Exercises 7.1 Transformations of Graphs

1.

$y=-3+2\sin x$

2.

$y=4-3\cos x$

3.

$y=-\cos 4x$

4.

$y=-\sin 3x$

5.

$y=-5\sin {\displaystyle \frac{x}{3}}$

6.

$y=6\cos {\displaystyle \frac{x}{2}}$

7.

$y=1-\cos \pi x$

8.

$y=2+\sin 2\pi x$

9.

1. ${\displaystyle y=\sin x}$
2. ${\displaystyle y=2\sin x}$
3. ${\displaystyle y=-3+2\sin x}$

10.

1. ${\displaystyle y=\cos x}$
2. ${\displaystyle y=-3\cos x}$
3. ${\displaystyle y=4-3\cos x}$

11.

1. ${\displaystyle y=\cos x}$
2. ${\displaystyle y=\cos 4x}$
3. ${\displaystyle y=-\cos x}$

12.

1. ${\displaystyle y=\sin x}$
2. ${\displaystyle y=\sin 3x}$
3. ${\displaystyle y=-\sin 3x}$

13.

1. ${\displaystyle y=\sin x}$
2. ${\displaystyle y=\sin {\displaystyle \frac{x}{3}}}$
3. ${\displaystyle y=-5\sin {\displaystyle \frac{x}{3}}}$

14.

1. ${\displaystyle y=\cos x}$
2. ${\displaystyle y=\cos {\displaystyle \frac{x}{2}}}$
3. ${\displaystyle y=6\cos {\displaystyle \frac{x}{2}}}$

15.

1. ${\displaystyle y=\cos x}$
2. ${\displaystyle y=\cos \pi x}$
3. ${\displaystyle y=1-\cos \pi x}$

16.

1. ${\displaystyle y=\sin x}$
2. ${\displaystyle y=\sin 2\pi x}$
3. ${\displaystyle y=2+\sin 2\pi x}$

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

$y=2-5\cos 2t$

32.

$y=-2+4\sin 3t$

33.

$y=1+3\cos {\displaystyle \frac{t}{2}}$

34.

$y=-2-3\sin {\displaystyle \frac{t}{4}}$

35.

$y=-3+2\sin {\displaystyle \frac{t}{3}}$

36.

$y=-1+4\cos {\displaystyle \frac{t}{6}}$

37.

$y=3-4\sin 2x$

38.

$y=2\cos 5x+2$

39.

$y={\displaystyle \frac{1}{2}}\sin 3x+{\displaystyle \frac{3}{2}}$

40.

$y={\displaystyle \frac{2}{5}}\cos 6x+{\displaystyle \frac{4}{5}}$

41.

$50-30\sin {\displaystyle \frac{x}{4}}$

42.

$25\cos {\displaystyle \frac{x}{3}}+15$

43.

$y=4\sin \pi x-3$

44.

$y={\displaystyle \frac{1}{2}}\cos {\displaystyle \frac{\pi x}{2}}+2$

45.

The height of the tide in Cabot Cove can be approximated by a sinusoidal function. At 5 am on July 23, the water level reached its high mark at the 20-foot line on the pier, and at 11 am, the water level was at its lowest at the 4-foot line.

1. Sketch a graph of $W(t),$ the water level as a function of time, from 5 am on July 23 to 5 am on July 24.
2. Write an equation for the function.

46.

The population of mosquitoes at Marsh Lake is a sinusoidal function of time. The population peaks around June 1, at about 6000 mosquitoes per square kilometer, and is smallest on December 1, at 1000 mosquitoes per square kilometer.

1. Sketch a graph of $M(t),$ the number of mosquitoes as a function of the month, where $t=0$ on June 1.
2. Write an equation for the function.

47.

The paddlewheel on the Delta Queen steamboat is 28 feet in diameter and is rotating once every ten seconds. The bottom of the paddlewheel is 4 feet below the surface of the water.

1. The ship’s logo is painted on one of the paddlewheel blades. At $t=0,$ the blade with the logo is at the top of the wheel. Sketch a graph of the logo’s height above the water as a function of $t.$
2. Write an equation for the function.

48.

Delbert’s bicycle wheel is 24 inches in diameter, and he has a light attached to the spokes 10 inches from the center of the wheel. It is dark, and he is cycling home slowly from work. The bicycle wheel makes one revolution every second.

1. At $t=0,$ the light is at its highest point the bicycle wheel. Sketch a graph of the light’s height as a function of $t.$
2. Write an equation for the function.

49.

The number of hours of daylight in Salt Lake City varies from a minimum of 9.6 hours on the winter solstice to a maximum of 14.4 hours on the summer solstice. Time is measured in months, starting at the winter solstice.

50.

A weight is 6.5 feet above the floor, suspended from the ceiling by a spring. The weight is pulled down to 5 feet above the floor and released, rising past 6.5 feet in 0.5 seconds before attaining its maximum height of feet. The weight oscillates between its minimum and maximum height.

51.

The voltage used in U.S. electrical current changes from 155V to 155V and back 60 times each second.

52.

Although the moon is spherical, what we see from earth looks like a disk, sometimes only partly visible. The percentage of the moon’s disk that is visible varies between 0 (at new moon) and 100 (at full moon) over a 28-day cycle.

53.

$y=\tan 2x$

54.

$y=\tan 4x$

55.

$y=4+2\tan 3x$

56.

$y=3+{\displaystyle \frac{1}{2}}\tan 2x$

57.

$y=3-\tan {\displaystyle \frac{x}{4}}$

58.

$y=1-2\tan {\displaystyle \frac{x}{3}}$

59.

$3\cos 4x=1.5$

60.

$2\sin 3x=-\sqrt{2}$

61.

$2+3\sin 2x=0.5$

62.

$2+4\cos 2x=4$

63.

$-3+\tan 3x=-2$

64.

$2+\tan 4x=3$

65.

1. ${\displaystyle f(x)=3\sin 2x}$
2. ${\displaystyle 3\sin 2x=-1.5}$

66.

1. ${\displaystyle g(x)=-2\cos 3x}$
2. ${\displaystyle -2\cos 3x=1}$

67.

1. ${\displaystyle h(x)=2-4\cos {\displaystyle \frac{x}{4}}}$
2. ${\displaystyle 2-4\cos {\displaystyle \frac{x}{4}}=0}$

68.

1. ${\displaystyle H(x)=3+2\sin {\displaystyle \frac{x}{2}}}$
2. ${\displaystyle 3+2\sin {\displaystyle \frac{x}{2}}=5}$

69.

1. ${\displaystyle G(x)=-1+3\cos 3x}$
2. ${\displaystyle -1+3\cos 3x=1}$

70.

1. ${\displaystyle F(x)=4-3\sin 2x}$
2. ${\displaystyle 4-3\sin 2x=2.5}$