Question

the graph of g(x) is a translation of $y = \sqrt3{x}$. which equation represents g(x)? \bigcirc $g(x) = \sqrt3{x - 4}$ \bigcirc $g(x) = \sqrt3{x + 4}$ \bigcirc $g(x) = \sqrt3{x} + 1.5$ \bigcirc $g(x) = \sqrt3{x} - 1.5$

Explanation:

Step1: Recall Translation Rules

For a cube root function \( y = \sqrt[3]{x} \), horizontal translation: \( y=\sqrt[3]{x - h} \) (shift right \( h \) units) or \( y=\sqrt[3]{x + h} \) (shift left \( h \) units); vertical translation: \( y=\sqrt[3]{x}+k \) (shift up \( k \) units) or \( y=\sqrt[3]{x}-k \) (shift down \( k \) units). The parent function \( y = \sqrt[3]{x} \) passes through \( (0,0) \).

Step2: Analyze the Graph of \( g(x) \)

The graph of \( g(x) \) passes through \( (4, -1) \)? Wait, no, looking at the graph, the key point: the parent \( y=\sqrt[3]{x} \) has a point at \( (0,0) \). The graph of \( g(x) \) seems to have a vertical shift? Wait, no, let's check the y - intercept or the point. Wait, the parent function \( y = \sqrt[3]{x} \): when \( x = 0 \), \( y = 0 \). For \( g(x) \), when \( x = 0 \), what's \( y \)? Wait, the graph of \( g(x) \) at \( x = 0 \): looking at the grid, when \( x = 0 \), \( y\approx - 1.5 \)? Wait, no, let's check the options. Wait, another approach: the parent function \( y=\sqrt[3]{x} \), and the graph of \( g(x) \) is a vertical translation? Wait, no, let's check the value when \( x = 0 \). For \( y=\sqrt[3]{x} \), at \( x = 0 \), \( y = 0 \). For \( g(x) \), when \( x = 0 \), from the graph, the y - coordinate is around \( - 1.5 \). So let's test the options at \( x = 0 \):

• Option 1: \( g(0)=\sqrt[3]{0 - 4}=\sqrt[3]{-4}\approx - 1.587\) (not - 1.5)
• Option 2: \( g(0)=\sqrt[3]{0 + 4}=\sqrt[3]{4}\approx 1.587\) (positive, not matching)
• Option 3: \( g(0)=\sqrt[3]{0}+1.5 = 1.5\) (positive, not matching)
• Option 4: \( g(0)=\sqrt[3]{0}-1.5=-1.5\) (matches the y - intercept of the graph)

Wait, also, check the point where the graph crosses the x - axis? Wait, the graph of \( g(x) \) seems to have a vertical shift down. Let's confirm with the vertical translation. The parent function \( y = \sqrt[3]{x} \) at \( x = 0 \) is \( 0 \), here at \( x = 0 \), \( g(0)=-1.5 \), so the vertical shift is down 1.5 units, so \( g(x)=\sqrt[3]{x}-1.5 \).

Answer:

\( g(x)=\sqrt[3]{x}-1.5 \) (the option: \( g(x)=\sqrt[3]{x}-1.5 \))