Write a cosine function that has a midline of y=4 an amplitude of 2, a period of 4pi , and a horizontal shift of 2pi to the left. Answer Attemptiout of 2 f(x)=

## Step 1<br />The general form of a cosine function is given by:<br />### \(f(x) = A \cos(B(x - C)) + D\)<br />where:<br />- \(A\) is the amplitude of the function,<br />- \(B\) is the frequency of the function,<br />- \(C\) is the horizontal shift of the function, and<br />- \(D\) is the midline of the function.<br /><br />## Step 2<br />In this problem, we are given that the amplitude \(A = 2\), the midline \(D = 4\), the period \(T = 4\pi\), and the horizontal shift \(C = 2\pi\).<br /><br />## Step 3<br />The frequency \(B\) is related to the period \(T\) by the formula:<br />### \(B = \frac{2\pi}{T}\)<br />Substituting the given period \(T = 4\pi\) into the formula, we get:<br />### \(B = \frac{2\pi}{4\pi} = \frac{1}{2}\)<br /><br />## Step 4<br />Substituting the values of \(A\), \(B\), \(C\), and \(D\) into the general form of the cosine function, we get:<br />### \(f(x) = 2 \cos\left(\frac{1}{2}(x + 2\pi)\right) + 4\)