\dfrac{\pi }{2} (x-3)\right)+7$$

13. $$y=2\sin (3x-21)+4$$

14. $$y=5\sin (5x+20)-2$$

15. $$y=\sin \left(\dfrac{\pi }{6} x+\pi \right)-3$$

16. $$y=8\sin \left(\dfrac{7\pi }{6} x+\dfrac{7\pi }{2} \right)+6$$

Find a formula for each of the functions graphed below.

17.

18.

19.

20.

21. Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature during the day are 57 and 43 degrees, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t.

22. Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68 degrees at midnight and the high and low temperature during the day are 80 and 56 degrees, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t.

23. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function $$h(t)$$ gives your height in meters above the ground t minutes after the wheel begins to turn.

a. Find the amplitude, midline, and period of $$h\left(t\right)$$.
b. Find a formula for the height function $$h\left(t\right)$$.
c. How high are you off the ground after 5 minutes?

24. A Ferris wheel is 35 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function $$h(t)$$ gives your height in meters above the ground t minutes after the wheel begins to turn.

a. Find the amplitude, midline, and period of $$h\left(t\right)$$.
b. Find a formula for the height function $$h\left(t\right)$$.
c. How high are you off the ground after 4 minutes?

Answer

1.

3.

5. Amp: 3. Period = 2. Midline: $$y = -4$$. $$f(t) = 3\sin(\pi t) - 4$$

7. Amp: 2. Period = $$4\pi$$. Midline: $$y = 1$$. $$f(t) = 2\cos(\dfrac{1}{2} t) + 1$$

9. Amp: 2. Period = 5. Midline: $$y = 3$$. $$f(t) = -2\cos(\dfrac{2\pi}{5} t) + 3$$

11. Amp: 3, Period = $$\dfrac{\pi}{4}$$, Shift: 4 left, Midline: $$y = 5$$

13. Amp: 2, Period = $$\dfrac{2\pi}{3}$$, Shift: 7 left, Midline: $$y = 4$$

15. Amp: 1, Period = 12, Shift: 6 left, Midline: $$y = -3$$

17. $$f(x) = 4\sin(\dfrac{\pi}{5} (x + 1))$$

19. $$f(x) = \cos(\dfrac{\pi}{5} (x + 2))$$

21. $$D(t) = 50 - 7 \sin(\dfrac{\pi}{12}t)$$

23. a. Amp: 12.5. Midline: $$y = 13.5$$. Period: 10
b. $$h(t) = -12.5 \cos(\dfrac{\pi}{5}t) + 13.5$$
c. $$h(t) = 26$$ meters