Writing Trig Equations from Graphs

Show your work for each problem.

1.
A sinusoid has a maximum of $8$ and a minimum of $2$ . What is its midline?
1. a) $y = 10$
2. b) $y = 6$
3. c) $y = 3$
4. d) $y = 5$
Answer: $y = 5$
$D = \frac{8 + 2}{2} = 5$ , so the midline is $y = 5$ . ( $3$ is the amplitude and $6$ is the total height.)
2.
A wave goes from a minimum of $1$ up to a maximum of $9$ . What is its amplitude?
Answer: 4
$A = \frac{9 - 1}{2} = \frac{8}{2} = 4$ . The amplitude is half of the total height.
3.
A sine wave has a period of $4\pi$ . What is $B$ in $y = \sin(Bx)$ ?
Answer: 1/2
$B = \frac{2\pi}{4\pi} = \frac{1}{2}$ . A long period gives a small $B$ .
4.
A wave has a maximum at $x = 0$ . Which form describes it with $C = 0$ ?
1. a) $y = A\cos(Bx) + D$
2. b) $y = A\sin(Bx) + D$
3. c) $y = Ax + D$
4. d) $y = A\tan(Bx) + D$
Answer: $y = A\cos(Bx) + D$
$\cos(0) = 1$ is the largest value of cosine, so cosine starts at a maximum. Sine starts on the midline, tangent and a line do not repeat between a maximum and a minimum.
5.
A wave has a maximum of $6$ and a minimum of $-2$ . Find the midline $D$ and the amplitude $A$ .
Answer: $D = 2$ , $A = 4$
1. What is the midline $D$ ? 2
2. What is the amplitude $A$ ? 4
6.
The water level in a harbor repeats every $12$ hours. What is $B$ in the model $y = A\cos(B(x - C)) + D$ with $x$ in hours? (Type pi for $\pi$ , like pi/4.)
Answer: pi/6
$B = \frac{2\pi}{12} = \frac{\pi}{6}$ . Not $12$ : the period goes into the denominator.
7.
A wave has amplitude $2$ , midline $y = 0$ , period $2\pi$ and a maximum at $x = \frac{\pi}{3}$ . Which equation fits?
1. a) $y = 2\sin\left(x - \frac{\pi}{3}\right)$
2. b) $y = 2\cos\left(x + \frac{\pi}{3}\right)$
3. c) $y = \cos\left(2\left(x - \frac{\pi}{3}\right)\right)$
4. d) $y = 2\cos\left(x - \frac{\pi}{3}\right)$
Answer: $y = 2\cos\left(x - \frac{\pi}{3}\right)$
$y = 2\cos\left(x - \frac{\pi}{3}\right)$ gives $2\cos(0) = 2$ at $x = \frac{\pi}{3}$ . With $(x + \frac{\pi}{3})$ the maximum is at $-\frac{\pi}{3}$ , the sine version is on the midline there, and the last one has amplitude $1$ and period $\pi$ .
8.
A wave has a maximum of $5$ at $x = \frac{\pi}{6}$ , a minimum of $-1$ and a period of $\frac{2\pi}{3}$ . Write it as $y = A\cos(B(x - C)) + D$ .
Answer: $y = 3\cos\left(3\left(x - \frac{\pi}{6}\right)\right) + 2$
1. What is $D$ ? 2
2. What is $A$ ? 3
3. What is $B$ ? 3
4. What is $C$ ? (Type pi for $\pi$ , like 3pi/4.) pi/6
9.
A wave crosses its midline $y = 0$ going up at $x = \frac{\pi}{4}$ , has a maximum of $3$ and a period of $\pi$ . Write it as $y = A\sin(B(x - C)) + D$ .
Answer: $y = 3\sin\left(2\left(x - \frac{\pi}{4}\right)\right)$
1. What is $A$ ? 3
2. What is $D$ ? 0
3. What is $B$ ? 2
4. What is $C$ ? (Type pi for $\pi$ , like 3pi/4.) pi/4
10.
Which equation does NOT draw the same graph as $y = 2\cos(x)$ ?
1. a) $y = 2\cos(x - 2\pi)$
2. b) $y = -2\cos(x - \pi)$
3. c) $y = 2\sin\left(x - \frac{\pi}{2}\right)$
4. d) $y = 2\sin\left(x + \frac{\pi}{2}\right)$
Answer: $y = 2\sin\left(x - \frac{\pi}{2}\right)$
At $x = 0$ : $2\sin\left(\frac{\pi}{2}\right) = 2$ , $-2\cos(-\pi) = 2$ , $2\cos(-2\pi) = 2$ , but $2\sin\left(-\frac{\pi}{2}\right) = -2$ . That last graph is the upside-down wave $y = -2\cos(x)$ .
11.
The graph of $y = 4\sin(2x) + 1$ can also be written as $y = 4\cos(2(x - C)) + 1$ . What is the smallest positive $C$ ? (Type pi for $\pi$ , like pi/3.)
Answer: pi/4
$\sin(2x)$ is largest when $2x = \frac{\pi}{2}$ , so $x = \frac{\pi}{4}$ . That is a quarter of the period $\pi$ . So $C = \frac{\pi}{4}$ .
12.
A Ferris wheel has a diameter of $30$ m, its center is $18$ m above the ground and it turns once every $6$ minutes. You board at the bottom at $t = 0$ . Use $h(t) = A\cos(Bt) + D$ .
Answer: 33
1. What is $D$ in m? 18
2. You start at the bottom, so $A$ is negative. What is $A$ ? -15
3. What is $B$ ? (Type pi for $\pi$ , like 3pi/4.) pi/3
4. How high are you after $3$ minutes (in m)? 33
13.
A wave has a maximum at $(1, 7)$ and the next minimum at $(4, 1)$ . Write it as $y = A\cos(B(x - C)) + D$ .
Answer: $y = 3\cos\left(\frac{\pi}{3}(x - 1)\right) + 4$
1. What is the period? 6
2. What is $B$ ? (Type pi for $\pi$ , like 3pi/4.) pi/3
3. What is $A$ ? 3
4. What is $D$ ? 4
5. What is $C$ ? 1
14.
High tide in a harbor is $9$ m at 4:00 and the next low tide is $1$ m at 10:00. Model the depth with $y = A\cos(B(x - C)) + D$ ( $x$ in hours after midnight) and find the depth at 6:00.
Answer: 7
1. What is the period in hours? 12
2. What is $A$ ? 4
3. What is $D$ ? 5
4. What is $B$ ? (Type pi for $\pi$ , like 3pi/4.) pi/6
5. What is the depth at 6:00 (in m)? 7
15.
A wave has a maximum of $10$ at $x = 3$ and the next minimum of $2$ at $x = 7$ . In $y = A\sin(B(x - C)) + D$ with $A > 0$ , what is the smallest positive $C$ ?
Answer: 1
Half period $= 7 - 3 = 4$ , so the period is $8$ and a quarter period is $2$ . The wave crosses the midline going up at $x = 3 - 2 = 1$ , so $C = 1$ .
16.
A weight on a spring starts at its lowest point, $2$ cm above the floor, at $t = 0$ . Its highest point is $12$ cm and one full bounce takes $4$ seconds. Which model is right?
1. a) $h = -5\cos\left(\frac{\pi}{2}t\right) + 7$
2. b) $h = -7\cos\left(\frac{\pi}{2}t\right) + 5$
3. c) $h = 5\cos\left(\frac{\pi}{2}t\right) + 7$
4. d) $h = -5\cos(4t) + 7$
Answer: $h = -5\cos\left(\frac{\pi}{2}t\right) + 7$
$D = \frac{12 + 2}{2} = 7$ , amplitude $= \frac{12 - 2}{2} = 5$ , $B = \frac{2\pi}{4} = \frac{\pi}{2}$ , and $A = -5$ because it starts at the bottom: $h = -5\cos\left(\frac{\pi}{2}t\right) + 7$ . Check: $t = 0$ gives $-5 + 7 = 2$ .
17.
A graph has a minimum of $-5$ at $x = 0$ , a maximum of $3$ and a period of $4\pi$ . Write it as $y = A\cos(Bx) + D$ (so $C = 0$ ).
Answer: $y = -4\cos\left(\frac{x}{2}\right) - 1$
1. What is $D$ ? -1
2. What is the amplitude? 4
3. The graph starts at a minimum. What is $A$ ? -4
4. What is $B$ ? 1/2
18.
A city has $17$ hours of daylight on its longest day (day $172$ ) and $7$ hours on its shortest. The pattern repeats every $365$ days. Find $A$ , $D$ , $B$ and $C$ in $y = A\cos(B(d - C)) + D$ .
Answer: $y = 5\cos\left(\frac{2\pi}{365}(d - 172)\right) + 12$
1. What is $A$ in hours? 5
2. What is $D$ in hours? 12
3. What is $B$ ? (Type pi for $\pi$ , like 3pi/4.) 2pi/365
4. What is $C$ ? 172