Exponential Models (Growth & Decay)
Master exponential models of growth and decay for NSW Year 12 Mathematics Standard 2. In this topic you construct and analyse models of the form \(y=k\,a^{x}\) and \(y=k\,a^{-x}\), reading the initial value \(k\) off the \(y\)-intercept and using the base \(a\) as the growth or decay factor.
You will learn to decide growth \((a>1)\) from decay \((0<a<1)\), find the percentage change each period, evaluate a model at a given value, and explain the limitations of using an exponential model to predict real quantities such as populations, savings, medicine and depreciation far into the future — a core Standard 2 skill.
Theory
Exponential models \(y=k\,a^{x}\) and \(y=k\,a^{-x}\) describe quantities that change by a fixed factor each step. This Year 12 Standard 2 (NSW) guide shows how to read the initial value \(k\), use the growth or decay factor \(a\), find the percentage change per period, and explain the limitations of an exponential model.
An exponential model has the form \(y=k\,a^{x}\) (or \(y=k\,a^{-x}\)), where the variable sits in the exponent. It describes a quantity that is multiplied by the same factor \(a\) every time \(x\) increases by 1.
The number \(k\) is the initial value: because \(a^{0}=1\), putting \(x=0\) gives \(y=k\), so \(k\) is the \(y\)-intercept of the graph. The base \(a\) is the growth or decay factor.
If \(a>1\) the amount rises each step — exponential growth. If \(0<a<1\) the amount falls each step — exponential decay. A negative index form \(y=k\,a^{-x}\) equals \(k\left(\tfrac{1}{a}\right)^{x}\), so it also models decay. This is a core NSW Year 12 Mathematics Standard 2 skill.
The two exponential model forms are:
The initial value is read at \(x=0\), using \(a^{0}=1\):
The percentage change per step comes straight from the base \(a\):
How to build and read an exponential model
- Find \(k\) — the starting amount is the initial value and the \(y\)-intercept.
- Find \(a\) — a \(p\%\) rise gives \(a=1+\tfrac{p}{100}\); a \(p\%\) fall gives \(a=1-\tfrac{p}{100}\). From a table, divide consecutive \(y\)-values.
- Write the model \(y=k\,a^{x}\) (or \(y=k\,a^{-x}\)).
- Evaluate by working out the power first, then multiplying by \(k\).
- Analyse — decide growth or decay, interpret the intercept, and remember the model has limits far into the future.
The initial balance is the \(y\)-intercept \(k\); evaluate the power for \(t=10\).
| \(A(0)\) | \(=\) | \(4000\times(1.05)^{0}=\$4000\) |
| \(A(10)\) | \(=\) | \(4000\times(1.05)^{10}\) |
| \(\) | \(=\) | \(4000\times 1.6289\) |
| \(\) | \(=\) | \(\$6515.58\) |
| \(\text{rate}\) | \(=\) | \(1.05-1=5\%\) |
Substitute \(t=0\) then \(t=3\); the percentage lost is \(1-a\).
| \(D(0)\) | \(=\) | \(60\times(0.5)^{0}=60\text{ mg}\) |
| \(D(3)\) | \(=\) | \(60\times(0.5)^{3}\) |
| \(\) | \(=\) | \(60\times 0.125\) |
| \(\) | \(=\) | \(7.5\text{ mg}\) |
| \(\text{lost}\) | \(=\) | \(1-0.5=50\%\) |
A negative index means a reciprocal power; simplify \(3^{-2}\) first.
| \(y\) | \(=\) | \(500\times 3^{-2}\) |
| \(3^{-2}\) | \(=\) | \(\dfrac{1}{3^{2}}=\dfrac{1}{9}\) |
| \(y\) | \(=\) | \(500\times\dfrac{1}{9}\) |
| \(y\) | \(\approx\) | \(55.56\) |
Since \(3^{-x}=\left(\tfrac{1}{3}\right)^{x}\) and \(\tfrac{1}{3}<1\), this is decay.
A \(8\%\) rise means \(a=1.08\); build \(N=k\,a^{t}\) then evaluate.
| \(a\) | \(=\) | \(1+0.08=1.08\) |
| \(N\) | \(=\) | \(150\times(1.08)^{t}\) |
| \(N(12)\) | \(=\) | \(150\times 2.5182\) |
| \(N(12)\) | \(\approx\) | \(378\text{ customers}\) |
Limitation: a fixed \(8\%\) growth cannot continue forever — seating and the local market will slow it, so the model overestimates far ahead.
Common pitfalls
Frequently asked questions
What is an exponential model?
It is a relationship of the form y equals k times a to the power x, where the variable is in the exponent. The quantity is multiplied by the same factor a each time x increases by one, which makes it grow or decay much faster than a straight line.
What do k and a mean in y equals k a to the x?
k is the initial value, the amount when x is zero, and it is the y-intercept of the graph because a to the power zero is one. a is the growth or decay factor, the number you multiply by each step.
How do you tell growth from decay?
Look only at the base a. If a is greater than 1 the amount rises each step, which is exponential growth. If a is between 0 and 1 the amount falls each step, which is exponential decay. The size of k does not change this.
How do you find the percentage change per period?
For growth the increase is a minus 1 as a percentage, so a equals 1.05 means 5 percent growth. For decay the decrease is 1 minus a as a percentage, so a equals 0.8 means 20 percent decay each period.
What does y equals k a to the minus x mean?
A negative index is a reciprocal power, so y equals k a to the minus x is the same as k times one over a all to the power x. Because one over a is less than 1 when a is greater than 1, this form models decay.
What is a limitation of an exponential model?
A constant percentage growth cannot continue forever. Real limits such as space, food, resources or market size eventually slow the growth, so an exponential model tends to overestimate a quantity far into the future.