Graph transformations: $e^{-4x}$ to $e^{8x}$

Question

Say if the parent function was $e^{-4x}$, what steps would be required to get to $e^{8x}$? I know a reflection across the $y$ axis is required to change the sign, but not too sure what to do with the constants. Thanks a lot.

Answer 1

Let $F(x) = e^{(-4x)}$.

$G(x) = F(2x) = e^{(-8x)}$

$H(x) = G(-x) = e^{(8x)}$

Answer 2

$Let$ $us$ $consider$, $$F(x) = e^{(-4x)}$$

$Then$, $$F(-2x)=e^{(8x)}$$

Answer 3

To answer this question, We need to consider the following facts:

First. The graph of $y=f(-x)$ is the graph of $y=f(x)$ reflected about the $y$-axis.

Second. There exist two cases:

• $c \gt 1$: The graph of $y=f(cx)$ is the graph of $y=f(x)$ horizontally shrunk by $c$.
• $0 \lt c \lt 1$: The graph of $y=f(cx)$ is the graph of $y=f(x)$ horizontally stretched by $c$.

Thus, according to the above facts, the graph of the function $y=e^{8x}=e^{-(2(-4x))}=e^{2(-(-4x))}$ is the graph of the function $e^{-4x}$ horizontally shrunk by $2$ and then reflected about the $y$-axis, or equivalently, is the graph of the function $e^{-4x}$ reflected about the $y$-axis and then horizontally shrunk by $2$.

Notice: Please note that shrinking by "$-2$" is meaningless because the scale factor must be a positive number.