# 6.1E: Sinusoidal Graphs (Exercises)

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

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

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

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

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

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