Question

In Problems 39 and 40 , find a function whose graph is given.

Short answer

The function is$y=-7\sin \left(\frac{\pi }{4}x\right)$

Step-by-step solution

Given information

Given graph

Identify the amplitude

From the graph that it has an amplitude of $7$. We can also see that it starts from $(0,0)$, therefore, we can say that it is a sine function. We can also see that instead of the normal sine that it goes up, it goes down which means that it is a negative sine function. A sine function takes the form $y=A\sin (\omega x-\varphi )+B$. We can see that the period of cosine is 8 therefore we are going to use the formula $T=\frac{2\mathrm{\pi }}{\omega }$

$$\begin{gathered} T=\frac{2\pi }{\omega } \\ 8=\frac{2\pi }{\omega } \\ \omega =\frac{\pi }{4} \end{gathered}$$

Checking the phase shift and forming the function

We can safely assume that $\varphi$and $B$is 0 since there is no phase shift in the graph and there are no vertical translations. Listing the values that we have and $B=0$. The function of the graph is

$y=-7\sin \left(\frac{\pi }{4}x\right)$