Sure! When applying a horizontal stretch to a function, the effect on the function’s equation can be understood through the transformation rules for functions.
Given the parent function [tex]\( f(x) = x^2 \)[/tex], we want to apply a horizontal stretch by a factor of 5. A horizontal stretch affects the input variable [tex]\( x \)[/tex] in a specific way:
1. Horizontal Stretch by a factor of 5: In mathematical terms, stretching a function horizontally by a factor of 5 means that each [tex]\( x \)[/tex]-value is multiplied by 5. To achieve this, we actually divide the [tex]\( x \)[/tex]-value by 5 inside the function.
Thus, the transformed function [tex]\( g(x) \)[/tex] can be written as:
[tex]\[ g(x) = f\left(\frac{x}{5}\right) \][/tex]
Given that the parent function is [tex]\( f(x) = x^2 \)[/tex], we substitute [tex]\(\frac{x}{5}\)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ g(x) = \left(\frac{x}{5}\right)^2 \][/tex]
Simplifying this expression:
[tex]\[ g(x) = \left(\frac{1}{5} x\right)^2 \][/tex]
Hence, the equation of the new function after applying the horizontal stretch by a factor of 5 is:
[tex]\[ g(x) = \left(\frac{1}{5} x\right)^2 \][/tex]
Now, let's match this with the given choices:
A. [tex]\( g(x) = \left(\frac{1}{5} x\right)^2 \)[/tex]
B. [tex]\( g(x) = 5 x^2 \)[/tex]
C. [tex]\( g(x) = (5 x)^2 \)[/tex]
D. [tex]\( g(x) = \frac{1}{5} x^2 \)[/tex]
The correct answer is option A:
[tex]\[ g(x) = \left(\frac{1}{5} x\right)^2 \][/tex]
