the curve above is the graph of a sinusoidal function. it goes through the points ( - 7, - 1) and (3, - 1)…

Explanation:

Step1: Determine the period

The points $(-7,-1)$ and $(3, - 1)$ are at the same $y -$ value. The distance between them is half of the period $T$. So, $d=\vert-7 - 3\vert=10$, and $T = 20$. Using the formula $T=\frac{2\pi}{\omega}$, we get $\omega=\frac{2\pi}{T}=\frac{2\pi}{20}=\frac{\pi}{10}$.

Step2: Determine the amplitude

The general form of a sinusoidal function is $y = A\sin(\omega x+\varphi)+k$ or $y = A\cos(\omega x+\varphi)+k$. The mid - line is between the maximum and minimum values. The maximum value is around $y = 1$ and the minimum is around $y=-1$, so the mid - line is $y = 0$ and $k = 0$. The amplitude $A = 1$.

Step3: Determine the phase shift

Let's use the cosine form $y=A\cos(\omega x+\varphi)+k$. Substituting $A = 1$, $\omega=\frac{\pi}{10}$, $k = 0$ and the point $(3,-1)$ into $y=\cos(\frac{\pi}{10}x+\varphi)$. We have $-1=\cos(\frac{3\pi}{10}+\varphi)$. Then $\frac{3\pi}{10}+\varphi=(2n + 1)\pi$. Let $n = 0$, we get $\varphi=\frac{7\pi}{10}$.

Answer:

$f(x)=\cos(\frac{\pi}{10}x+\frac{7\pi}{10})$