Question

review the graph of $f(x) = \sqrt3{x}$ and the graph of the transformed function $g(x)$.
if $g(x) = a \cdot f(x + b)$, how is $f(x)$ transformed to get $g(x)$?
$\boldsymbol{-2f(x + 4)}$
$\boldsymbol{-2f(x - 4)}$
$\boldsymbol{-f(x + 4)}$
$\boldsymbol{-f(x - 4)}$

Explanation:

Step1: Identify vertical reflection

The graph of $g(x)$ is a vertical reflection of $f(x)$ (it is flipped over the x-axis), so $a=-1$.

Step2: Identify horizontal shift

For $f(x)=\sqrt[3]{x}$, its inflection point is at $(0,0)$. The inflection point of $g(x)$ is at $(-4,0)$, which means the graph is shifted left 4 units. For a horizontal shift, $f(x+b)$ shifts left when $b>0$. Here, shifting left 4 units gives $f(x+4)$.

Step3: Combine transformations

Putting the reflection and shift together, $g(x)=-1\cdot f(x+4)=-f(x+4)$.

Answer:

• $-f(x + 4)$