which graph represents a reflection of $f(x) = 6(0.5)^x$ across the $x$-axis?
which graph represents a reflection of $f(x) = 6(0.5)^x$ across the $x$-axis?
Answer
Explanation:
Step1: Recall Reflection Rule
Reflecting a function ( y = f(x) ) across the ( x )-axis gives ( y = -f(x) ). So for ( f(x)=6(0.5)^x ), the reflected function is ( g(x)=-6(0.5)^x ).
Step2: Analyze Original Function
The original function ( f(x)=6(0.5)^x ) is an exponential decay function (since ( 0 < 0.5 < 1 )) with ( y )-intercept at ( x = 0 ): ( f(0)=6(0.5)^0 = 6 ), so it starts at ( (0, 6) ) and decreases towards ( y = 0 ) as ( x \to \infty ).
Step3: Analyze Reflected Function
The reflected function ( g(x)=-6(0.5)^x ) will have a ( y )-intercept at ( x = 0 ): ( g(0)=-6(0.5)^0=-6 ), so it starts at ( (0, -6) ) and as ( x \to \infty ), ( (0.5)^x \to 0 ), so ( g(x) \to 0 ) (approaches the ( x )-axis from below). Also, since ( 0.5^x ) is always positive, ( g(x) ) will be negative (below the ( x )-axis) for all ( x ), and as ( x \to -\infty ), ( (0.5)^x = 2^{-x} \to \infty ), so ( g(x) \to -\infty ) (but we focus on the shape: it's a reflection, so the curve should be a mirror image across the ( x )-axis of the original. The original is above the ( x )-axis, decreasing; the reflected should be below the ( x )-axis, and since the base function's shape (exponential) is preserved but flipped, we look for a graph with ( y )-intercept at ( -6 ) and curving towards the ( x )-axis from below. Looking at the options, the third graph (assuming left to right: first is original, second maybe growth, third is reflected with ( y )-intercept negative, curving up towards ( x )-axis? Wait, no: wait, ( g(x)=-6(0.5)^x ), when ( x ) increases, ( (0.5)^x ) decreases, so ( -6(0.5)^x ) increases (since multiplying by a negative and a decreasing positive makes it increase towards 0). So the reflected function is increasing (from ( -\infty ) as ( x \to -\infty ) to 0 as ( x \to \infty )) with ( y )-intercept at ( -6 ). So we need a graph that starts at ( (0, -6) ) and increases towards ( y = 0 ) (since as ( x ) increases, ( g(x) ) increases from ( -6 ) to 0). Looking at the four graphs, the third graph (the one with ( y )-intercept negative, curving upwards towards the ( x )-axis) should be the reflection.
Answer:
The third graph (the one in the middle - right position, with the curve below the ( x )-axis, starting at ( (0, -6) ) and curving towards the ( x )-axis as ( x ) increases).