Question

if the parent function is $f(x) = x^2$ and its transformed function is $g(x) = 2(x - 1)^2 + 2$, how is the graph of $g(x)$ transformed from the graph of its parent function? \
\
\\(\bigcirc\\) a. stretch, shift along the $y$-axis, and shift along the $x$-axis \
\\(\bigcirc\\) b. shift along the $x$-axis and the $y$-axis \
\\(\bigcirc\\) c. reflection only \
\\(\bigcirc\\) d. stretch along the $y$-axis and reflection through the origin

Explanation:

Step1: Identify vertical stretch

The coefficient $2$ in front of $(x-1)^2$ vertically stretches $f(x)$ by factor $2$.

Step2: Identify horizontal shift

The $(x-1)$ term shifts $f(x)$ right 1 unit along the x-axis.

Step3: Identify vertical shift

The $+2$ shifts the stretched/shifted graph up 2 units along the y-axis.

Answer:

A. stretch, shift along the y-axis, and shift along the x-axis