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

Explanation:

Step1: Determine the amplitude A

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is 7 and the minimum is 3. So, $A=\frac{7 - 3}{2}=\frac{4}{2}=2$.

Step2: Determine the vertical - shift D

The vertical - shift D is the mid - value between the maximum and minimum values. So, $D=\frac{7 + 3}{2}=5$.

Step3: Determine the period and B

The period of the cosine function is the horizontal distance between two consecutive maxima or minima. Here, the period $T = 2\pi$. The formula for the period of $y = A\cos(Bx + C)+D$ is $T=\frac{2\pi}{|B|}$. Since $T = 2\pi$, then $B = 1$.

Step4: Determine the phase - shift C

The graph of $y = A\cos(Bx + C)+D$ is a cosine graph with no phase - shift (the cosine function starts at its maximum value as in the standard $y=\cos(x)$ graph), so $C = 0$.

Answer:

$f(x)=2\cos(x)+5$