To determine the equation of the new function when vertically stretching the quadratic parent function [tex]\( F(x) = x^2 \)[/tex] by multiplying by 7, follow these steps:
1. Identify the parent function: [tex]\( F(x) = x^2 \)[/tex].
2. Understand what a vertical stretch means. When you vertically stretch a function by a factor, you multiply the entire function by that factor.
Given that the vertical stretch factor is 7, we need to multiply the parent function [tex]\( F(x) \)[/tex] by 7:
[tex]\[ F(x) \rightarrow G(x) = 7 \cdot F(x) \][/tex]
Since [tex]\( F(x) = x^2 \)[/tex], the new function will be:
[tex]\[ G(x) = 7 \cdot x^2 \][/tex]
This is a direct vertical stretch of the parent function by a factor of 7.
Therefore, the equation of the new function is [tex]\( G(x) = 7x^2 \)[/tex].
Matching this with the given options, we see that:
- Option A: [tex]\( G(x) = (x+7)^2 \)[/tex] is incorrect, as it represents a horizontal shift, not a vertical stretch.
- Option B: [tex]\( G(x) = 7x^2 \)[/tex] is correct, as explained above.
- Option C: [tex]\( G(x) = (7x)^2 \)[/tex] represents a horizontal stretch, not a vertical stretch.
- Option D: [tex]\( G(x) = x^2 - 7 \)[/tex] represents a vertical shift downwards.
Thus, the correct answer is:
[tex]\[ B. \, G(x) = 7x^2 \][/tex]
