explain how to rewrite the function shown in order to determine the transformation of the parent function…

Explanation:

Step1: Rewrite the function

First, factor out - 8 from the expression inside the cube - root. We have $y=\sqrt[3]{-8(x + \frac{1}{2})}$. Since $\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}$ for real - valued $a$ and $b$, and $\sqrt[3]{-8}=-2$, the function can be rewritten as $y = - 2\sqrt[3]{x+\frac{1}{2}}$.

Step2: Identify the parent function and transformations

The parent function of a cube - root function is $y=\sqrt[3]{x}$. The transformation $y=-2\sqrt[3]{x + \frac{1}{2}}$ has the following effects compared to the parent function $y = \sqrt[3]{x}$:

• Horizontal shift: The value of $h=-\frac{1}{2}$ in the form $y = a\sqrt[3]{x - h}$. So, there is a horizontal shift of $\frac{1}{2}$ unit to the left.
• Vertical stretch and reflection: The coefficient $a=-2$. The absolute value of $a = 2$ means a vertical stretch by a factor of 2, and the negative sign means a reflection across the $x$ - axis.

Answer:

The function $y=\sqrt[3]{-8x - 4}$ can be rewritten as $y=-2\sqrt[3]{x+\frac{1}{2}}$. Compared to the parent function $y = \sqrt[3]{x}$, the graph of $y=\sqrt[3]{-8x - 4}$ is shifted $\frac{1}{2}$ unit to the left, vertically stretched by a factor of 2, and reflected across the $x$ - axis.