Question

62 write an equation that represents the transformation described. the cube root parent function is reflected across the x-axis and has a turning point at (8, 6).

Explanation:

Step1: Recall the parent function

The parent function for the cube root is \( y = \sqrt[3]{x} \), and its turning point (vertex) is at \( (0, 0) \).

Step2: Apply reflection across the x - axis

A reflection across the \( x \)-axis changes the function to \( y=-\sqrt[3]{x} \) (since reflecting a function \( y = f(x) \) across the \( x \)-axis gives \( y=-f(x) \)).

Step3: Apply the horizontal and vertical shifts

We want the turning point to be at \( (8, 6) \). For a cube root function \( y = a\sqrt[3]{x - h}+k \), the vertex (turning point) is at \( (h,k) \). Here, \( h = 8 \) and \( k = 6 \), and \( a=- 1 \) (from the reflection). Substituting these values into the general form of the transformed cube root function \( y=a\sqrt[3]{x - h}+k \), we get \( y=-\sqrt[3]{x - 8}+6 \).

Answer:

\( y = -\sqrt[3]{x - 8}+6 \)