6.1E: Sinusoidal Graphs (Exercises)
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Section 6.1 Exercises
1. Sketch a graph of f(x)=−3sin(x).
2. Sketch a graph of f(x)=4sin(x).
3. Sketch a graph of f(x)=2cos(x).
4. Sketch a graph of f(x)=−4cos(x).
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=3sin(8(x+4))+5
12. y=4sin(π2(x−3))+7
13. y=2sin(3x−21)+4
14. y=5sin(5x+20)−2
15. y=sin(π6x+π)−3
16. y=8sin(7π6x+7π2)+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(t).
b. Find a formula for the height function h(t).
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(t).
b. Find a formula for the height function h(t).
c. How high are you off the ground after 4 minutes?
- Answer
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1.
3.
5. Amp: 3. Period = 2. Midline: y=−4. f(t)=3sin(πt)−4
7. Amp: 2. Period = 4π. Midline: y=1. f(t)=2cos(12t)+1
9. Amp: 2. Period = 5. Midline: y=3. f(t)=−2cos(2π5t)+3
11. Amp: 3, Period = π4, Shift: 4 left, Midline: y=5
13. Amp: 2, Period = 2π3, Shift: 7 left, Midline: y=4
15. Amp: 1, Period = 12, Shift: 6 left, Midline: y=−3
17. f(x)=4sin(π5(x+1))
19. f(x)=cos(π5(x+2))
21. D(t)=50−7sin(π12t)
23. a. Amp: 12.5. Midline: y=13.5. Period: 10
b. h(t)=−12.5cos(π5t)+13.5
c. h(t)=26 meters