Exponential Functions
Exponential Functions in NSW Year 10 Core maths studies graphs like \(y = 2^x\), where the variable is in the index (NESA outcomes MA5‑NLI‑C‑01 and MA5‑NLI‑C‑02, with moved curves from MA5‑NLI‑P‑01). These curves pass through \((0, 1)\), either grow faster and faster or decay towards zero, and approach the \(x\)-axis without ever touching it.
You will complete tables, sketch growth and decay curves such as \(y = 2^{-x}\), find points on exponential curves and their equations, and describe how \(y = 3^x - 1\) is moved from \(y = 3^x\). You will also model a numbat colony, cooling water, bacteria, medicine in the blood, the value of a unit in Newcastle and the populations of two towns. It is part of Year 10 Core Non-Linear Relationships.
Theory
In an exponential function such as \(y = 2^x\), the pronumeral is in the index. This Year 10 maths page shows how to draw exponential graphs from a table of values, their key features (the \(y\)-intercept and the asymptote), how to find an equation from points, and how exponential growth and decay model populations, prices and medicines.
An exponential function has the form \(y = a^x\), where the base \(a\) is a positive number other than \(1\) and \(x\) is the index (power). \(y = 10^x\) is exponential; \(y = x^{10}\) is not.
Since \(a^0 = 1\), every graph of \(y = a^x\) passes through \((0, 1)\). Its values are always positive.
An asymptote is a line that a curve gets closer and closer to without touching. For \(y = a^x\), the asymptote is the \(x\)-axis, \(y = 0\).
If \(a > 1\), the graph is increasing, and grows faster and faster: this is exponential growth. If \(0 < a < 1\), or the index is \(-x\) with \(a > 1\), the graph is decreasing: exponential decay.
Index facts used with exponentials, and the growth and decay model:
\(A_0\) is the starting amount. The multiplier \(r\) is \(1 +\) the growth rate (\(1.2\) for \(20\%\) growth) or \(1 -\) the decay rate (\(0.85\) for a \(15\%\) loss).
How to sketch \(y = a^x\)
- Make a table for \(x = -2\) to \(2\). Use \(a^{-1} = \dfrac{1}{a}\) and \(a^0 = 1\).
- Plot the points and mark the \(y\)-intercept \((0, 1)\).
- Draw a smooth curve that gets closer and closer to the asymptote but never touches or crosses it.
How to use a growth or decay model
- Identify the starting amount (put \(t = 0\)) and the multiplier.
- Substitute the value of \(t\). Use a calculator for the power, then multiply. Round only at the end.
- To find the time for a given amount, try whole-number values of \(t\) in a table (or write both sides as powers of the same base).
(i) Complete a table of values for \(y = 4^x\) for \(x = -2\) to \(2\).
(ii) Describe the graph.
(i) A negative index means a fraction: \(4^{-1} = \dfrac{1}{4}\) and \(4^{-2} = \dfrac{1}{16}\). Any number to the power \(0\) is \(1\).
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(y\) | \(\dfrac{1}{16}\) | \(\dfrac{1}{4}\) | \(1\) | \(4\) | \(16\) |
(ii) Each step of \(1\) to the right multiplies \(y\) by \(4\).
| \(y\text{-intercept}\) | \(\) | \(\boldsymbol{(0, 1)}\) |
| \(\text{shape}\) | \(\) | \(\text{increasing, faster and faster}\) |
| \(\text{asymptote}\) | \(\) | \(\boldsymbol{y = 0}\) \(\text{(as } x \text{ becomes very negative)}\) |
(i) The curve \(y = a^x\) passes through \((2, 36)\). Find \(a\).
(ii) The curve \(y = k \times a^x\) passes through \((0, 5)\) and \((1, 15)\). Find its equation.
(i) Substitute the point.
| \(a^2\) | \(=\) | \(36\) |
| \(a\) | \(=\) | \(6 \quad (a > 0)\) |
| \(y\) | \(=\) | \(\boldsymbol{6^x}\) |
(ii) At \(x = 0\), \(a^0 = 1\), so the \(y\)-intercept is \(k\). Then use the second point.
| \(5\) | \(=\) | \(k \times a^0 = k\) |
| \(15\) | \(=\) | \(5 \times a^1\) |
| \(a\) | \(=\) | \(3\) |
| \(y\) | \(=\) | \(\boldsymbol{5 \times 3^x}\) |
(i) Find the population after \(4\) years.
(ii) After how many whole years does it first exceed \(2000\)?
(i) Substitute \(t = 4\). Round to a whole number of wombats.
| \(P\) | \(=\) | \(500 \times 1.2^4\) |
| \(\) | \(=\) | \(500 \times 2.0736 = 1036.8\) |
| \(\) | \(\approx\) | \(\mathbf{1037}\) |
(ii) Make a table of values:
| \(t\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|
| \(P\) | \(1493\) | \(1792\) | \(2150\) |
It first exceeds \(2000\) after \(\mathbf{8}\) years.
Losing \(15\%\) leaves \(85\%\), so the multiplier is \(0.85\), which is less than \(1\).
| \(V\) | \(=\) | \(30\,000 \times 0.85^3\) |
| \(\) | \(=\) | \(30\,000 \times 0.614125\) |
| \(\) | \(=\) | \(\mathbf{\$18\,423.75}\) |
The value falls each year but never reaches \(\$0\): the graph has the asymptote \(V = 0\).
Common pitfalls
Frequently asked questions
What is an exponential function?
It is a function where the pronumeral is in the index, such as \(y = 2^x\). Each step of \(1\) in \(x\) multiplies \(y\) by the same number.
Why does y = a^x always pass through (0, 1)?
Any non-zero number to the power \(0\) equals \(1\), so when \(x = 0\), \(y = 1\).
What is an asymptote?
A line that a graph approaches more and more closely but never reaches. The graph of \(y = 2^x\) has the asymptote \(y = 0\).
What is the difference between exponential growth and decay?
In growth the multiplier is more than \(1\), so the amount increases faster and faster. In decay it is between \(0\) and \(1\), so the amount decreases towards \(0\).
Where are exponential functions used?
Population growth, compound interest, the spread of a virus, the cooling of a drink and how a medicine leaves the body are all modelled with exponentials.