Answer
$y=5\cos(\frac{\pi}{4}x)$
Work Step by Step
If we think backward, first, we have to observe the period, the amplitude of the function and the y-intercept, also decide whether it is a sine or cosine function.
Then, apply the rules that we know to construct the function itself:
$y=A\cos(wx)+c$
where
Amplitude=$|A|$
Period: $T=\frac{2\pi}{w}$
We can see that the graph has the following features:
Amplitude=$5$
Period=$8$ (as one full-cycle runs from $x=0$ to $x=8$), and the $y$-intercept is $5$.
Since the graph's $y$-intercept is not $0$, then we take the cosine function, as its $y$-intercept, without any transformation is $1$. We can also use the fact that the graph is symmetric only with respect to the $x$-axis so it must involve the cosine function.
Here, $A$ could either be $-5$ or $5$.
However, since the $y$-intercept is positive, then $A$ must be positive, so $A=5$..
With a period of $8$, then
$T=\dfrac{2\pi}{w}\\
8=\dfrac{2\pi}{w}\\
8w=2\pi\\
w=\dfrac{2\pi}{8}\\
w=\dfrac{\pi}{4}$
Therefore, with $A=5$ and $w=\frac{\pi}{4}$, the equation represented by the given graph is:
$y=5\cos(\frac{\pi}{4}x)$
