You're likely familiar with linear functions, where things change by a constant amount. If you save $10 every week, your savings grow linearly. Add 10, add 10, add 10. Simple and steady.

Exponential functions are different. They change by a constant factor, not a constant amount. Instead of adding the same number over and over, you're multiplying by it. This creates a pattern of growth or decay that speeds up over time.

Notice the difference. The linear function plods along, increasing by 2 each time. The exponential function starts slower but quickly overtakes it by doubling at each step. This multiplicative growth is also known as a constant percentage change. For example, multiplying by 1.05 is the same as increasing by 5%.

The key feature of an exponential function is that the variable is in the exponent. This is different from functions like $f(x) = x^2$, where the variable is the base. The standard form for an exponential function looks like this:

Let's break down the roles of $a$ and $b$.

The value of $a$ gives you your starting point. It's what you have before any of the multiplicative change begins.

The base $b$ determines how the function behaves. If $b$ is greater than 1, the function shows growth. If $b$ is between 0 and 1, it shows decay. We don't use a base of 1 because $1^x$ is always 1, which is just a constant function.

So, when you see a function where the variable is sitting up in the exponent, you know you're dealing with an exponential relationship.