Question

the graph of h(x) is a translation of f(x) = sqrt3{x}. which equation represents h(x)? the graph has points (-3, -1), (-1, 1) and the curve h(x). options: h(x) = sqrt3{x - 2}, h(x) = sqrt3{x + 2}, h(x) = sqrt3{x} - 2, h(x) = sqrt3{x} + 2

Explanation:

Step1: Recall translation rules for cube root functions

For a cube root function \( f(x)=\sqrt[3]{x} \), the horizontal translation is given by \( f(x - h)=\sqrt[3]{x - h} \) (shift right by \( h \)) or \( f(x + h)=\sqrt[3]{x + h} \) (shift left by \( h \)), and vertical translation is \( f(x)+k=\sqrt[3]{x}+k \) (shift up by \( k \)) or \( f(x)-k=\sqrt[3]{x}-k \) (shift down by \( k \)).

Step2: Analyze the parent function's key point

The parent function \( f(x)=\sqrt[3]{x} \) has a key point at \( (0,0) \). The graph of \( h(x) \) has a key point at \( (-3,-1) \) and \( (-1,1) \). Let's check the horizontal shift first.

Step3: Test each option with a point

Let's take the point \( (-1,1) \) and test each option:

• Option 1: \( h(x)=\sqrt[3]{x - 2} \). Plug \( x=-1 \): \( \sqrt[3]{-1 - 2}=\sqrt[3]{-3}\approx - 1.44

eq1 \). Eliminate.

• Option 2: \( h(x)=\sqrt[3]{x + 2} \). Plug \( x=-1 \): \( \sqrt[3]{-1 + 2}=\sqrt[3]{1}=1 \). This matches. Let's check another point \( (-3,-1) \): \( \sqrt[3]{-3 + 2}=\sqrt[3]{-1}=-1 \). This also matches.
• Option 3: \( h(x)=\sqrt[3]{x}-2 \). Plug \( x=-1 \): \( \sqrt[3]{-1}-2=-1 - 2=-3

eq1 \). Eliminate.

• Option 4: \( h(x)=\sqrt[3]{x}+2 \). Plug \( x=-1 \): \( \sqrt[3]{-1}+2=-1 + 2=1 \). Wait, but check \( x=-3 \): \( \sqrt[3]{-3}+2\approx - 1.44 + 2=0.56

eq - 1 \). So this is incorrect.

Answer:

\( h(x)=\sqrt[3]{x + 2} \) (the second option: \( h(x)=\sqrt[3]{x + 2} \))