Exponential Graphs
Master exponential graphs for NSW Year 12 Mathematics Standard 2. In this topic you recognise an exponential relationship across its equation, table, ordered pairs and graph, then sketch \(y=a^{x}\) and \(y=a^{-x}\) for a base \(a>0\).
You will learn to read growth and decay curves, interpret the \(y\)-intercept \((0,1)\) that every \(y=a^{x}\) shares, and describe how the curve approaches the \(x\)-axis — the horizontal asymptote \(y=0\) — a core Standard 2 skill for modelling real-world growth and decay.
Theory
An exponential relationship has the variable in the index, \(y=a^{x}\) with \(a>0\). This Year 12 Standard 2 (NSW) guide shows how to recognise one across an equation, table, ordered pairs or graph, graph \(y=a^{x}\) (growth) and \(y=a^{-x}\) (decay), and interpret the \(y\)-intercept \((0,1)\) and the \(x\)-axis asymptote.
An exponential relationship has the variable in the index (the exponent), so its equation has the form \(y=a^{x}\) where the base \(a>0\). This is different from \(y=x^{a}\) (a power) or \(y=ax\) (linear), where the variable is the base or a simple multiplier.
The graph of \(y=a^{x}\) is a smooth curve that always passes through the \(y\)-intercept \((0,1)\), because \(a^{0}=1\) for any \(a>0\). When \(a>1\) the curve increases (growth); the curve \(y=a^{-x}\) decreases (decay) and is the reflection of \(y=a^{x}\) in the \(y\)-axis.
You can spot an exponential in a table or set of ordered pairs: the \(y\)-values change by a constant ratio for equal steps in \(x\), and that ratio is the base \(a\). The \(x\)-axis \((y=0)\) is a horizontal asymptote — the curve gets ever closer but never touches it, so there is no \(x\)-intercept.
An exponential relationship has the variable as the index:
Reflecting in the \(y\)-axis (replacing \(x\) with \(-x\)) gives the decay curve:
Every such curve shares the same \(y\)-intercept, because a base to the power zero is one:
How to recognise and graph an exponential
- Check the form. If the variable is in the index, \(y=a^{x}\) (with \(a>0\)), it is exponential — not \(y=x^{a}\) or \(y=ax\).
- From a table or pairs, test the ratio of successive \(y\)-values. A constant ratio confirms an exponential, and that ratio is the base \(a\).
- Plot the \(y\)-intercept \((0,1)\) and one or two more points from \(y=a^{x}\).
- Join with a smooth curve: \(a>1\) rises (growth); \(y=a^{-x}\) falls (decay).
- Show the asymptote. The curve approaches the \(x\)-axis \((y=0)\) but never crosses it.
An exponential has the variable in the index (exponent).
| \(y=6x\) | \(\) | \(\text{linear}\) |
| \(y=x^{6}\) | \(\) | \(\text{power (variable is base)}\) |
| \(y=6x-2\) | \(\) | \(\text{linear}\) |
| \(y=6^{x}\) | \(\) | \(\text{index is } x\ \checkmark\) |
The exponential relationship is \(y=6^{x}\).
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(y\) | 1 | 4 | 16 | 64 |
Test the ratio between successive \(y\)-values.
| \(\dfrac{4}{1}=\dfrac{16}{4}=\dfrac{64}{16}\) | \(=\) | \(4\) |
| \(\text{constant ratio}\) | \(\Rightarrow\) | \(\text{exponential}\) |
| \(\text{intercept }1,\ \text{base }4\) | \(\Rightarrow\) | \(y=4^{x}\) |
| \(y\) | \(=\) | \(4^{4}=256\) |
Base \(a=4\), equation \(y=4^{x}\), and \(y=256\) at \(x=4\).
Use \(a^{0}=1\) for the intercept, then read the trend from the base.
| \(\text{At } x=0:\ y\) | \(=\) | \(4^{0}=1\) |
| \(\text{y-intercept}\) | \(=\) | \((0,1)\) |
| \(\text{At } x=1:\ y\) | \(=\) | \(4^{1}=4\) |
| \(\text{as } x\to\text{large }-ve:\ y\) | \(\to\) | \(0\) |
Intercept \((0,1)\); \(y=4\) at \(x=1\); the curve approaches the \(x\)-axis (asymptote \(y=0\)).
Read each feature from the two curves and the index laws.
| \(y=5^{-x}\) | \(\) | \(\text{decreasing (decay)}\) |
| \(5^{0}=5^{-0}\) | \(=\) | \(1\) |
| \(\text{common point}\) | \(=\) | \((0,1)\) |
| \(x\to -x\) | \(\Rightarrow\) | \(\text{reflection in } y\text{-axis}\) |
\(y=5^{-x}\) decreases; both pass through \((0,1)\); they are reflections in the \(y\)-axis.
Common pitfalls
Frequently asked questions
What makes a relationship exponential?
The variable is in the index (the exponent), as in y equals a to the power x with base a greater than 0. In a table, the y-values change by a constant ratio for equal steps in x, and that ratio is the base.
Why does every graph of y = a^x pass through (0,1)?
Because any positive base raised to the power 0 equals 1. So when x is 0, y is 1 for every base a, which means all these curves share the y-intercept (0,1).
What is the difference between y = a^x and y = a^(-x)?
y equals a to the power x increases as x increases (growth) when a is greater than 1, while y equals a to the power minus x decreases (decay). The decay curve is the reflection of the growth curve in the y-axis, and both still pass through (0,1).
Does an exponential graph ever touch the x-axis?
No. For a base greater than 0, a to the power x is always positive, so the curve never reaches the x-axis. The x-axis (y = 0) is a horizontal asymptote: the curve gets ever closer to it but never crosses it, so there is no x-intercept.
How do you find the base from a table of values?
Divide each y-value by the one before it, using equal steps in x. If the ratio is the same every time, the relationship is exponential and that constant ratio is the base a, giving the equation y equals a to the power x.
Is y = x^2 an exponential relationship?
No. In y equals x squared the variable is the base and the index is a fixed number, so it is a power (quadratic) relationship. An exponential has the variable in the index instead, like y equals 2 to the power x.