Some things change by adding or subtracting. If you add $100 to a savings account each month, its growth is constant and predictable. This is linear growth. Other things change by multiplying. A single bacterium splits into two, those two split into four, and those four become eight. This is exponential growth, and it's driven by multiplication.

An exponential function has the variable in the exponent. Its general form looks like this:

Let's break that down:

• $a$ is the initial amount. It's the value of the function when $x=0$. For our bacteria, the initial amount might be 1.
• $b$ is the growth factor. It's the number you multiply by in each step. Since our bacteria double, the growth factor is 2.
• $x$ is the variable, often representing time or the number of steps. The key is that it's the exponent.

The value of the base, $b$, tells you whether you're looking at growth or decay.

When $b$ is greater than 1, we get exponential growth. The function's value increases, and it does so faster and faster over time. Think of a viral video. The more people who have seen it, the more people they can share it with, causing the view count to explode.

When $b$ is between 0 and 1, we get exponential decay. The function's value decreases, with the biggest drops happening at the beginning. A common example is radioactive decay. A substance like carbon-14 has a half-life, meaning half of it decays over a certain period. The remaining amount keeps getting halved, shrinking toward zero but never quite reaching it.

Notice how both curves pass through the point $(0, 1)$. This is because any number raised to the power of 0 is 1. Also, see how the decay curve gets closer and closer to the x-axis ($y=0$) but never touches it. This line is called an asymptote.

There's a special number that often appears as the base in exponential functions: Euler's number, denoted by the letter $e$. It's an irrational number, like $\pi$, and its value is approximately 2.718.

Why is $e$ so important? It's the base for continuous growth. Imagine a savings account. You could get interest calculated once a year. Or once a month. Or once a day. What if it were calculated every second? Every millisecond? The logical end point is to compound the interest continuously, at every single moment. The number $e$ is the key to modeling that kind of smooth, continuous change.

Because of this, you'll see $e$ used in many scientific and financial models, from population growth to the decay of radioactive materials. The function $f(x) = e^x$ is often called the natural exponential function.

Let's check your understanding of these new concepts.

Exponential functions are a powerful tool for describing how the world works, from the microscopic level of bacteria to the global spread of information.