Question

which graph represents a reflection of $f(x) = 6(0.5)^x$ across the $x$-axis?

Explanation:

Step1: Define reflection over x-axis

A reflection of $f(x)$ across the $x$-axis gives the new function $g(x) = -f(x)$.

Step2: Substitute the given function

Substitute $f(x)=6(0.5)^x$ into the reflection formula:
$g(x) = -6(0.5)^x$

Step3: Analyze key points of $f(x)$

For $f(x)=6(0.5)^x$:

• When $x=0$, $f(0)=6(0.5)^0=6$
• As $x\to+\infty$, $f(x)\to0$; as $x\to-\infty$, $f(x)\to+\infty$

This is a decreasing exponential curve above the $x$-axis, matching the top graph.

Step4: Analyze key points of $g(x)$

For $g(x)=-6(0.5)^x$:

• When $x=0$, $g(0)=-6(0.5)^0=-6$
• As $x\to+\infty$, $g(x)\to0$; as $x\to-\infty$, $g(x)\to-\infty$

This is an increasing exponential curve below the $x$-axis, matching the middle graph.

Answer:

The middle graph (the second one, with the curve below the x-axis passing through (0, -6)) represents the reflection.