Question

how does the graph of $y = 3^x$ compare to the graph of $y = 3^{-x}$? the graphs are reflected across the $x$-axis. the graphs are the same. the graphs are reflected across the $y$-axis.

Explanation:

Step1: Recall reflection rules

For a function \( y = f(x) \), reflecting over the \( y \)-axis gives \( y = f(-x) \), and reflecting over the \( x \)-axis gives \( y=-f(x) \).

Step2: Analyze the given functions

We have \( y = 3^{x} \) and \( y = 3^{-x} \). Notice that \( 3^{-x}=(3^{x}) \) with \( x \) replaced by \( -x \), i.e., if \( f(x)=3^{x} \), then \( 3^{-x}=f(-x) \).

Step3: Determine the reflection

From the reflection rule, \( y = f(-x) \) is the reflection of \( y = f(x) \) over the \( y \)-axis. So the graph of \( y = 3^{-x} \) is the reflection of \( y = 3^{x} \) over the \( y \)-axis. The first option is reflection over \( x \)-axis (which would be \( y=-3^{x} \)), and the second option is incorrect as \( 3^{x}
eq3^{-x} \) in general (e.g., at \( x = 1 \), \( 3^{1}=3 \), \( 3^{-1}=\frac{1}{3} \)).

Answer:

The graphs are reflected across the \( y \)-axis.