Question

which graph represents a reflection of f(x) = \frac{1}{3}(9)^x across the x - axis?

Explanation:

Step1: Recall reflection rule

When a function \(y = f(x)\) is reflected across the \(x -\)axis, the new function is \(y=-f(x)\).

Step2: Find the reflected function

Given \(f(x)=\frac{1}{3}(9)^{x}\), the reflected - function is \(y =-\frac{1}{3}(9)^{x}\). The original function \(y=\frac{1}{3}(9)^{x}\) is an exponential growth function (\(a=\frac{1}{3}>0\) and \(b = 9>1\)) passing through the point \((0,\frac{1}{3})\). The reflected function \(y =-\frac{1}{3}(9)^{x}\) is an exponential decay - like function (in the sense of its general shape) passing through the point \((0,-\frac{1}{3})\) and has the opposite \(y\) - values for the same \(x\) - values as the original function.

Answer:

The graph of \(y =-\frac{1}{3}(9)^{x}\) (the one with the opposite \(y\) - values compared to the graph of \(y=\frac{1}{3}(9)^{x}\) for the same \(x\) - values, passing through \((0,-\frac{1}{3})\) instead of \((0,\frac{1}{3})\)) represents the reflection of \(f(x)=\frac{1}{3}(9)^{x}\) across the \(x\) - axis. Since the provided graph of \(y = f(x)=\frac{1}{3}(9)^{x}\) is an increasing exponential function passing through \((0,\frac{1}{3})\), the reflected graph should be a decreasing - looking exponential function passing through \((0,-\frac{1}{3})\).