Question

determine which graph represents a reflection across the x - axis of $f(x)=3(1.5)^x$.

Explanation:

Step1: Find reflected function

A reflection across the $x$-axis transforms $f(x)$ to $-f(x)$.
$g(x) = -3(1.5)^x$

Step2: Analyze original function behavior

For $f(x)=3(1.5)^x$: when $x=0$, $f(0)=3$; as $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to0^+$.

Step3: Analyze reflected function behavior

For $g(x)=-3(1.5)^x$: when $x=0$, $g(0)=-3$; as $x\to+\infty$, $g(x)\to-\infty$; as $x\to-\infty$, $g(x)\to0^-$. This matches the bottom graph, which has a $y$-intercept at negative $y$, decreases downward as $x$ increases, and approaches 0 from below as $x$ decreases.

Answer:

The bottom graph (the third graph shown) represents the reflection across the $x$-axis of $f(x)=3(1.5)^x$.