Question

cube and cube root functions and equations
the graph of the cube root parent function $y = \sqrt3{x}$ is translated to form $f(x)$ shown on the graph.
which equation represents $f(x)$?
$f(x) = \sqrt3{x+6}-1$
$f(x) = \sqrt3{x+6}+1$
$f(x) = \sqrt3{x-6}-1$
$f(x) = \sqrt3{x-6}+1$

Explanation:

Step1: Identify parent function point

The parent function $g(x)=\sqrt[3]{x}$ has a key point at $(0,0)$.

Step2: Find translated key point

From the graph, the corresponding key point of $f(x)$ is $(-6,-1)$.

Step3: Determine translation rules

For a horizontal shift: $x \to x - h$, here $0 \to -6$, so $x = x' - h \implies 0 = -6 - h \implies h = -6$, meaning shift left 6: $\sqrt[3]{x+6}$.
For a vertical shift: $y \to y + k$, here $0 \to -1$, so $0 = -1 + k \implies k = -1$, meaning shift down 1: $\sqrt[3]{x+6}-1$.

Step4: Match to option

This matches $f(x) = \sqrt[3]{x+6}-1$.

Answer:

$f(x) = \sqrt[3]{x+6}-1$