Question

consider the following two functions.
original: $f(x)=|x| + 1$
final: $f(x)=\frac{1}{2}|x| + \frac{1}{2}$
which option best describes the change from the original to the final function?
option #1: a vertical stretch
option #2: a vertical compression
option #3: no change
(1 point)

Explanation:

Step1: Recall vertical transformation rules

For a function $g(x) = a\cdot f(x) + b$, if $0<|a|<1$, it is a vertical compression.

Step2: Rewrite final function

Factor out $\frac{1}{2}$:
$$f_{\text{final}}(x) = \frac{1}{2}(|x| + 1) = \frac{1}{2}\cdot f_{\text{original}}(x)$$

Step3: Match to transformation type

Here $a=\frac{1}{2}$, where $0<\frac{1}{2}<1$, so this is a vertical compression.

Answer:

Option #2: a vertical compression