Question

what radical function is represented in the graph?

f(x) =
(simplify your answer.)

Explanation:

Step1: Identify the parent radical function

The general form of a cube root function is \( f(x) = \sqrt[3]{x - h} + k \), where \((h, k)\) is the vertex. From the graph, the vertex is at \((-1, -4)\), so \( h = -1 \) and \( k = -4 \). So the function becomes \( f(x) = \sqrt[3]{x - (-1)} - 4=\sqrt[3]{x + 1}-4 \). Let's check the point \((0, -3)\). Substitute \( x = 0 \) into the function: \( f(0)=\sqrt[3]{0 + 1}-4=1 - 4=-3 \), which matches the point \((0, -3)\).

Step2: Confirm the function

Since the point \((0, -3)\) satisfies \( f(x)=\sqrt[3]{x + 1}-4 \) and the vertex is \((-1, -4)\) which fits the cube - root function form, this is the correct radical function.

Answer:

\( \sqrt[3]{x + 1}-4 \)