Question

describe the transformation:
$g(x) = \frac{1}{2}f(7x)$
options:
vertical compression of 1/2 ; horizontal stretch of 1/7
vertical compression of 1/2 ; horizontal compression of 1/7
vertical stretch of 2 ; horizontal stretch of 7
vertical stretch of 2 ; horizontal compression of 1/7

Explanation:

Step1: Analyze vertical transformation

For $g(x) = \frac{1}{2}f(7x)$, the coefficient $\frac{1}{2}$ outside $f$ scales the output of $f$. A coefficient $0 < a < 1$ causes a vertical compression by factor $a$. Here, $a=\frac{1}{2}$, so vertical compression of $\frac{1}{2}$.

Step2: Analyze horizontal transformation

The $7x$ inside $f$ scales the input of $f$. For $f(kx)$ where $k>1$, it causes a horizontal compression by factor $\frac{1}{k}$. Here, $k=7$, so horizontal compression of $\frac{1}{7}$.

Answer:

Vertical compression of 1/2 ; Horizontal compression of 1/7