Question

which graph represents $y = \sqrt3{x} + 2$?

Explanation:

Step1: Recall the parent function

The parent function is \( y = \sqrt[3]{x} \), which has a point at \((0,0)\) and passes through \((1,1)\) and \((-1,-1)\), with a cubic - like shape (symmetric about the origin).

Step2: Analyze the transformation

The function \( y=\sqrt[3]{x}+2 \) is a vertical shift of the parent function \( y = \sqrt[3]{x} \) upward by 2 units. So, we need to find the graph of \( y=\sqrt[3]{x} \) shifted up 2.

• For the parent function \( y = \sqrt[3]{x} \), when \( x = 0 \), \( y=0 \). For \( y=\sqrt[3]{x}+2 \), when \( x = 0 \), \( y=0 + 2=2 \). So the graph should pass through \((0,2)\).
• Let's check the end - behavior: As \( x

ightarrow+\infty \), \( \sqrt[3]{x}
ightarrow+\infty \), so \( y=\sqrt[3]{x}+2
ightarrow+\infty \); as \( x
ightarrow-\infty \), \( \sqrt[3]{x}
ightarrow-\infty \), so \( y=\sqrt[3]{x}+2
ightarrow-\infty \).

• Now let's analyze the graphs:
• The first graph: Check the y - intercept. If we assume the first graph is the top - most one, when \( x = 0 \), the y - value does not seem to be 2.
• The second graph: When \( x = 0 \), let's see the y - coordinate. It does not seem to be 2.
• The third graph: When \( x = 0 \), the y - coordinate is 2. Also, the shape is a vertical shift of the cube - root function. As \( x\) increases, \( y\) increases and as \( x\) decreases, \( y\) decreases, which matches the end - behavior of \( y=\sqrt[3]{x}+2 \).
• The fourth graph: The y - intercept is not 2, and the shape and direction of the graph do not match the transformed cube - root function.

Answer:

The third graph (the one in the middle - lower position among the four graphs)