Question

which statement explains the difference between the graphs of $f(x) = 4x^2$ and $g(x) = -8x^2$?
a the graph of $g(x)$ is obtained by flipping $f(x)$ over the $y$-axis and stretching vertically by a factor of 2.
b the graph of $g(x)$ is obtained by flipping $f(x)$ over the $x$-axis and compressing vertically by a factor of 2.
c the graph of $g(x)$ is obtained by flipping $f(x)$ over the $x$-axis and stretching vertically by a factor of 2.
d the graph of $f(x)$ is obtained by flipping $g(x)$ over the $y$-axis and compressing vertically by a factor of 2.

Explanation:

Step1: Analyze sign change effect

The negative sign in $g(x) = -8x^2$ compared to $f(x)=4x^2$ reflects $f(x)$ over the x-axis, since multiplying a function by $-1$ flips it across the x-axis.

Step2: Analyze vertical scale factor

Take the absolute ratio of coefficients: $\frac{|-8|}{|4|} = 2$. A factor greater than 1 means vertical stretching by 2.

Step3: Match to options

This matches the description in option C.

Answer:

C. The graph of g(x) is obtained by flipping f(x) over the x-axis and stretching vertically by a factor of 2.