Understanding exponential functions

An exponential function is any function where x is raised to a power, for example f(x) = 1.5 \times {2^{x}} + 10. For most of this blog I will concentrate on 2^x, but the same rules apply whatever the base of the power is.

2x is not the same as x2

They feel like the same sort of thing, don’t they, but they’re completely different functions. Look at the graphs: yes, both functions increase rapidly for x > 0, but they’re very different to the left of the y-axis – we’ll see why in a moment. Every time x increases by 1, x2 increases by 3, then 5, then 7 and so on. But 2x doubles each time, and so increases much faster (although this happens further on in the graph). Work out 210 and 10and compare.

Graph 1

Powers of 2

2^3 = 2 \times 2 \times \ 22^5 = 2 \times 2 \times \ 2 \times \ 2 \times 2; but for powers such as 2^0 and 2^{1.5}, repeated multiplication doesn’t make sense. What you need to know to calculate any power is:

a^1 = a
a^0 = 1
a^{-n} = \frac{1}{{{a^n}}}

So, for example:  2^1 = 2, 4^0 = 1, and  3^{-2} = \frac{1}{{{3^2}}} = \frac{1}{9}

Also, fractional powers will always give roots, since a^\frac{1}{n} =  nth root of a. This means, for example, that 2^{1.5} = 2^\frac{3}{2} = \sqrt{2^3} = \sqrt{8} \approx 2.83.

So now we see that all powers of 2 (or any other number) have a value, not just the whole number powers, and that for negative powers 2^x < 1.

The graphs of y = 2x and y = 2x

Graph 2

The graphs of all exponential functions of the form yax have these essential properties:

  • They cross the y-axis at (0, 1) (since a0 = 1)
  • The y values multiply at the same rate for each fixed increase in x. This is called exponential growth. For example, the number of cells in an organism growing exponentially may double every 10 hours.
  • At the left-hand end of the graph, the y values get ever closer to zero without reaching it – try working out 2-100 on your calculator. The x-axis is called an asymptote of the graph.

The graphs of exponential functions of the form y = a-x have the same properties except that they reduce rather than increase. An example of this is radioactive decay whereby the mass halves over a fixed time period, known as the half-life.

More complex exponential functions

If a function has an exponential term in it, the graph will show exponential growth, and there will be an asymptote. If the function is modelling a real-life situation then the x-axis usually represents time (so the function will use the variable t rather than x) and the  y-intercept will represent a “start point”, ie when t = 0.

For a function of the form ykatthe start value will be k, since when t = 0, a0 = 1. So for y=60 \times 1.2^t, the function will have value 60 when t = 0, and will multiply by 1.2 every time t increases by 1. Its graph will look similar to y = 2x, with the x-axis as an asymptote, the intercept at (0, 60), and displaying exponential growth.

If there is a constant term as well, this will simply shift the whole graph up or down, changing both the y-intercept and the asymptote. Thus, for y=20+60 \times 1.2^t the y-intercept will be at (0, 80) and the asymptote will be y = 20.

Graph 3

Occasionally you might come across an exponential function which, instead of increasing from, or decreasing to, an asymptote, actually decreases from or increaseto an asymptote. For example, a Mathematical Studies paper set a question about the electrical charge in a mobile phone battery whilst it was being charged, and used the model C(t)=2.5-2^{-t}t being the time in hours. The graph of this function looks like this:

Graph 4

As you might expect, the rate of charge slows as the amount of charge reaches its maximum. In the model, it never quite gets there – in practice it probably will. With functions like these, the easiest way to work out the intercept and the asymptote is to put appropriate values into your calculator. In this case:

  • For the intercept, put in t = 0, to get C = 0.5 (ie the amount of charge was already 0.5 when the phone was plugged into the charger)
  • For the asymptote, put in a large value of t, say 1000, to get C = 2.5 (the larger t is, the closer 2-t gets to zero).

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