match the amplitude, midline, period, and frequency for the cosine equation. f(x)=5cos(2x)+3 a. period b…

Explanation:

Step1: Recall cosine - function form

The general form of a cosine function is $y = A\cos(Bx - C)+D$, where $A$ is the amplitude, $B$ affects the period, $C$ is the phase - shift, and $D$ is the vertical shift (mid - line). For the function $f(x)=5\cos(2x)+3$, we have $A = 5$, $B = 2$, and $D = 3$.

Step2: Calculate the amplitude

The amplitude of the cosine function $y = A\cos(Bx - C)+D$ is given by $|A|$. Here, $A = 5$, so the amplitude is $5$.

Step3: Determine the mid - line

The mid - line of the cosine function $y = A\cos(Bx - C)+D$ is $y = D$. Here, $D = 3$, so the mid - line is $y = 3$.

Step4: Calculate the period

The period of the cosine function $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here, $B = 2$, so $T=\frac{2\pi}{2}=\pi$.

Step5: Calculate the frequency

The frequency $f$ of a periodic function is the reciprocal of the period. Since $T=\pi$, then $f=\frac{1}{T}=\frac{1}{\pi}$. But among the given options, we match the values we have calculated for period, amplitude, mid - line.

Answer:

a. 4. $\pi$ b. 2. 5 c. 3. $y = 3$ d. 1. 2 (Note: The value 2 does not correctly represent the frequency for this function based on standard calculations, but we are matching from the given options)