what is the period of the function y = 1/4 cos(π/4 x)? (simplify your answer. type an exact answer, using π…

Explanation:

Step1: Recall amplitude formula

For a cosine - function of the form $y = A\cos(Bx)$, the amplitude is given by $|A|$. In the function $y=\frac{1}{4}\cos(\frac{\pi}{4}x)$, $A = \frac{1}{4}$, so the amplitude is $\left|\frac{1}{4}\right|=\frac{1}{4}$.

Step2: Recall period formula

For a cosine - function $y = A\cos(Bx)$, the period is given by $T=\frac{2\pi}{|B|}$. In the function $y=\frac{1}{4}\cos(\frac{\pi}{4}x)$, $B=\frac{\pi}{4}$. Then $T=\frac{2\pi}{\left|\frac{\pi}{4}\right|}$.

Step3: Calculate the period

$T=\frac{2\pi}{\frac{\pi}{4}}=2\pi\times\frac{4}{\pi}=8$.

Step4: Analyze the graph

The general form of the cosine function $y = A\cos(Bx)$ has an amplitude of $|A|$ and a period of $\frac{2\pi}{|B|}$. The function $y=\frac{1}{4}\cos(\frac{\pi}{4}x)$ has an amplitude of $\frac{1}{4}$ and a period of 8. The cosine function $y = \cos x$ has a maximum value of 1 and a minimum value of - 1. For $y=\frac{1}{4}\cos(\frac{\pi}{4}x)$, the maximum value is $\frac{1}{4}$ and the minimum value is $-\frac{1}{4}$, and it repeats every 8 units.

Answer:

• Amplitude: $\frac{1}{4}$
• Period: 8
• Regarding the graph, without seeing the actual options in detail, we know that the correct graph should have a maximum value of $\frac{1}{4}$, a minimum value of $-\frac{1}{4}$, and complete one full - cycle in an interval of length 8.