Question

match each transformation of function f with a feature of the transformed function.

( h(x) = 3f(x) - 3 )
( j(x) = f(x + 3) )
( g(x) = -f(x - 3) )

Explanation:

Step1: Analyze $h(x)=3f(x)-3$

Vertical stretch by 3, shift down 3. If original vertex is $(2,1)$, new vertex: $x=2$, $y=3(1)-3=0$. So vertex $(2,0)$.

Step2: Analyze $j(x)=f(x+3)$

Horizontal shift left 3. Original vertex $(2,1)$: new $x=2-3=-1$, $y=1$. Domain shifts left 3: original domain $(1,\infty)$ becomes $(1-3,\infty)=(-2,\infty)$.

Step3: Analyze $g(x)=-f(x-3)$

Reflect over x-axis, shift right 3. Original vertex $(2,1)$: new $x=2+3=5$, $y=-1$. Range: original range $(1,\infty)$, reflected becomes $(-\infty,-1)$.

Answer:

$h(x) = 3f(x) - 3$ $\boldsymbol{\longrightarrow}$ vertex at $(2, 0)$
$j(x) = f(x + 3)$ $\boldsymbol{\longrightarrow}$ domain of $(-2, \infty)$
$g(x) = -f(x - 3)$ $\boldsymbol{\longrightarrow}$ range of $(-\infty, -1)$