Phase Shift

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Show your work for each problem.

1.
How is the graph of $y = \sin\left(x - \frac{\pi}{6}\right)$ shifted compared with $y = \sin(x)$ ?
1. a) $\frac{\pi}{6}$ up
2. b) $\frac{\pi}{6}$ to the right
3. c) $\frac{\pi}{6}$ to the left
4. d) $6$ to the right
2.
Which equation is $y = \cos(x)$ shifted $\frac{\pi}{4}$ to the left?
1. a) $y = \cos(x) - \frac{\pi}{4}$
2. b) $y = \cos(x) + \frac{\pi}{4}$
3. c) $y = \cos\left(x - \frac{\pi}{4}\right)$
4. d) $y = \cos\left(x + \frac{\pi}{4}\right)$
3.
What is the phase shift of $y = \sin(3x - \pi)$ ?
1. a) $\frac{\pi}{3}$ to the left
2. b) $\pi$ to the right
3. c) $3\pi$ to the right
4. d) $\frac{\pi}{3}$ to the right
4.
Which equation has exactly the same graph as $y = \cos(x)$ ?
1. a) $y = \sin(x + \pi)$
2. b) $y = \sin\left(x + \frac{\pi}{2}\right)$
3. c) $y = \sin\left(x - \frac{\pi}{2}\right)$
4. d) $y = \sin(x - \pi)$
5.
A sine curve has period $\pi$ and starts a cycle (rising through its midline) at $x = \frac{\pi}{3}$ . Which equation fits?
1. a) $y = \sin\left(x - \frac{\pi}{3}\right)$
2. b) $y = \sin\left(2x - \frac{2\pi}{3}\right)$
3. c) $y = \sin\left(2x + \frac{2\pi}{3}\right)$
4. d) $y = \sin\left(2x - \frac{\pi}{3}\right)$
6.
In $y = \sin\left(x - \frac{\pi}{5}\right)$ , what is the phase shift $C$ ? (Type pi for $\pi$ , like 3pi/4.)
7.
In $y = \cos\left(x + \frac{\pi}{2}\right)$ , what is the phase shift $C$ ? Give a negative number for a shift to the left. (Type pi for $\pi$ , like -3pi/4.)
8.
What is the phase shift $C$ of $y = \sin\left(2x - \frac{\pi}{3}\right)$ ? (Type pi for $\pi$ , like 3pi/4.)
9.
What is the phase shift $C$ of $y = \cos(4x + \pi)$ ? Give a negative number for a shift to the left. (Type pi for $\pi$ , like -3pi/4.)
10.
What is the phase shift $C$ of $y = \sin\left(\frac{x}{2} - \frac{\pi}{4}\right)$ ? (Type pi for $\pi$ , like 3pi/4.)
11.
For $y = \sin\left(x - \frac{\pi}{3}\right)$ , find where the first cycle starts and where its maximum is. (Type pi for $\pi$ , like 3pi/4.)
12.
For $y = \cos\left(x + \frac{\pi}{6}\right)$ , find the phase shift and the new position of the maximum that $\cos(x)$ has at $x = 0$ . (Type pi for $\pi$ , like -3pi/4.)
13.
For $y = \sin\left(2x - \frac{\pi}{2}\right)$ , find the phase shift, the period and where the first cycle ends. (Type pi for $\pi$ , like 3pi/4.)
14.
For $y = \cos\left(3x + \frac{\pi}{2}\right)$ , find the phase shift and the period. (Type pi for $\pi$ , like -3pi/4.)
15.
Find $c$ so that $y = \sin(2x - c)$ is the graph of $y = \sin(2x)$ shifted $\frac{\pi}{6}$ to the right. (Type pi for $\pi$ , like 3pi/4.)
16.
Find $C$ with $-\pi \le C \le \pi$ so that $\sin(x - C) = \cos(x)$ for every $x$ . (Type pi for $\pi$ , like -3pi/4.)
17.
The height of a Ferris wheel seat above the hub is $h(t) = 10\sin\left(\frac{\pi}{15}(t - 5)\right)$ meters, with $t$ in seconds. At what time $t$ does the seat first reach its highest point after $t = 0$ ?
18.
Show that $y = \sin\left(x - \frac{\pi}{4}\right)$ and $y = \sin\left(x + \frac{7\pi}{4}\right)$ have the same graph: how many whole periods apart are their shifts? (Type pi for $\pi$ , like -3pi/4.)