Graphs of Exponentials
Every question with a fully worked solution.
- Graphs of Exponentials - Video - Exponential Graphs Watch
Theory
In an exponential function such as \(y = 2^x\), the pronumeral is in the index. This Year 10A maths page covers tables of values, the \(y\)-intercept, asymptote and range, translations and reflections, finding the rule \(y = k \times a^x\) from two points, solving by writing both sides with the same base, and growth and decay models.
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.
The larger the base, the steeper the curve for \(x > 0\): \(4^x\) is above \(2^x\) there. For \(x < 0\) the order swaps, because \(4^{-1} = \dfrac{1}{4}\) is less than \(2^{-1} = \dfrac{1}{2}\).
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).
(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 \(4\).
| \(y\text{-intercept}\) | \(\) | \(\boldsymbol{(0, 1)}\) |
| \(\text{shape}\) | \(\) | increasing, faster and faster |
| \(\text{asymptote}\) | \(\) | \(\boldsymbol{y = 0}\) (as \(x\) becomes very negative) |
(i) Find \(a\) and \(k\).
(ii) Find \(x\) when \(y = 1250\).
(i) Substitute both points:
| \(ka\) | \(=\) | \(10\quad (1)\) |
| \(ka^3\) | \(=\) | \(250\quad (2)\) |
Divide (2) by (1) so that \(k\) cancels:
| \(a^2\) | \(=\) | \(25\) |
| \(a\) | \(=\) | \(5\quad (a \gt 0)\) |
| \(k\) | \(=\) | \(10 \div 5 = 2\) |
So \(y = \boldsymbol{2 \times 5^x}\).
(ii)
| \(2 \times 5^x\) | \(=\) | \(1250\) |
| \(5^x\) | \(=\) | \(625 = 5^4\) |
| \(x\) | \(=\) | \(\mathbf{4}\) |
(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.
(i) \(5^{2x - 1} = 125\)
(ii) \(8^x = \dfrac{1}{4}\).
(iii) Find where the curves \(y = 9^x\) and \(y = 3^{x + 3}\) meet.
Write both sides as powers of the same base, then set the indices equal.
(i) \(125 = 5^3\):
| \(2x - 1\) | \(=\) | \(3\) |
| \(x\) | \(=\) | \(\mathbf{2}\) |
(ii) \(8 = 2^3\) and \(\dfrac{1}{4} = 2^{-2}\):
| \(2^{3x}\) | \(=\) | \(2^{-2}\) |
| \(3x\) | \(=\) | \(-2\) |
| \(x\) | \(=\) | \(\mathbf{-\dfrac{2}{3}}\) |
(iii) \(9^x = (3^2)^x = 3^{2x}\):
| \(3^{2x}\) | \(=\) | \(3^{x + 3}\) |
| \(2x\) | \(=\) | \(x + 3\) |
| \(x\) | \(=\) | \(3\) |
| \(y\) | \(=\) | \(9^3 = 729\) |
The curves meet at \(\mathbf{(3,\ 729)}\).
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.