Writing Trig Equations from Graphs

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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) $y = 5$
2.
A wave goes from a minimum of $1$ up to a maximum of $9$ . What is its amplitude?
3.
A sine wave has a period of $4\pi$ . What is $B$ in $y = \sin(Bx)$ ?
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) $y = A\tan(Bx) + D$
5.
A wave has a maximum of $6$ and a minimum of $-2$ . Find the midline $D$ and the amplitude $A$ .
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.)
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) $y = 2\cos\left(x - \frac{\pi}{3}\right)$
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 $y = A\cos(B(x - C)) + D$ .
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 $y = A\sin(B(x - C)) + D$ .
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) $y = 2\sin\left(x + \frac{\pi}{2}\right)$
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.)
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 $h(t) = A\cos(Bt) + D$ .
13.
A wave has a maximum at $(1, 7)$ and the next minimum at $(4, 1)$ . Write it as $y = A\cos(B(x - C)) + D$ .
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.
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 $C$ ?
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) $h = -5\cos(4t) + 7$
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 $C = 0$ ).
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 $y = A\cos(B(d - C)) + D$ .