Vertical Shift

Show your work for each problem.

1.
What is the midline of $y = 4\sin(x) - 3$ ?
1. a) $y = 4$
2. b) $y = -3$
3. c) $y = 0$
4. d) $y = 3$
Answer: $y = -3$
In $y = 4\sin(x) - 3$ we have $D = -3$ , so the midline is $y = -3$ . The 4 is the amplitude, not the shift.
2.
Which equation shifts $y = \cos(x)$ up by 5 units?
1. a) $y = \cos(x) + 5$
2. b) $y = 5\cos(x)$
3. c) $y = \cos(x + 5)$
4. d) $y = \cos(x) - 5$
Answer: $y = \cos(x) + 5$
Adding 5 outside the cosine raises every output by 5: $y = \cos(x) + 5$ . $\cos(x + 5)$ moves sideways, $5\cos(x)$ changes the amplitude and $\cos(x) - 5$ moves down.
3.
What is the maximum value of $y = 2\sin(x) + 5$ ?
Answer: 7
$D = 5$ and $|A| = 2$ , so the maximum is $5 + 2 = 7$ .
4.
The midline of $y = \cos(x) - 6$ is $y = \square$ . What number goes in the box?
Answer: -6
$y = \cos(x) - 6 = \cos(x) + (-6)$ , so $D = -6$ and the midline is $y = -6$ .
5.
What is the range of $y = 2\cos(x) + 3$ ?
1. a) $[3, 5]$
2. b) $[-1, 5]$
3. c) $[-2, 2]$
4. d) $[1, 5]$
Answer: $[1, 5]$
$D = 3$ and $|A| = 2$ : minimum $3 - 2 = 1$ , maximum $3 + 2 = 5$ , so the range is $[1, 5]$ .
6.
For $y = 3\cos(x) + 1$ , find the midline value, the amplitude, the maximum and the minimum.
Answer: D = 1, |A| = 3, max = 4, min = -2
1. What is D (the midline is $y = D$ )? 1
2. What is the amplitude $|A|$ ? 3
3. What is the maximum $D + |A|$ ? 4
4. What is the minimum $D - |A|$ ? -2
7.
The graph of $y = \sin(x)$ is shifted down by 3 units. Find the new midline, maximum and minimum.
Answer: D = -3, max = -2, min = -4
1. The new function is $y = \sin(x) + D$ . What is D? -3
2. What is the new maximum? -2
3. What is the new minimum? -4
8.
What is the minimum value of $y = -4\cos(x) + 1$ ?
Answer: -3
$|A| = |-4| = 4$ and $D = 1$ , so the minimum is $1 - 4 = -3$ . It is reached at $x = 0$ : $-4\cos(0) + 1 = -3$ .
9.
What is the maximum value of $y = -3\sin(x) + 1$ ?
1. a) $-4$
2. b) $1$
3. c) $4$
4. d) $-2$
Answer: $4$
$|A| = 3$ and $D = 1$ , so the maximum is $1 + 3 = 4$ . It is reached at $x = \frac{3\pi}{2}$ : $-3\sin\left(\frac{3\pi}{2}\right) + 1 = -3 \cdot (-1) + 1 = 4$ .
10.
A sine wave has a maximum of $8$ and a minimum of $-2$ . Find D and the amplitude $|A|$ .
Answer: D = 3, |A| = 5
1. What is max + min? 6
2. What is $D = \frac{\text{max} + \text{min}}{2}$ ? 3
3. What is max - min? 10
4. What is $|A| = \frac{\text{max} - \text{min}}{2}$ ? 5
11.
A wave swings between a maximum of $10$ and a minimum of $2$ . Its midline is $y = \square$ . What number goes in the box?
Answer: 6
$D = \frac{10 + 2}{2} = 6$ , so the midline is $y = 6$ .
12.
Find the range of $y = -2\sin(x) - 1$ .
Answer: [-3, 1]
1. What is D? -1
2. What is the amplitude $|A|$ ? 2
3. What is the maximum? 1
4. What is the minimum? -3
13.
The temperature in a town is modeled by $T(t) = 5\cos\left(\frac{\pi}{12}t\right) + 15$ in °C. Find the average temperature, the highest and lowest temperature, and the difference between them.
Answer: 15, 20, 10, 10
1. What is the average temperature (the midline value D)? 15
2. What is the highest temperature? 20
3. What is the lowest temperature? 10
4. How many degrees apart are the highest and lowest temperature? 10
14.
A wave has a maximum of $7$ and a minimum of $-1$ . Which equation could describe it?
1. a) $y = 4\sin(x) - 3$
2. b) $y = 8\sin(x) + 3$
3. c) $y = 4\sin(x) + 3$
4. d) $y = 3\sin(x) + 4$
Answer: $y = 4\sin(x) + 3$
$D = \frac{7 + (-1)}{2} = 3$ and $|A| = \frac{7 - (-1)}{2} = 4$ , so $y = 4\sin(x) + 3$ . The others have ranges $[1, 7]$ , $[-5, 11]$ and $[-7, 1]$ .
15.
A sinusoid has midline $y = -2$ and a maximum of $3$ . What is its minimum?
Answer: -7
$|A| = 3 - (-2) = 5$ , so the minimum is $-2 - 5 = -7$ .
16.
A Ferris wheel has a diameter of 30 m and its center is 18 m above the ground. A seat starts at the bottom, and one turn takes 12 minutes, so $h(t) = -15\cos\left(\frac{\pi}{6}t\right) + 18$ . Analyze the model.
Answer: |A| = 15, D = 18, max = 33, min = 3, h(6) = 33
1. What is the amplitude $|A|$ , in meters? 15
2. What is D, in meters? 18
3. What is the greatest height, in meters? 33
4. What is the smallest height, in meters? 3
5. What is the height after 6 minutes, $h(6)$ , in meters? 33
17.
At a harbor, high tide is 9.4 m and low tide is 1.6 m. Find the midline D and the amplitude $|A|$ of a tide model.
Answer: D = 5.5, |A| = 3.9
1. What is high tide + low tide, in meters? 11
2. What is the midline $D$ ? 5.5
3. What is high tide - low tide, in meters? 7.8
4. What is the amplitude $|A|$ ? 3.9
18.
The function $y = A\sin(x) + D$ with $A > 0$ has range $[-6, 2]$ . Find $D$ and $A$ , then the value of $y$ at $x = \frac{7\pi}{6}$ .
Answer: y = 4sin(x) - 2, y = -4
1. What is $D = \frac{\text{max} + \text{min}}{2}$ ? -2
2. What is $A = \frac{\text{max} - \text{min}}{2}$ ? 4
3. What is $\sin\left(\frac{7\pi}{6}\right)$ ? -0.5
4. What is $y = 4\sin\left(\frac{7\pi}{6}\right) - 2$ ? -4