Question

identify the equation for the graph. $y + 3 = sqrt3{x - 1}$ $y - 3 = sqrt3{x - 1}$ $y - 3 = sqrt3{x + 1}$

Explanation:

Step1: Recall the parent cube root function

The parent cube root function is \( y = \sqrt[3]{x} \), which passes through the origin \((0,0)\) and has a point at \((1,1)\), \((-1,-1)\) etc.

Step2: Analyze the graph's key point

Looking at the graph, when \( x = -1 \), let's check the y - value. From the graph, when \( x=-1\), \( y = 3\)? Wait, no, let's look at the graph again. Wait, the graph has a point around \( x = - 1\)? Wait, the graph seems to have a point at \( x=-1\) (since the graph has a "corner" or the inflection point? Wait, the cube root function's inflection point is at \((h,k)\) for the transformed function \( y - k=\sqrt[3]{x - h}\).

Let's check the options. Let's find a point on the graph. Let's see, when \( x=-1\), let's plug into each option:

Option 1: \( y + 3=\sqrt[3]{x - 1}\). If \( x=-1\), then \( y+3=\sqrt[3]{-2}\), \( y=\sqrt[3]{-2}-3\approx - 1.26 - 3=-4.26\), not matching the graph (graph at \( x=-1\) is around \( y = 3\)? Wait, maybe I misread the graph. Wait the graph has a point at \( x = - 1\)? Wait the graph is a cube root function shifted. Let's recall the transformation: \( y - k=\sqrt[3]{x - h}\) shifts the parent function \( y=\sqrt[3]{x}\) horizontally by \( h\) and vertically by \( k\).

Let's check the third option: \( y - 3=\sqrt[3]{x + 1}\). Let's find when \( x=-1\): \( y - 3=\sqrt[3]{-1 + 1}=\sqrt[3]{0}=0\), so \( y=3\). Now check another point. Let's take \( x = 0\): \( y-3=\sqrt[3]{0 + 1}=1\), so \( y=4\). Looking at the graph, when \( x = 0\), \( y\) is 4 (from the graph, the y - intercept is 4). That matches.

Check the second option: \( y - 3=\sqrt[3]{x - 1}\). When \( x=1\), \( y - 3=\sqrt[3]{0}=0\), \( y = 3\). But when \( x = 0\), \( y-3=\sqrt[3]{-1}=-1\), \( y=2\), which doesn't match the graph (graph at \( x = 0\) is \( y = 4\)).

First option: \( y + 3=\sqrt[3]{x - 1}\). When \( x=1\), \( y+3=\sqrt[3]{0}=0\), \( y=-3\), not matching.

Third option: \( y - 3=\sqrt[3]{x + 1}\). When \( x = 0\), \( y-3=\sqrt[3]{1}=1\), so \( y=4\) (matches the graph's y - intercept, since the graph crosses the y - axis at \( y = 4\)). When \( x=-1\), \( y - 3=\sqrt[3]{0}=0\), \( y=3\) (which is a point on the graph). So this matches.

Answer:

\( y - 3=\sqrt[3]{x + 1} \) (the third option)