Phase Shift

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
Answer: $\frac{\pi}{6}$ to the right
$y = \sin(x - C)$ with $C = \frac{\pi}{6} > 0$ moves the graph $\frac{\pi}{6}$ to the right. The number $6$ is not the shift, and a shift up would be a number added outside the sine.
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) $y = \cos\left(x + \frac{\pi}{4}\right)$
Answer: $y = \cos\left(x + \frac{\pi}{4}\right)$
Left means $C = -\frac{\pi}{4}$ , so $\cos\left(x - \left(-\frac{\pi}{4}\right)\right) = \cos\left(x + \frac{\pi}{4}\right)$ . Adding or subtracting $\frac{\pi}{4}$ outside the cosine moves the graph up or down instead.
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
Answer: $\frac{\pi}{3}$ to the right
$3x - \pi = 3\left(x - \frac{\pi}{3}\right)$ , so $C = \frac{\pi}{3}$ : the graph moves $\frac{\pi}{3}$ to the right. The shift is $\frac{c}{B} = \frac{\pi}{3}$ , not $\pi$ .
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) $y = \sin(x - \pi)$
Answer: $y = \sin\left(x + \frac{\pi}{2}\right)$
The cosine is the sine moved $\frac{\pi}{2}$ to the left: $\sin\left(x + \frac{\pi}{2}\right) = \cos(x)$ . At $x = 0$ the other choices give $-1$ , $0$ and $0$ .
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) $y = \sin\left(2x - \frac{\pi}{3}\right)$
Answer: $y = \sin\left(2x - \frac{2\pi}{3}\right)$
Period $\pi$ gives $B = \frac{2\pi}{\pi} = 2$ , and the start gives $C = \frac{\pi}{3}$ . So $y = \sin\left(2\left(x - \frac{\pi}{3}\right)\right) = \sin\left(2x - \frac{2\pi}{3}\right)$ . The choice $\sin\left(2x - \frac{\pi}{3}\right)$ only starts at $\frac{\pi}{6}$ .
6.
In $y = \sin\left(x - \frac{\pi}{5}\right)$ , what is the phase shift $C$ ? (Type pi for $\pi$ , like 3pi/4.)
Answer: pi/5
$x - \frac{\pi}{5}$ is $x - C$ with $C = \frac{\pi}{5}$ : a shift of $\frac{\pi}{5}$ to the right.
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.)
Answer: -pi/2
$x + \frac{\pi}{2} = x - \left(-\frac{\pi}{2}\right)$ , so $C = -\frac{\pi}{2}$ : the cosine moves $\frac{\pi}{2}$ to the left.
8.
What is the phase shift $C$ of $y = \sin\left(2x - \frac{\pi}{3}\right)$ ? (Type pi for $\pi$ , like 3pi/4.)
Answer: pi/6
$2x - \frac{\pi}{3} = 2\left(x - \frac{\pi}{6}\right)$ , so $C = \frac{\pi}{6}$ to the right, not $\frac{\pi}{3}$ .
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.)
Answer: -pi/4
$4x + \pi = 4\left(x + \frac{\pi}{4}\right) = 4\left(x - \left(-\frac{\pi}{4}\right)\right)$ , so $C = -\frac{\pi}{4}$ : $\frac{\pi}{4}$ to the left.
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.)
Answer: pi/2
$\frac{x}{2} - \frac{\pi}{4} = \frac{1}{2}\left(x - \frac{\pi}{2}\right)$ , so $C = \frac{\pi/4}{1/2} = \frac{\pi}{2}$ to the right. Here the shift is larger than $c$ because $B < 1$ .
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.)
Answer: 5pi/6
1. What is the phase shift $C$ ? pi/3
2. At which $x$ does the first cycle start? pi/3
3. $\sin(x)$ has its maximum at $\frac{\pi}{2}$ . At which $x$ is the new maximum? 5pi/6
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.)
Answer: -pi/6
1. Write $x + \frac{\pi}{6}$ as $x - C$ . What is $C$ ? -pi/6
2. At which $x$ is the new maximum? -pi/6
3. What is the maximum value $y$ ? 1
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.)
Answer: 5pi/4
1. What is $B$ ? 2
2. Factor: $2x - \frac{\pi}{2} = 2(x - C)$ . What is $C$ ? pi/4
3. What is the period? pi
4. The cycle starts at $x = C$ . At which $x$ does it end? 5pi/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.)
Answer: 2pi/3
1. What is $B$ ? 3
2. Divide: what is $\frac{\pi}{2} \div 3$ ? pi/6
3. What is the phase shift $C$ ? (Negative for left.) -pi/6
4. What is the period? 2pi/3
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.)
Answer: pi/3
1. What is the phase shift $C$ ? pi/6
2. What is $B$ ? 2
3. $\sin(2(x - C)) = \sin(2x - 2C)$ . What is $c = 2C$ ? pi/3
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.)
Answer: -pi/2
1. At which $x$ does $\cos(x)$ have a maximum (the one closest to $0$ )? 0
2. At which $x$ does $\sin(x)$ have its first positive maximum? pi/2
3. The sine peak must move from $\frac{\pi}{2}$ to $0$ . What is $C$ ? (Negative for left.) -pi/2
4. Check: what is $\sin\left(0 - \left(-\frac{\pi}{2}\right)\right)$ ? 1
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 $t = 0$ ?
Answer: 12.5
1. What is the phase shift $C$ in seconds? 5
2. What is the period in seconds? 30
3. What is a quarter of the period? 7.5
4. At what time $t$ is the seat at the top? 12.5
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.)
Answer: 1
1. What is the shift $C_1$ of $\sin\left(x - \frac{\pi}{4}\right)$ ? pi/4
2. What is the shift $C_2$ of $\sin\left(x + \frac{7\pi}{4}\right)$ ? -7pi/4
3. What is $C_1 - C_2$ ? 2pi
4. The period of $\sin(x)$ is $2\pi$ . How many whole periods is that? 1