write your answer in the form f(x) = a cos (bx + c) + d, where a, b, c, and d are real numbers.\nf(x) =

Explanation:

Step1: Determine the amplitude A

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is - 3 and the minimum value is - 5. The vertical distance is $|-3-(-5)| = 2$, so $A=\frac{2}{2}=1$.

Step2: Determine the period and find B

The period $T$ of the cosine - function is the horizontal distance between two consecutive maxima or minima. From the graph, $T=\pi$. Since the formula for the period of $y = A\cos(Bx + C)+D$ is $T=\frac{2\pi}{|B|}$, and $T = \pi$, we have $\pi=\frac{2\pi}{|B|}$, so $|B| = 2$. Since the graph has a normal - looking cosine - wave shape (not reflected horizontally), $B = 2$.

Step3: Determine the phase - shift C

The standard cosine function $y=\cos(x)$ has a maximum at $x = 0$. Our function has a maximum at $x=\frac{\pi}{2}$. For the function $y=\cos(Bx + C)$, when $Bx + C=0$ gives the location of the maximum. Substituting $x=\frac{\pi}{2}$ and $B = 2$ into $2x + C=0$, we get $2\times\frac{\pi}{2}+C = 0$, so $C=-\pi$.

Step4: Determine the vertical shift D

The mid - line of the cosine function is the average of the maximum and minimum values. The mid - line is $y=\frac{-3+( - 5)}{2}=-4$, so $D=-4$.

Answer:

$f(x)=\cos(2x-\pi)-4$