Graphs Of Exponential Functions
Understand the graphs of exponential functions for Queensland Year 11 Mathematical Methods (QCAA). An exponential graph shows a quantity multiplying by the same factor each step, giving the curve of growth or decay.
You will learn to sketch these curves, identify the intercept and asymptote, state the domain and range, and describe the effect of the parameters that shift and scale the graph — building intuition for growth, decay and logarithms.
Every question with a fully worked solution.
- Graphs Of Exponential Functions - Video - Graphs of exponential functions Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 2), an exponential function \(y=r^{x}\) with base \(r>0\) grows when \(r>1\) and decays when \(0
An exponential function has the variable in the index, \(y=r^{x}\), with a positive base \(r\). If \(r>1\) the graph shows exponential growth (rising to the right); if \(0
Every graph of \(y=a\,r^{x}\) passes through its \(y\)-intercept (put \(x=0\)) and hugs a horizontal asymptote — the line the curve approaches but never touches. For \(y=r^{x}\) the asymptote is \(y=0\); a vertical translation \(y=r^{x}+c\) lifts it to \(y=c\). The domain is all real \(x\); the range is \(y>c\).
The basic exponential function and its features:
After a vertical translation by \(c\):
How to read an exponential graph
- Shape: decide growth or decay from the base — \(r>1\) rises, \(0
- Asymptote: read the constant \(c\) to get the horizontal asymptote \(y=c\); draw it as a dashed line.
- Intercept and range: put \(x=0\) for the \(y\)-intercept, then state the range relative to the asymptote (\(y>c\) for an upright curve).
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(2^{0}\) |
| \(=\) | \(1\) |
So the \(y\)-intercept is \((0,\,1)\).
Asymptote — as \(x\) becomes large and negative, \(2^{x}\to 0\):
| \(\text{asymptote}\) | \(=\) | \(y=0\) |
Since the base \(2>1\), the curve rises to the right: exponential growth.
\(y\)-intercept \((0,1)\); asymptote \(y=0\); exponential growth.
Asymptote — the \(+3\) lifts \(y=0\) to:
| \(\text{asymptote}\) | \(=\) | \(y=3\) |
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(2^{0}+3\) |
| \(=\) | \(1+3\) | |
| \(=\) | \(4\) |
Range — the curve stays above its asymptote:
| \(\text{range}\) | \(=\) | \(y>3\) |
Asymptote \(y=3\); \(y\)-intercept \((0,4)\); range \(y>3\).
Base check — \(0<\tfrac13<1\), so the graph decays (falls to the right).
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(\left(\tfrac13\right)^{0}\) |
| \(=\) | \(1\) |
Asymptote — as \(x\to\infty\), \(\left(\tfrac13\right)^{x}\to 0\):
| \(\text{asymptote}\) | \(=\) | \(y=0\) |
Exponential decay; \(y\)-intercept \((0,1)\); asymptote \(y=0\); range \(y>0\).
Asymptote — as \(x\to-\infty\), \(2^{x}\to 0\), so \(y\to 4\):
| \(\text{asymptote}\) | \(=\) | \(y=4\) |
\(y\)-intercept — put \(x=0\):
| \(y\) | \(=\) | \(-2^{0}+4\) |
| \(=\) | \(-1+4\) | |
| \(=\) | \(3\) |
Shape — the leading minus reflects the curve below \(y=4\), so:
| \(\text{range}\) | \(=\) | \(y<4\) |
\(y\)-intercept \((0,3)\); asymptote \(y=4\); range \(y<4\).
Common pitfalls
Frequently asked questions
What is the horizontal asymptote of an exponential graph?
For \(y=r^{x}\) it is the line \(y=0\). A vertical translation \(y=r^{x}+c\) moves the asymptote to \(y=c\); the curve approaches it but never touches it.
How do you tell growth from decay?
Look at the base \(r\). If \(r>1\) the graph grows (rises to the right); if \(0
How do you find the y-intercept of y = r^x?
Substitute \(x=0\). Since \(r^{0}=1\), the graph of \(y=r^{x}\) has \(y\)-intercept \((0,1)\); for \(y=r^{x}+c\) it is \((0,1+c)\).
What are the domain and range of an exponential function?
The domain is all real numbers, \(x\in\mathbb{R}\). For an upright curve \(y=r^{x}+c\) the range is \(y>c\).
What does a minus sign do, as in y = -2^x?
It reflects the graph in the \(x\)-axis, so the curve sits below its asymptote. The base stays positive; only the whole power is negated.