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Describe the transformation of the function [tex]\( h(x) = x^3 \)[/tex] after a reflection in the [tex]\( x \)[/tex]-axis and a vertical translation of 2 units down.

Sagot :

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.
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