What Makes a Function Exponential

In this lesson you will: Tell an exponential table from a linear one by dividing, not subtracting; Write f(x)=a* b^x from a starting value and a growth or decay factor; Connect an exponential function to the geometric sequence it extends.

Linear adds. Exponential multiplies.

Two savings habits. Kiana puts $8 a week into a jar that already holds $5. Devon starts with $5 and somehow doubles his money every week. For the first week they look similar. By week four they are not close.

The difference is in the operation. Kiana's totals go up by the same amount each week. Devon's go up by the same factor. To spot which one a table is showing, subtract consecutive outputs; if that gets you nowhere, divide them.

The form, and what each letter does

An exponential function can always be written as a starting value times a repeated factor.

Put x=0 into f(x)=a* b^x. Since b^0=1, you get f(0)=a. So a is the value before any multiplying has happened — the starting amount, the initial population, the price on day zero.

The base b is the factor. It must be positive and it must not be 1. If b=1 nothing ever changes and the function is the constant a. A negative base would make the outputs flip sign every step, which is not a growth or decay pattern at all.

Which way it bends is decided by b

If b>1, each step multiplies by something larger than one, so the outputs climb. That is growth.

If 0<b<1, each step multiplies by a proper fraction, so the outputs shrink. That is decay. A decaying quantity gets smaller and smaller without ever turning negative, because a positive number times a positive fraction is still positive.

Common mistake — Multiplying a and b before applying the exponent

Asked for f(4) when f(x)=3* 2^x, a student multiplies 3 by 2 first and computes 6^4=1296.

The reasoning is not lazy, it is just the wrong order. The two numbers are sitting next to each other, multiplication feels like the obvious first move, and nothing on the page shouts otherwise.

But the order of operations settles it: the exponent applies to 2 and only to 2. Work out 2^4=16 first, then multiply by 3. The answer is 48, not 1296.

The same slip turns 100* 3^t into 300^t, which at t=2 is off by a factor of a hundred.

How to catch it: Substitute x=0. A correct exponential returns its starting value: 3* 2^0=3* 1=3. The merged version returns 6^0=1. If your expression does not give back the starting amount at x=0, you combined a and b when you should have left them apart.