Some things change by adding a constant amount. If you save $5 every week, your savings grow linearly. Other things change by multiplying. Imagine a single bacterium that splits into two every hour. After one hour, you have 2. After two hours, you have 4. After three hours, 8. This pattern of change, based on repeated multiplication, is called exponential growth.

An exponential function describes this kind of relationship. It has a standard form:

Let's break down this formula:

• $f(x)$ is the output, or the final amount.
• $a$ is the initial value. It's the amount you start with when $x=0$.
• $b$ is the base of the function. It's the constant factor you multiply by. For an exponential function, the base $b$ must be a positive number and not equal to 1.
• $x$ is the exponent, which is often a measure of time or the number of intervals.

The value of the base, $b$, tells you everything you need to know about the function's direction. It determines whether the quantity is growing or shrinking.

When $b > 1$, we have exponential growth. Each step multiplies the total by a number greater than one, so the amount gets bigger, faster and faster over time. Think of an investment earning compound interest. The interest earns interest, and the investment snowballs.

For example, if a population of 100 birds grows by 20% each year, the growth factor is $1 + 0.20 = 1.2$. The function modeling this would be $P(t) = 100 \cdot (1.2)^t$.

When $0 < b < 1$, we have exponential decay. Multiplying by a fraction makes the total smaller. This models things like radioactive decay, where a substance loses a certain percentage of its mass over a fixed period, or the way a medication's concentration decreases in the bloodstream.

If a car bought for $20,000 loses 15% of its value each year, the decay factor is $1 - 0.15 = 0.85$. Its value could be modeled by the function $V(t) = 20000 \cdot (0.85)^t$.

The graphs of exponential functions have a distinct and recognizable curve. They start off changing slowly and then shoot up (growth) or plummet down (decay) dramatically.

Two features are critical for understanding these graphs:

1. The Y-Intercept: This is the point where the graph crosses the y-axis. For any function in the form $f(x) = a \cdot b^x$, this always happens at the point $(0, a)$. Why? Because any number raised to the power of 0 is 1, so $f(0) = a \cdot b^0 = a \cdot 1 = a$. This is the starting value.

2. The Horizontal Asymptote: Notice how both curves get incredibly close to the x-axis ($y=0$) but never actually touch it. This line is a horizontal asymptote. It's a value that the function approaches but never reaches as $x$ gets very large (for decay) or very small in the negative direction (for growth).

Understanding these functions is the first step toward modeling many real-world situations, from finance to biology.

Ready to check your understanding?

Now you have a solid foundation in what exponential functions are and how they behave.