Question

question 11 (1 point)
give an equation for a cosine function that has a maximum at (\\(\frac{\pi}{4}, 8\\)), followed immediately by another maximum at (\\(\frac{9\pi}{4}, 8\\)), and has a minimum value of 4.
\\(\bigcirc\\) a) \\(y = 8\cos(x + \frac{\pi}{4})\\)
\\(\bigcirc\\) b) \\(y = 2\cos(x - \frac{\pi}{4}) + 6\\)
\\(\bigcirc\\) c) \\(y = 8\cos(x - \frac{\pi}{4})\\)
\\(\bigcirc\\) d) \\(y = 2\cos(x + \frac{\pi}{4}) + 6\\)

Explanation:

Step1: Find the amplitude

The formula for the amplitude \(A\) of a cosine function \(y = A\cos(B(x - C))+D\) is \(A=\frac{\text{Max}-\text{Min}}{2}\). Given \(\text{Max} = 8\) and \(\text{Min}=4\), then \(A=\frac{8 - 4}{2}=2\).

Step2: Find the vertical shift

The formula for the vertical shift \(D\) is \(D=\frac{\text{Max}+\text{Min}}{2}\). So \(D=\frac{8 + 4}{2}=6\).

Step3: Find the period and \(B\)

The period \(T\) is the distance between two consecutive maxima. Here \(T=\frac{9\pi}{4}-\frac{\pi}{4}=2\pi\). The formula for the period of \(y = A\cos(B(x - C))+D\) is \(T=\frac{2\pi}{|B|}\). Since \(T = 2\pi\), then \(|B| = 1\) (we can take \(B = 1\) for simplicity).

Step4: Find the phase shift

We know that the general form is \(y=A\cos(B(x - C))+D\). Let's use the point \((\frac{\pi}{4},8)\). Substitute \(A = 2\), \(B = 1\), \(D=6\) into \(y=A\cos(B(x - C))+D\), we get \(8=2\cos(1\times(\frac{\pi}{4}-C))+6\). Then \(\cos(\frac{\pi}{4}-C)=1\). So \(\frac{\pi}{4}-C = 2k\pi\) (\(k\in\mathbb{Z}\)), taking \(k = 0\), we have \(C=\frac{\pi}{4}\). The function is \(y = 2\cos(x-\frac{\pi}{4})+6\).

Answer:

B. \(y = 2\cos(x-\frac{\pi}{4})+6\)