find an equation for the cosine graph, f(x): write your answer in the form f(x) = a cos (bx + c) + d, where…

Answer:

$f(x)=\cos(x)$

Explanation:

Step1: Determine amplitude A

The amplitude is half - the distance between the maximum and minimum values. Here, max = 1, min=-1. So $A=\frac{1 - (-1)}{2}=1$.

Step2: Determine period and B

The period of the given cosine - like function is $2\pi$. The formula for the period of $y = A\cos(Bx + C)+D$ is $T=\frac{2\pi}{|B|}$. Since $T = 2\pi$, then $|B| = 1$. We can take $B = 1$ for the standard form.

Step3: Determine phase - shift C

The graph has no horizontal shift. For a cosine function $y=\cos(x)$, when there is no phase - shift, $C = 0$.

Step4: Determine vertical shift D

The mid - line of the graph is $y = 0$. The formula for the mid - line of $y=A\cos(Bx + C)+D$ is $y = D$. So $D = 0$. So the function is $f(x)=\cos(x)$.