An exponential function is any function where *x* is raised to a power, for example . For most of this blog I will concentrate on , but the same rules apply whatever the base of the power is.

**2^{x}^{ }is not the same as x^{2}**

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, *x*^{2} increases by 3, then 5, then 7 and so on. But 2^{x} *doubles* each time, and so increases much faster (although this happens further on in the graph). Work out 2^{10} and 10^{2 }and compare.

**Powers of 2**

, ; but for powers such as and , repeated multiplication doesn’t make sense. What you need to know to calculate any power is:

So, for example: , , and

Also, fractional powers will always give roots, since *n*^{th} root of *a*. This means, for example, that .

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 .

**The graphs of y = 2^{x} and y = 2^{–x}**

The graphs of all exponential functions of the form *y* = *a ^{x} *have these essential properties:

- They cross the
*y*-axis at (0, 1)*(since a*^{0}= 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 *y* = *ka ^{t}, *the start value will be

*k*, since when

*t*= 0,

*a*

^{0}= 1. So for , 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*= 2

^{x}

*,*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 the *y*-intercept will be at (0, 80) and the asymptote will be *y* = 20.

Occasionally you might come across an exponential function which, instead of increasing from, or decreasing to, an asymptote, actually decreases *from* or increase*s **to *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 , *t* being the time in hours. The graph of this function looks like this:

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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