Free for students · Ad-free · WCAG 2.1 AA Compliant · Accessibility

Exponential Functions

Lesson ~12 min read 8 MCQs

In simple terms: In simple terms, exponential functions are about quantities that grow or shrink by multiplying by the same number over and over, like compound interest or a bouncing ball losing height.

Why this matters

Imagine you post a short video of your dog doing a funny trick. In the first hour, it gets 100 views. A friend shares it, and in the second hour, you have 200 views. Then a popular account shares it, and by the third hour, you're at 400 views. It's doubling every hour! This isn't slow, steady, linear growth. This is explosive, rapid change.

This pattern is called exponential growth, and it's everywhere. It describes how savings grow with compound interest, how a viral trend spreads across the country, or even how a population of bacteria multiplies in a lab. Understanding the rules of these functions is key to making sense of rapid change in the world around you.

In this lesson, we'll break down the anatomy of exponential functions, learn to tell the difference between growth and decay, and master their key features so you can spot them with confidence.

Exponential growth shows rapid increase, unlike linear growth.

Concept overview

flowchart TD
    A[Start with f(x) = ab^x] --> B{Is b > 0 and b != 1?};
    B -- No --> C[Not an exponential function];
    B -- Yes --> D{Is b > 1?};
    D -- Yes --> E[Exponential Growth];
    D -- No --> F{Is 0 < b < 1?};
    F -- Yes --> G[Exponential Decay];
    E --> H[Graph is always increasing & concave up];
    G --> I[Graph is always decreasing & concave up];
    H --> J[lim x->inf is inf\nlim x->-inf is 0];
    I --> K[lim x->inf is 0\nlim x->-inf is inf];

Core explanation

Hello! Let's dive into one of the most powerful function types you'll encounter: exponential functions. They might seem intimidating with their curvy graphs, but their underlying rule is beautifully simple.

The Anatomy of an Exponential Function

The general form you need to know is: f(x) = ab^x

Let's break down the parts. Think of it like a recipe.

  • a is the initial value. This is your starting point, the value of the function when x = 0. Why? Because any number b raised to the power of 0 is 1, so f(0) = a * b^0 = a * 1 = a. On a graph, (0, a) is the y-intercept. If you start a savings account with $500, a is 500.
  • b is the base, or the growth/decay factor. This is the number you multiply by over and over. It controls the speed and direction of the change. For this to be an exponential function, we have two firm rules for b: b must be positive (b > 0) and b cannot be 1.
  • x is the input, usually time or the number of intervals. It tells you how many times to apply the multiplier b to the initial value a.

Growth vs. Decay: The Story of b

The value of b tells you the whole story.

Exponential Growth: This happens when b > 1. Every time x increases by 1, the output f(x) gets multiplied by a number bigger than 1, so it grows. Think of a savings account with a 5% annual interest rate. Each year, your money is multiplied by 1.05. Since b = 1.05 is greater than 1, your money grows. The function f(x) = 2^x is a classic example. When x goes from 2 to 3, the output goes from 2^2=4 to 2^3=8. It doubled.

Compare exponential growth (b>1) with exponential decay (0<b<1).

Exponential Decay: This happens when 0 < b < 1. Every time x increases by 1, the output f(x) gets multiplied by a fraction, so it shrinks. Think of a new car you bought for $30,000 that loses 15% of its value each year. Its value is multiplied by 1 - 0.15 = 0.85 annually. Since b = 0.85 is between 0 and 1, the car's value decays. The function g(x) = (1/2)^x is a perfect example. When x goes from 2 to 3, the output goes from (1/2)^2 = 1/4 to (1/2)^3 = 1/8. It was cut in half.

Table of values for an exponential decay function.

Key Characteristics You Must Know

1. Domain and Range: The domain (all possible x-values) of an exponential function f(x) = ab^x is all real numbers. You can plug in any real number for x—positive, negative, or zero. The range (all possible y-values), assuming a > 0, is all positive real numbers, (0, ∞). The function's graph gets incredibly close to the x-axis (y=0) but never actually touches or crosses it. That line, y=0, is a horizontal asymptote.

2. Constant Shape (Concavity): Look at the graphs for growth and decay. You'll notice they are smooth curves that are always bending in the same direction.

  • Exponential functions are always concave up (if a > 0). Imagine the curve is a bowl. It's always shaped to hold water.
  • Because they are always increasing (for growth) or always decreasing (for decay), and always concave up, they do not have any local maximums or minimums (extrema) on their full domain. There's no "peak" or "valley."
  • They also do not have any points of inflection. The concavity never changes.

3. Proportional Outputs: This is a fancy way of saying what we've already discovered: for equal steps in x, the y values are multiplied by the same factor. If you have a table of values and you see that for every Δx = 1, the y-value is multiplied by 3, you're looking at an exponential function with b=3. If it's multiplied by 0.5, you have b=0.5. This is the core difference from linear functions, where you add a constant amount for each step in x.

4. End Behavior (The Limits): What happens as x gets huge or tiny?

  • For Growth (f(x) = 2^x):
    • As x → ∞ (goes to the right), f(x) → ∞ (shoots up forever).
    • As x → -∞ (goes to the left), f(x) → 0 (gets closer and closer to the asymptote y=0).
  • For Decay (g(x) = (1/2)^x):
    • As x → ∞ (goes to the right), g(x) → 0 (gets closer to the asymptote y=0).
    • As x → -∞ (goes to the left), g(x) → ∞ (shoots up forever).

5. A Note on Transformations: What if you see a function like g(x) = ab^x + k? This is just our standard exponential function shifted vertically by k units. The horizontal asymptote moves from y=0 to y=k. If you were given a table of values for g(x) and noticed the differences weren't being multiplied by a constant factor, but the values minus k were, that's your clue! If g(x) - k shows that proportional change, then the underlying function is exponential. This is a subtle but important concept for identifying these functions in disguise.

Worked examples

Let's put this theory into practice with a couple of typical AP-style problems.

Example 1

Identifying Characteristics from an Equation

Problem: A function is given by f(x) = 150(0.75)^x. Identify the initial value, determine if the function represents growth or decay, and state its end behavior using limits.

Solution:

  1. 1
    Analyze the form
    The function is in the form f(x) = ab^x.
    • a is the coefficient in front: a = 150.
    • b is the base being raised to the power of x: b = 0.75.
  2. 2
    Identify the initial value
    The initial value is a.
    • Answer: The initial value is 150. This is the y-intercept, so the graph passes through the point (0, 150).
  3. 3
    Determine growth or decay
    We need to look at the base, b.
    • Here, b = 0.75. Since 0 < 0.75 < 1, the function represents exponential decay.
    • Why? With each increase in x, the output is multiplied by 0.75, making it 75% of its previous value. It's shrinking.
  4. 4
    State the end behavior
    Since this is a decay function (with a > 0), we know what its graph looks like. It starts high on the left and approaches the x-axis on the right.
    • As x approaches infinity (moves to the right), the function's value gets closer and closer to zero.
      • Limit: lim_{x→∞} 150(0.75)^x = 0.
    • As x approaches negative infinity (moves to the left), the function's value gets infinitely large.
      • Limit: lim_{x→-∞} 150(0.75)^x = ∞.
Example 2

Finding the Equation from a Table

Problem: The population of a town, P(t), is recorded in the table below, where t is the number of years since 2020. Determine if the population growth is exponential, and if so, find the function that models it.

t (years) P(t) (population)
0 12,000
1 12,600
2 13,230
3 13,891.5

Solution:

  1. Check for a common ratio: To see if it's exponential, we check if the output is multiplied by the same factor for each equal step in the input. Here, the time step is Δt = 1.

    • From t=0 to t=1: 12,600 / 12,000 = 1.05
    • From t=1 to t=2: 13,230 / 12,600 = 1.05
    • From t=2 to t=3: 13,891.5 / 13,230 ≈ 1.05
    • Yes! The ratio is constant. The population is multiplied by 1.05 each year. This confirms the behavior is exponential.

  2. Identify a and b:

    • The common ratio is our base, b. So, b = 1.05.
    • The initial value, a, is the population when t=0. From the table, this is 12,000. So, a = 12,000.

  3. Write the function: Now we plug a and b into the general form P(t) = ab^t.

    • Answer: The function is P(t) = 12,000(1.05)^t.

This model tells us the town started with 12,000 people and is growing by 5% per year.

Visualizing the decay of f(x) = 150(0.75)^x with its y-intercept and asymptote.
Population growth showing a constant ratio between consecutive P(t) values.

Try it yourself

Ready to try on your own? Don't worry about getting it perfect, just focus on applying the steps we covered.

Problem 1: A substance decays such that the amount remaining, A(t) in grams, after t hours is given by the function A(t) = 50(0.9)^t.

  • Is this growth or decay?
  • What is the initial amount of the substance?
  • What percentage of the substance decays each hour?

Problem 2: A scientist observes a bacterial culture. At the start (t=0), there are 800 bacteria. One hour later (t=1), there are 2400. Assuming the growth is exponential, find the function B(t) that models the number of bacteria after t hours.

Back to AP Precalculus Take a mock exam