Question

the graph of h(x) is a translation of f(x) = \sqrt3{x}. which equation represents h(x)? \
\
\bigcirc h(x) = \sqrt3{x - 2} \
\bigcirc h(x) = \sqrt3{x + 2} \
\bigcirc h(x) = \sqrt3{x} - 2 \
\bigcirc h(x) = \sqrt3{x} + 2 \
(graph with points (-1,1) and (-3,-1) for h(x), grid from x=-7 to 7 and y=-7 to 7)

Explanation:

Step1: Recall parent function's point

The parent function \( f(x) = \sqrt[3]{x} \) has a point \((0,0)\) (since \( \sqrt[3]{0}=0 \)).

Step2: Analyze translation using given points

For \( h(x) \), we have points like \((-1,1)\) and \((-3,-1)\). Let's check the transformation. The parent function \( f(x)=\sqrt[3]{x} \): when \( x = -1 \), \( f(-1)=\sqrt[3]{-1}=-1 \); but in \( h(x) \), at \( x = -1 \), \( h(-1)=1 \). The difference is \( 1 - (-1)=2 \)? Wait, no, let's check the vertical or horizontal shift. Alternatively, let's test each option:

• Option 1: \( h(x)=\sqrt[3]{x - 2} \). At \( x=-1 \), \( \sqrt[3]{-1 - 2}=\sqrt[3]{-3}

eq1 \).

• Option 2: \( h(x)=\sqrt[3]{x + 2} \). At \( x=-1 \), \( \sqrt[3]{-1 + 2}=\sqrt[3]{1}=1 \). At \( x=-3 \), \( \sqrt[3]{-3 + 2}=\sqrt[3]{-1}=-1 \). This matches the points \((-1,1)\) and \((-3,-1)\).
• Option 3: \( h(x)=\sqrt[3]{x}-2 \). At \( x=-1 \), \( \sqrt[3]{-1}-2=-1 - 2=-3

eq1 \).

• Option 4: \( h(x)=\sqrt[3]{x}+2 \). At \( x=-1 \), \( \sqrt[3]{-1}+2=-1 + 2=1 \), but at \( x=-3 \), \( \sqrt[3]{-3}+2\approx -1.442 + 2 = 0.558

eq -1 \). So option 2 works.

Answer:

\( \boldsymbol{h(x) = \sqrt[3]{x + 2}} \) (corresponding to the option \( h(x) = \sqrt[3]{x + 2} \))