Exponential Graphs of Practical Situations
Explore exponential graphs of practical situations for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). When an amount is multiplied by the same number each step — doubling, tripling or halving — its graph is a curve, not a straight line, like algae spreading across a pond.
You will draw an exponential graph from a table of values, find the starting value and the multiplier, tell growth from decay, and read the graph to make predictions — for example bacteria in a dish, an online post spreading, or a medicine leaving the body.
Theory
When a quantity is multiplied by the same number each step — doubling, tripling or halving — it is exponential. This NSW Year 12 Mathematics Standard 1 guide shows how to find the multiplier, read growth and decay curves, and tell exponential patterns from linear ones.
In an exponential situation the amount is multiplied by the same number each step — it doubles, triples or halves each hour, day or year. Plotting a table of values gives a smooth curve, not a straight line.
The starting value is the amount at time 0, where the curve meets the vertical axis. The number you multiply by each step is the multiplier.
Growth (multiplier more than 1, such as doubling) makes the curve rise more and more steeply. Decay (multiplier less than 1, such as halving) makes the curve fall, then flatten towards 0 — it gets closer and closer to 0 but never reaches it.
Each step multiplies by the same number:
Table test — divide each value by the one before it:
| \(t\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Growth \(N\) (\(\times 2\)) | 7 | 14 | 28 | 56 | 112 | |
| Decay \(M\) (\(\div 2\)) | 96 | 48 | 24 | 12 | 6 | 3 |
How to work with an exponential situation
- Find the starting value at time 0 (divide back if you only know later values).
- Test the table: divide each value by the one before. The same answer each time is the multiplier.
- Predict by continuing to multiply (or divide) step by step, or read across and down on the curve.
- Describe the graph: growth rises more and more steeply; decay falls and flattens towards 0.
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(y\) | 4 | 12 | 36 | 108 | ? |
Divide each value by the one before.
| \(12\div 4\) | \(=\) | \(3\) |
| \(36\div 12\) | \(=\) | \(3\) |
| \(108\div 36\) | \(=\) | \(3\) |
| \(y\) | \(=\) | \(108\times 3=324\) |
Same multiplier, so the pattern is exponential (growth). Multiplier 3; the missing value is 324.
Keep doubling from the start. For (b), go across from 40 to the curve, then down.
| \(\text{day } 0,1,2,3,4\) | \(\colon\) | \(5,\ 10,\ 20,\ 40,\ 80\) |
(a) 80 cm\(^2\); (b) after 3 days.
Keep halving from the start.
| \(t=1\colon\ M\) | \(=\) | \(200\div 2=100\) |
| \(t=2\colon\ M\) | \(=\) | \(100\div 2=50\) |
| \(t=3\colon\ M\) | \(=\) | \(50\div 2=25\) |
| \(t=4\colon\ M\) | \(=\) | \(25\div 2=12.5\) |
(a) 25 mg; (b) 12.5 mg; (c) a curve that falls and flattens towards 0 (exponential decay).
| \(t\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Table A | 2 | 5 | 8 | 11 |
| Table B | 2 | 6 | 18 | 54 |
A: \(5-2=8-5=11-8=3\), the same amount is added. B: \(6\div 2=18\div 6=54\div 18=3\), the same multiplier.
| \(\text{A next}\) | \(=\) | \(11+3=14\) |
| \(\text{B next}\) | \(=\) | \(54\times 3=162\) |
A is linear (next 14); B is exponential (next 162).
Common pitfalls
Frequently asked questions
What is an exponential graph in real life?
It shows an amount that is multiplied by the same number each step, such as bacteria doubling each hour or a medicine halving each hour. The graph is a curve, not a straight line.
How do you tell if a table is exponential or linear?
Divide each value by the one before it. If you get the same answer each time, the table is exponential and that answer is the multiplier. If instead the same amount is added each time, it is linear.
What is the difference between exponential growth and decay?
In growth the multiplier is more than 1, so the curve rises more and more steeply. In decay the multiplier is less than 1, so the curve falls and levels off towards 0.
Does exponential decay ever reach zero?
No. Halving keeps making the amount smaller, so the curve gets closer and closer to 0 but never reaches it or goes below it.
How do you find the starting value of an exponential pattern?
The starting value is the amount at time 0, where the curve meets the vertical axis. If you only know later values, divide back by the multiplier, for example 28, then 14, then 7.