To transform the function [tex]\( h(x) = x^3 \)[/tex] by reflecting it in the [tex]\( x \)[/tex]-axis and then translating it vertically 2 units down, follow these steps:
1. Reflect the function in the [tex]\( x \)[/tex]-axis:
Reflecting a function [tex]\( f(x) \)[/tex] in the [tex]\( x \)[/tex]-axis means transforming it into [tex]\( -f(x) \)[/tex]. For the function [tex]\( h(x) = x^3 \)[/tex], the reflection would be:
[tex]\[
h_{\text{reflected}}(x) = -h(x) = -x^3
\][/tex]
2. Translate the function 2 units down:
Translating a function [tex]\( f(x) \)[/tex] downward by 2 units implies altering the function to [tex]\( f(x) - 2 \)[/tex]. Applying this translation to [tex]\( h_{\text{reflected}}(x) \)[/tex], we get:
[tex]\[
h_{\text{transformed}}(x) = h_{\text{reflected}}(x) - 2 = -x^3 - 2
\][/tex]
Thus, the resulting transformed function after reflecting [tex]\( h(x) = x^3 \)[/tex] in the [tex]\( x \)[/tex]-axis and then translating it downward by 2 units is:
[tex]\[
h_{\text{transformed}}(x) = -x^3 - 2
\][/tex]
This is the final form of the new function.
