Transformations of Trig Graphs

Show your work for each problem.

1.
What is the amplitude of $y = -4\sin(2x) + 1$ ?
1. a) $5$
2. b) $-4$
3. c) $2$
4. d) $4$
Answer: $4$
$A = -4$ , so the amplitude is $|-4| = 4$ . The minus sign only reflects the wave across its midline.
2.
What is the period of $y = 3\cos(4x) - 2$ ? (Type pi for $\pi$ , like 3pi/4.)
Answer: pi/2
$B = 4$ , so the period is $\frac{2\pi}{4} = \frac{\pi}{2}$ . $A = 3$ and $D = -2$ do not change the period.
3.
How is the graph of $y = \cos(x + \frac{\pi}{3})$ shifted compared with $y = \cos(x)$ ?
1. a) $\frac{\pi}{3}$ down
2. b) $\frac{\pi}{3}$ to the right
3. c) $\frac{\pi}{3}$ to the left
4. d) $\frac{\pi}{3}$ up
Answer: $\frac{\pi}{3}$ to the left
$x + \frac{\pi}{3} = x - (-\frac{\pi}{3})$ , so $C = -\frac{\pi}{3}$ : the graph moves $\frac{\pi}{3}$ to the left. A shift inside the bracket is never vertical.
4.
The midline of $y = 2\sin(x - \pi) + 5$ is the line $y = D$ . What is $D$ ?
Answer: 5
The vertical shift is $D = 5$ , so the midline is $y = 5$ and the wave runs between $5 - 2 = 3$ and $5 + 2 = 7$ .
5.
Find the maximum and the minimum of $y = 3\sin(x) + 2$ .
Answer: 5, -1
1. What is the amplitude $|A|$ ? 3
2. What is the midline value $D$ ? 2
3. What is the maximum $D + |A|$ ? 5
4. What is the minimum $D - |A|$ ? -1
6.
For $y = \sin(4x)$ , find the period, the key point step and the x-value of the first maximum after $x = 0$ . (Type pi for $\pi$ , like 3pi/4.)
Answer: pi/8
1. What is the period $\frac{2\pi}{B}$ ? pi/2
2. What is the step $\frac{\text{period}}{4}$ ? pi/8
3. At which x is the first maximum? pi/8
7.
What is the phase shift of $y = \sin(2x - \frac{\pi}{2})$ ?
1. a) $\frac{\pi}{4}$ to the left
2. b) $\frac{\pi}{4}$ to the right
3. c) $\frac{\pi}{2}$ to the right
4. d) $\pi$ to the right
Answer: $\frac{\pi}{4}$ to the right
$2x - \frac{\pi}{2} = 2(x - \frac{\pi}{4})$ , so $C = \frac{\pi}{4}$ : a shift of $\frac{\pi}{4}$ to the right, not $\frac{\pi}{2}$ .
8.
For $y = -2\cos(x) + 3$ , find the amplitude, the maximum, the minimum and the y-value at $x = 0$ .
Answer: 1
1. What is the amplitude? 2
2. What is the maximum $D + |A|$ ? 5
3. What is the minimum $D - |A|$ ? 1
4. What is $y$ at $x = 0$ ? 1
9.
The key points of $y = \sin(2(x - \frac{\pi}{6}))$ start at $x = \frac{\pi}{6}$ . What is the x-value of the maximum (the second key point)? (Type pi for $\pi$ , like 3pi/4.)
Answer: 5pi/12
Period $\frac{2\pi}{2} = \pi$ , step $\frac{\pi}{4}$ . Maximum at $\frac{\pi}{6} + \frac{\pi}{4} = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12}$ .
10.
Use the key points of $y = 3\sin(x - \frac{\pi}{2}) - 1$ to find its first maximum and its minimum value. (Type pi for $\pi$ , like 3pi/4.)
Answer: -4
1. Where does the cycle start ( $x = C$ )? pi/2
2. What is the step (period divided by 4)? pi/2
3. At which x is the first maximum? pi
4. What is the maximum value? 2
5. What is the minimum value? -4
11.
Rewrite $y = \cos(3x + \pi)$ in the form $y = \cos(B(x - C))$ and find the phase shift and the period. (Type pi for $\pi$ , like -3pi/4.)
Answer: 2pi/3
1. What is $B$ ? 3
2. What is $C$ ? (a negative value means a shift to the left) -pi/3
3. What is the period? 2pi/3
12.
Which equation has amplitude $2$ , period $\pi$ and midline $y = -1$ ?
1. a) $y = 2\sin(\frac{1}{2}x) - 1$
2. b) $y = 2\sin(\pi x) - 1$
3. c) $y = 2\sin(2x) - 1$
4. d) $y = -\sin(2x) + 2$
Answer: $y = 2\sin(2x) - 1$
$\frac{2\pi}{B} = \pi$ gives $B = 2$ , with $A = 2$ and $D = -1$ : $y = 2\sin(2x) - 1$ . $\sin(\pi x)$ has period $2$ and $\sin(\frac{1}{2}x)$ has period $4\pi$ .
13.
Find the key point y-values of $y = -2\sin(x) + 1$ at $x = 0$ , $\frac{\pi}{2}$ , $\pi$ and $\frac{3\pi}{2}$ .
Answer: 3
1. What is $y$ at $x = 0$ ? 1
2. What is $y$ at $x = \frac{\pi}{2}$ ? -1
3. What is $y$ at $x = \pi$ ? 1
4. What is $y$ at $x = \frac{3\pi}{2}$ ? 3
14.
Graph $y = 4\cos(\frac{1}{2}(x - \frac{\pi}{2})) - 1$ with key points: find the period, the step, the x-value of the minimum and the minimum value. (Type pi for $\pi$ , like 3pi/4.)
Answer: -5
1. What is the period? 4pi
2. What is the step? pi
3. The cycle starts at $x = \frac{\pi}{2}$ . At which x is the minimum (two steps later)? 5pi/2
4. What is the minimum value? -5
15.
For $y = 3\sin(2x + \frac{\pi}{3}) + 1$ , find where the cycle starts, the step and the first maximum. (Type pi for $\pi$ , like -3pi/4.)
Answer: 4
1. At which x does the cycle start ( $C$ )? -pi/6
2. What is the step? pi/4
3. At which x is the first maximum? pi/12
4. What is the maximum value? 4
16.
For $y = -3\sin(\frac{\pi}{4}(x - 1)) + 2$ , what is the x-value of the first minimum to the right of $x = 1$ ?
Answer: 3
Period $8$ , step $2$ , start $x = 1$ . The sine is reflected ( $A = -3$ ), so the pattern is midline, min, ...: the first minimum is at $x = 1 + 2 = 3$ , where $y = -3\sin(\frac{\pi}{2}) + 2 = -1$ .
17.
The graph of $y = \sin(x)$ is first shifted $\frac{\pi}{2}$ to the right and THEN compressed horizontally by the factor $2$ . Which equation results?
1. a) $y = \sin(2(x - \frac{\pi}{2}))$
2. b) $y = \sin(2(x - \frac{\pi}{4}))$
3. c) $y = \sin(\frac{1}{2}(x - \frac{\pi}{2}))$
4. d) $y = \sin(\frac{x}{2} - \frac{\pi}{2})$
Answer: $y = \sin(2(x - \frac{\pi}{4}))$
After the shift: $\sin(x - \frac{\pi}{2})$ . Compressing replaces $x$ by $2x$ : $\sin(2x - \frac{\pi}{2}) = \sin(2(x - \frac{\pi}{4}))$ . The compression also halves the shift.
18.
What is the value of $y = 5\cos(2(x + \frac{\pi}{8})) - 2$ at $x = \frac{\pi}{8}$ ?
Answer: -2
$x = \frac{\pi}{8} = -\frac{\pi}{8} + \frac{\pi}{4}$ is the second key point; for cosine that is on the midline, so $y = D = -2$ . Directly: $5\cos(2 \cdot \frac{\pi}{4}) - 2 = 5\cos(\frac{\pi}{2}) - 2 = 0 - 2 = -2$ .