write a cosine function that has a midline of y = 4, an amplitude of 5 and a period of 4/7.

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine - function is $y = A\cos(Bx - C)+D$, where $|A|$ is the amplitude, $\frac{2\pi}{|B|}$ is the period, $C$ is the phase - shift, and $D$ is the mid - line.

Step2: Determine the value of $A$

Given that the amplitude is $5$, so $A = 5$ (we can choose the positive value for simplicity, since the negative value would just reflect the graph over the mid - line).

Step3: Determine the value of $B$

We know that the period $T=\frac{2\pi}{|B|}$. Given $T = \frac{4}{7}$, then $\frac{2\pi}{|B|}=\frac{4}{7}$. Solving for $B$: [ \begin{align*} |B|&=\frac{2\pi\times7}{4}\ |B|&=\frac{7\pi}{2} \end{align*} ] We can choose $B=\frac{7\pi}{2}$ for simplicity.

Step4: Determine the value of $D$

Given that the mid - line is $y = 4$, so $D = 4$. And since there is no mention of a phase - shift, we assume $C = 0$.

Answer:

$y = 5\cos\left(\frac{7\pi}{2}x\right)+4$