Exponential functions: growth and decay

Just like linear functions, exponential functions come with their own equation. The equation for an exponential function is:

$$f(x) = b \times a^x$$

Here $b$ is where the graph crosses the y-axis (the start value), and $a$ is what we call the growth factor. The growth factor $a$ shows how many per cent the function grows or falls by, because it's defined as

$$a = 1 + r$$

where $r$ is the percentage the function rises or falls by, written as a decimal. The reason we add $1$ is that we have to put this percentage rise or fall on top of the $100\%$ we already have. So we can also write the equation of an exponential function like this instead:

$$f(x) = b \times (1 + r)^x$$

The bank example

The best example of an equation like this is when you have money sitting in a bank. Let's say you start with £$1000$. The bank has been really kind and gives you $2\%$ interest a year, which means that every year you get $2\%$ added on top. Our start value is the £$1000$, and our r-value is the $2\%$, which is $0.02$ as a decimal:

$$f(x) = 1000 \times (1 + 0.02)^x$$

What we have on the x-axis is how many years go by, since you get $2\%$ interest per year. When you put a particular number of years in for $x$, you find out how much money you have in the account after that many years. Say you'd like to know for $2$ years, so when $x$ is $2$. We put $2$ in for $x$:

$$f(2) = 1000 \times (1 + 0.02)^2 = 1040.40$$

So we have £$1040.40$ in the account after $2$ years at $2\%$ interest. The guide exponential growth and decay takes this step by step, including what happens when something falls by a percentage instead. The same formula is what the compound interest guide uses with $K_0$ and $K_n$.

Growth or decay: what a tells you

As we saw, an exponential function grows by percentages. When $a$ is bigger than $1$, the function grows: that's the blue curve on the graph above, which starts near zero on the left and shoots up to the right. When $a$ is between $0$ and $1$, the function falls by a percentage every step instead: that's the red curve, which starts high and flattens out towards zero. Both are exponential functions. The growth factor tells you which one you have, and the start value $b$ is where both curves cross the y-axis.

When is it not an exponential function?

An exponential function grows with percentages, and that's one of its properties. That's also why not every a-value is allowed. If $a$ were $0$, we'd see:

$$f(x) = b \times 0$$

which makes this function stop growing or falling by any percentage. It's just $0$, and that means it's no longer an exponential function. Not an especially exciting function either. If the b-value were $0$, we'd also get a fairly dull function, namely

$$f(x) = 0 \times a^x$$

So we're no longer dealing with an exponential function when $a$ or $b$ is $0$. The same actually applies if $a$ is $1$:

$$f(x) = b \times 1^x$$

We can see that whatever x-value we have, we just get $f(x) = b \times 1$, since $1$ raised to anything at all is $1$. When $a$ is $1$, we simply get $f(x) = b$, which again isn't exponential growth, because it's just a constant function.

More formally, you can say that an exponential function holds when $a$ and $b$ are greater than $0$, and $a$ isn't equal to $1$:

$$a > 0, \quad a \neq 1, \quad b > 0$$

What the graph does far to the left

Are we allowed to put negative x-values in? Let's try putting $-2$ into the function:

$$f(-2) = b \times a^{-2}$$

When we have a negative index, we can also write it as a fraction:

$$f(-2) = b \times \frac{1}{a^2}$$

We've rewritten $a^{-2}$ as $\frac{1}{a^2}$, which is one of the laws of indices. Nothing wrong with that. So whatever negative number we put in, it's allowed. If we'd used $-700$, we'd have got

$$f(-700) = b \times \frac{1}{a^{700}}$$

As you can see, that gives a very small fraction, because the denominator is enormous. So we can say: the bigger the negative number gets, the smaller the y-value gets (when the a-value is bigger than $1$). How do we know the y-value gets smaller? Well, the b-value gets multiplied by this fraction, and the smaller the fraction is, the smaller the whole thing gets too. A good idea is to go back and look at the figure with the two exponential functions at the start. Then you can see that it makes sense.

And because $a$ and $b$ are positive, the y-value always comes out positive. It can get very close to $0$, but it never actually becomes $0$. The y-value can get infinitely big, but it can never be $0$. That's why the graph flattens out towards the x-axis without ever touching it.

Exponential functions and power functions

One thing to be aware of is the difference between exponential functions and power functions. In $f(x) = b \times a^x$ the $x$ sits in the index. In a power function, $f(x) = b \times x^a$, it doesn't: there the $a$ is the index instead. The graphs can look alike, but they're different kinds of growth.

Common misunderstandings

• "$a = 0.02$ for $2\%$ growth." The growth factor is $1 + r$, so $2\%$ gives $a = 1.02$. The $1$ is the $100\%$ you already have.
• "$a = 1$ is fine." With $a = 1$ you get $f(x) = b$, a constant, not an exponential function.
• "Only $a > 1$ counts as exponential." $0 < a < 1$ is exponential too. That's decay: the function falls by a fixed percentage every step.
• "You can't put negative $x$ in." You can. $a^{-2}$ is just $\frac{1}{a^2}$.
• "Far to the left the graph reaches $0$." It gets as close as you like but never gets there. The y-value can never be $0$.

Related guides

• Exponential growth and decay
• Compound interest: the interest formula
• Power functions