Welcome to Exponential Functions!

In our previous lessons, we looked at how sequences grow and compared linear change to exponential change. Now, we are diving deep into Topic 2.3: Exponential Functions. This is where we define the "rules of the game" for functions that grow (or shrink) by a constant multiplier. Whether it’s a population of bacteria doubling every hour or a new car losing value over time, exponential functions are the mathematical tools we use to describe these patterns.

Don't worry if these graphs look a bit different from the straight lines or parabolas you’ve seen before—by the end of these notes, you'll be able to spot an exponential function a mile away!

1. The General Form of an Exponential Function

An exponential function is a function where the input variable \(x\) is sitting up in the exponent. The standard analytical form is:

\(f(x) = a \cdot b^x\)

Let's break down what these letters actually mean:

  • \(a\) (The Initial Value): This is the value of the function when \(x = 0\). On a graph, this is the y-intercept. In a real-world scenario, this is your starting amount. (Note: In this course, we assume \(a \neq 0\)).
  • \(b\) (The Base): This is the constant ratio or growth factor. It tells us what we multiply by every time \(x\) increases by 1.
  • \(x\) (The Input): This is usually time or some other independent variable.

Important Rule: For a function to be exponential, the base \(b\) must be positive (\(b > 0\)) and usually we say \(b \neq 1\) (because \(1^x\) is just a flat line, which is boring!).

Key Takeaway:

In the function \(f(x) = a \cdot b^x\), the value \(a\) is where you start, and \(b\) is how you change.

2. Growth vs. Decay

The value of the base \(b\) determines the "vibe" of the graph. There are two main behaviors you need to recognize:

Exponential Growth (\(b > 1\))

When the base is greater than 1, the function values get larger as \(x\) increases. Example: \(f(x) = 3 \cdot (2)^x\). Here, the value doubles every time \(x\) goes up by 1. The graph starts flat on the left and shoots up rapidly to the right.

Exponential Decay (\(0 < b < 1\))

When the base is between 0 and 1 (like a fraction or a decimal), the function values get smaller as \(x\) increases. Example: \(f(x) = 10 \cdot (0.5)^x\). Here, the value is cut in half every time \(x\) goes up by 1. The graph starts high on the left and levels off toward the right.

Quick Review: If \(b = 1.25\), it's growth. If \(b = 0.75\), it's decay!

3. Graphical Characteristics

Exponential functions have very specific "personalities" when graphed. Here are the traits you need to know for the AP Exam:

  • Domain: The domain is all real numbers, written as \((-\infty, \infty)\). You can plug any value of \(x\) into an exponent!
  • Range: If \(a > 0\), the range is all positive real numbers, written as \((0, \infty)\). The graph stays above the x-axis.
  • Horizontal Asymptote: This is a huge concept! As \(x\) moves toward infinity or negative infinity (depending on growth or decay), the graph gets closer and closer to the line \(y = 0\) (the x-axis) but never actually touches it.
  • Intercepts: The y-intercept is always at the point \((0, a)\). There are no x-intercepts for the basic form \(f(x) = a \cdot b^x\).

Did you know? Even if you have a tiny fraction like \((\frac{1}{2})^{1,000,000}\), the result is still a positive number. That's why the graph never crosses below \(y = 0\)!

4. Comparing Parameters

On the AP Exam, you might be asked to compare two different exponential functions. Here is how to "read" them:

Example: Compare \(f(x) = 2 \cdot (3)^x\) and \(g(x) = 5 \cdot (1.5)^x\).

  • \(g(x)\) starts higher because its initial value (\(a = 5\)) is greater than \(f(x)\)'s initial value (\(a = 2\)).
  • \(f(x)\) grows faster because its base (\(b = 3\)) is larger than \(g(x)\)'s base (\(b = 1.5\)).
  • Eventually, the "faster" function (\(f\)) will always overtake the "slower" function (\(g\)), even if it started at a lower point!

5. Common Mistakes to Avoid

1. Multiplying \(a\) and \(b\) first: If you see \(f(x) = 3 \cdot 2^x\), DO NOT write \(6^x\). The order of operations (PEMDAS) says we must do exponents before multiplication.

2. Confusing Negative Bases: Remember, the base \(b\) must be positive. If you see \(f(x) = -2 \cdot 3^x\), the initial value is negative, but the base is still positive 3. This would be a graph that "grows" downward into negative numbers!

3. Asymptote Confusion: Don't forget that an asymptote is a line. Always write it as an equation: \(y = 0\), not just "0".

Summary Checklist

Before moving on to Topic 2.4: Exponential Function Manipulation, make sure you can:

  • Identify the initial value \(a\) and base \(b\) from an equation.
  • Determine if a function represents growth or decay based on \(b\).
  • State the domain, range, and horizontal asymptote.
  • Identify the y-intercept as the point \((0, a)\).