Question

1. (10 points) find the amplitude and period of the function, and sketch its gr

$y = 1 + \frac { 1 } { 2 } \cos \left( \frac { \pi } { 2 } x \
ight)$

Explanation:

Step1: Find the amplitude

For the function \(y = A\cos(Bx - C)+D\), the amplitude is \(|A|\).
In the function \(y = 1+\frac{1}{2}\cos(\frac{\pi}{2}x)\), \(A=\frac{1}{2}\).
So the amplitude \(|A|=\frac{1}{2}\).

Step2: Find the period

For the function \(y = A\cos(Bx - C)+D\), the period is \(T=\frac{2\pi}{|B|}\).
In the function \(y = 1+\frac{1}{2}\cos(\frac{\pi}{2}x)\), \(B = \frac{\pi}{2}\).
Then \(T=\frac{2\pi}{\frac{\pi}{2}}=4\).

Answer:

The amplitude is \(\frac{1}{2}\) and the period is \(4\).