Question

how would you describe the difference between the graphs of $f(x) = 2x^2$ and $g(x) = -2x^2$?

a. $g(x)$ is a reflection of $f(x)$ over the $x$-axis.

b. $g(x)$ is a reflection of $f(x)$ over the line $y = -1$.

c. $g(x)$ is a reflection of $f(x)$ over the $y$-axis.

d. $g(x)$ is a reflection of $f(x)$ over the line $y = x$.

Explanation:

To determine the transformation between \( f(x) = 2x^2 \) and \( g(x) = -2x^2 \), we recall the rule for reflecting a function over the \( x \)-axis: if we have a function \( y = f(x) \), its reflection over the \( x \)-axis is \( y = -f(x) \). Here, \( g(x) = -f(x) \) (since \( f(x)=2x^2 \), then \( -f(x)= -2x^2 \)), which matches the reflection over the \( x \)-axis. Option B is incorrect as reflecting over \( y = -1 \) would involve a vertical shift, not just a sign change. Option C (reflection over \( y \)-axis) would be \( f(-x) \), which is \( 2(-x)^2 = 2x^2 \), not \( -2x^2 \). Option D (reflection over \( y = x \)) is for inverse functions, and \( f(x) = 2x^2 \) is not one - to - one, so its inverse is not a function in the same form, and \( g(x) \) is not the inverse of \( f(x) \).

Answer:

A. \( g(x) \) is a reflection of \( f(x) \) over the \( x \)-axis.