To determine what happens when you vertically stretch the cubic parent function [tex]\(F(x) = x^3\)[/tex] by a factor, we'll take a series of logical steps.
1. Understanding Vertical Stretch:
Vertical stretching (or dilation) of a function [tex]\(F(x) = x^3\)[/tex] by a factor of 'a' means that every value of the function is multiplied by 'a'. So, the new function [tex]\(M(x)\)[/tex] after a vertical stretch will be [tex]\(M(x) = a \cdot F(x)\)[/tex].
2. Applying the Vertical Stretch:
Given that 'a' is 5, we apply this factor to the function [tex]\(F(x)\)[/tex].
- Original function: [tex]\(F(x) = x^3\)[/tex]
- After a vertical stretch by a factor of 5:
[tex]\[
M(x) = 5 \cdot F(x) = 5 \cdot x^3
\][/tex]
3. Resulting Function:
The new function after applying the vertical stretch is:
[tex]\[
M(x) = 5x^3
\][/tex]
4. Comparing with the Given Options:
- Option A: [tex]\(G(x) = \frac{1}{5}x^3\)[/tex] – This is a vertical compression.
- Option B: [tex]\(J(x) = 5x^3\)[/tex] – This matches the vertical stretch we calculated.
- Option C: [tex]\(K(x) = (5x)^3 = 125x^3\)[/tex] – This is a function resulting from scaling the input, not the output.
- Option D: [tex]\(H(x) = \left(\frac{1}{5}x\right)^3 = \frac{1}{125}x^3\)[/tex] – This again is a function resulting from scaling the input, not the output.
Based on our calculation and comparison with the given options, the correct answer for the equation of the new function after a vertical stretch of [tex]\(F(x) = x^3\)[/tex] by a factor of 5 is:
[tex]\[
\boxed{J(x) = 5x^3}
\][/tex]
