To graph the function [tex]\( h(x) = -\sqrt[3]{x+8} \)[/tex] by transforming the parent function, we need to follow these steps:
### Identify the Parent Function
The parent function here is [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
### Examine the Transformations
1. Reflection: The negative sign in front of the cube root function indicates that the graph of the parent function will be reflected across the [tex]\( x \)[/tex]-axis. This means that if a point [tex]\((a, b)\)[/tex] is on the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex], the point [tex]\((a, -b)\)[/tex] will be on the graph of [tex]\( h(x)\)[/tex].
2. Horizontal Translation: The term [tex]\((x + 8)\)[/tex] inside the cube root function indicates a horizontal translation. Specifically, [tex]\((x + 8)\)[/tex] implies a shift to the left by 8 units. This means that if a point [tex]\((a, b)\)[/tex] is on the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex], then the point [tex]\((a - 8, b)\)[/tex] will be on the graph of [tex]\( \sqrt[3]{x + 8} \)[/tex].
### Combining Transformations
First, we reflect the graph of the parent function over the [tex]\( x \)[/tex]-axis. After reflection, we then translate the graph 8 units to the left.
### Conclusion
Therefore, the correct description of how to graph [tex]\( h(x) = -\sqrt[3]{x+8} \)[/tex] by transforming the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is:
Reflect the parent function over the [tex]\( x \)[/tex]-axis, and translate it 8 units to the left.
The answer is:
Reflect the parent function over the [tex]\( x \)[/tex]-axis, and translate it 8 units to the left.
