Question

use what you know about reflections of functions to match the graph to the function rule.

\\(y = 3^x\\)

\\(y = 3^{-x}\\)

\\(y = -3^x\\)

Explanation:

Identify the base function

Using the Exponential Function Graphs knowledge point

$$ y = 3^x $$

This is an exponential growth function because the base \(3 > 1\). As \(x \to \infty\), \(y \to \infty\), and as \(x \to -\infty\), \(y \to 0\). The graph passes through \((0, 1)\) and \((1, 3)\). This matches curve \(b\).

Analyze the horizontal reflection

Using the Function Reflections knowledge point

$$ y = 3^{-x} $$

Replacing \(x\) with \(-x\) reflects the graph of \(y = 3^x\) horizontally across the \(y\)-axis. This results in an exponential decay curve that passes through \((0, 1)\) and \((-1, 3)\). This matches curve \(a\).

Analyze the vertical reflection

Using the Function Reflections knowledge point

$$ y = -3^{-x} $$

Multiplying the function \(y = 3^{-x}\) by \(-1\) reflects its graph vertically across the \(x\)-axis. All \(y\)-values become negative, so the graph passes through \((0, -1)\) and \((1, -3)\). This matches curve \(c\).

Answer:

Question 1

• \(y = 3^x\) matches graph <blank>b</blank>
• \(y = 3^{-x}\) matches graph <blank>a</blank>
• \(y = -3^{-x}\) matches graph <blank>c</blank>