Exponential Graphs (HSC SSCE Mathematics Advanced): Revision Notes

Exponential Graphs

Introduction

An exponential function is a special type of function where the variable appears in the exponent (the power). These functions have unique properties and create distinctive curved graphs that you'll learn to recognise and sketch. Understanding exponential functions is essential because they model many real-world situations, from population growth to radioactive decay.

Key terminology

Before exploring exponential graphs, you need to understand some important terms:

Exponential function: A function written in the form $y = a^x$, where $x$ is the variable and the base $a$ is a positive number.

Exponential growth: This occurs when a quantity increases at a rate that is proportional to its current value. The larger the quantity becomes, the faster it grows.

Exponential decay: This occurs when a quantity decreases at a rate that is proportional to its current value. As the quantity gets smaller, it decreases more slowly.

Asymptote: A line that a curve approaches but never actually touches, no matter how far the graph extends. For exponential functions, this is typically a horizontal line.

Domain: The complete set of input values ($x$-values) that are allowed in a function. For exponential functions, this includes all real numbers.

Range: The complete set of output values ($y$-values) that a function can produce.

Key features of exponential functions

The general form of an exponential function is $y = k(a^x)$, where $a > 0$, $a \neq 1$, and $k \neq 0$.

This function has several important characteristics:

Y-intercept: The graph crosses the $y$-axis at the point $(0, k)$. This is because when $x = 0$, we have $y = k(a^0) = k \times 1 = k$.

Horizontal asymptote: The line $y = 0$ (the $x$-axis) acts as a horizontal asymptote. As $x$ moves towards negative or positive infinity, the graph approaches this line but never touches it.

Domain: All real numbers. You can substitute any real number for $x$.

Range: All values where $y > 0$ (when $k > 0$). The function never produces negative values or zero.

Exponential growth

When the base $a$ is greater than 1 (that is, $a > 1$), the function shows exponential growth. As $x$ increases, the $y$-values increase at an accelerating rate.

Let's examine the function $y = 2^x$ as an example. Here's a table showing how the $y$-values change:

When we plot these points on a graph, we get a distinctive J-shaped curve:

The graph shows several key features:

• The curve passes through the point $(0, 1)$, which is the $y$-intercept
• As $x$ increases (moving right), the curve rises steeply
• As $x$ decreases (moving left), the curve approaches the $x$-axis but never touches it
• The horizontal asymptote is at $y = 0$

Worked example 1: Identifying features of $y = 2(3^x)$

Exponential decay

When the base $a$ is between 0 and 1 (that is, $0 < a < 1$), the function shows exponential decay. As $x$ increases, the $y$-values decrease and approach zero.

The function $y = k(a^{-x})$ is equivalent to $y = k\left(\frac{1}{a}\right)^x$. This form represents exponential decay and creates a graph that is a reflection of the growth function across the $y$-axis.

Worked example 2: Exploring $y = 3^x$

Worked example 3: Comparing growth and decay with $y = \left(\frac{1}{2}\right)^x$

Behaviour of exponential functions

The behaviour of exponential functions depends on the base $a$ and the scaling factor $k$.

For $y = k(a^x)$ where $k > 0$:

When $a > 1$ (exponential growth):

• As $x \to \infty$, $y \to \infty$ (the function increases without limit)
• As $x \to -\infty$, $y \to 0^+$ (the function approaches zero from above)

When $0 < a < 1$ (exponential decay):

• As $x \to \infty$, $y \to 0^+$ (the function approaches zero from above)
• As $x \to -\infty$, $y \to \infty$ (the function increases without limit)

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