Amplitude, period and phase shift of can be recovered from the graph by noticing the coordinates of the peaks and troughs of the wave.

Let $y_{peak}$ and $y_{trough}$ denote the $y$-coordinates of the peaks and troughs of the wave. Then for the amplitude we have

$$ A=\frac{1}{2}\left(y_{peak}-y_{trough}\right) $$

From your graph there is no indication of the vertical scale, but let us suppose that the horizontal dashed lines are one unit apart. Then we would have $y_{peak}=2$ and $y_{trough}=-6$. This would give $A=\frac{1}{2}(2-(-6))=4$.

We can also use $y_{peak}$ and $y_{trough}$ to find the vertical shift $D$.

$$ D=\frac{1}{2}\left(y_{peak}+y_{trough}\right) $$

So, for your example, $D=\frac{1}{2}(2+(-6))=-2$.

This leaves the values of $B$ and $C$. But first, we must find the period $P$, which is straightforward.

The period P is the horizontal distance between two successive peaks of the graph. For your graph this would be a distance $P=3\pi$.

The value of $B$ is then found from

$$ B=\frac{2\pi}{P} $$

For your graph, then, we have $B=\frac{2\pi}{3\pi}=\frac{2}{3}$.

Finally we have the phase shift $\phi$.

The phase shift for the sine and cosine are computed differently. The easiest to determine from the graph is the phase shift of the cosine.

Let $x_{peak}$ denote the $x$-coordinate of the peak **closest to the vertical axis**. For your graph, we have $x_{peak}=0$.

$$\phi=x_{peak}\text{ for the cosine graph}$$

$$\phi=\left(x_{peak}-\frac{P}{4}\right)\text{ for the sine graph}$$

For your graph this gives phase shift $\phi=0$ for the cosine graph and phase shift $\phi=0-\frac{3\pi}{4}=-\frac{3\pi}{4}$ for the sine graph.

But for your equations, we need the value of $C$. The value of $C$ is found from the values of $\phi$ and $B$.

The equation for $C$ in terms of the phase shift $\phi$ is

$$ C=B\phi $$

So if we use the cosine function to model your graph we have $C=0$ and for the sine graph we have $C=\frac{2}{3}\cdot\left(-\frac{3\pi}{4}\right)=-\frac{\pi}{2}$.

So we have for both sine and cosine

- $A=4$
- $D=-2$
- $B=\frac{2}{3}$

For cosine, $C=0$ and for sine, $C=-\frac{\pi}{2}$.

Thus your graph can be represented by either of the two equations

$$ y=4\cos\left(\frac{2}{3}x \right)-2 $$

$$ y=4\sin\left(\frac{2}{3}x+\frac{\pi}{2}\right)-2 $$

Since the question states that the Ferris Wheel must start half a meter off of the ground, then we can make our phase shift $d=0$. This allows us to assume that the minimum height is achieved at $x\equiv \frac{\pi}{2}n$ where $n$ is every other odd integer starting with $n=3$. This is because the sine function is $-1$ at those values, and is at a minimum.

Next, sine functions ,$y= a\sin{k(x-d)}+c$, are $2\pi$ periodic, meaning that it takes $2\pi$ radians, or $1$ period, to get back to your initial starting point. The period, $T$, is given as $60$ seconds. Using the formula for the period of a sine and cosine function, $T=\frac{2\pi}{|k|}$, we find that $|k|=\frac{\pi}{30}$. The absolute value signs are not really necessary, but period is typically always positive and $k$ can be positive or negative.

Now to find the amplitude. No speed was specified, nor was the radius of the Ferris Wheel, and the only way I see to solve this is to let $a=r$ where $r$ is the radius of the Ferris Wheel.

Finally, we need that when $\sin{k(x-d)}=-1$, $y=0.5$. Setting $y=-r+c=0.5$, we see that $c=r+0.5$.

$$y=r\sin{\frac{\pi}{30}x} \,+ r+\frac{1}{2}$$

## Best Answer

If the max value is $14$ and the min value is $4$, what is the midline value? What is the amplitude? Answering these questions will let you fix your errors--specifically, the $+2$ at the end and the $14$. The rest looks good.