Introduction

Welcome! Today we’ll be looking at exponential functions, which are functions where an initial value is multiplied repeatedly to produce the output. These functions are different from linear or polynomial ones because they grow or decay at a constant rate of multiplication. They are often used to model real-world situations like population growth, interest, or decay, and understanding them is an important part of precalculus.

Essential Knowledge from the CED

According to the AP Precalculus Course and Exam Description (CED), these are the key ideas you’ll be expected to understand about exponential functions for the exam:

Exponential functions follow the form $f (x) = a (b)^{x}$ , where $a > 0$ . Exponential growth occurs when $b > 1$ and decay when $0 < b < 1$ .
Their domain includes all real numbers.
Graphs of exponential functions are always increasing or always decreasing, with no extrema or inflection points.
Recognizing the different types of transformations for exponential functions while retaining that the function remains exponential regardless of the transformation.
Writing limit statements for exponential functions while emphasizing that the end behavior depends on the base: the values either increase without bound or approach zero.

Introduction to the General Form of Exponential Functions

In AP Precalculus, you will need to recognize and work with the general form of an exponential function. This form looks different from functions you may have studied earlier, such as linear or quadratic functions, because the variable appears in the exponent instead of in the base. Understanding this setup is key, since it tells us whether the function models exponential growth or exponential decay.

The general form of an exponential function is: $f (x) = a (b)^{x}$ where a is the initial value (not equal to $0$ ), and $b$ is the base (greater than $0$ and not equal to $1$ ).

If $b > 1$ , the function represents exponential growth.
If $0 < b < 1$ , the function represents exponential decay.

Example 1: Identify growth or decay

Suppose we have the function $f (x) = 2 (3)^{x}$

Here, $a = 2$ and $b = 3$ . Since $b > 1$ , this function shows exponential growth.

Example 2: Identify growth or decay

Now consider $f (x) = 5 (\frac{1}{2})^{x}$

Here, $a = 5$ and $b = \frac{1}{2}$ . Since $0 < b < 1$ , this function shows exponential decay.

Additional Note

It is important to be able to write statements that describe the behavior of an exponential function. For example, consider

$f (x) = 3 (5)^{x}$

Since $a > 0$ and $b > 1$ , we can write the statement:

“This exponential function represents exponential growth because $a > 0$ and $b > 1$ .”

The Domain of Exponential Functions

Another important property of exponential functions is their domain. The domain refers to all possible input values, or $x$ -values, where the function gives a value (instead of undefined).

For exponential functions of the form $f (x) = a (b)^{x}$ , the domain is all real numbers. This means that no matter what value of $x$ you choose, positive, negative, or zero, you will always get a number back and an output that is defined.

Quick Check 1: Why this is true

Take $f (x) = 2 (4)^{x}$

If $x$ is $- 2$ , $0$ , or $5$ , the function produces a defined output every time.

Quick Check 2: Another case

Take $f (x) = 4 (\frac{1}{3})^{x}$

Even though the base is less than one, you can still plug in any value for $x$ , and the function will always give a defined output.

Looking at the graph of an exponential function, we see that it exists everywhere.

Figure 2.3.1: A graph of an exponential function. Made using desmos.com. — Figure 2.3.1: A graph of an exponential function. Made using **desmos.com**.

The Graphs of Exponential Functions

Graphs of exponential functions have a very consistent shape. They are either always increasing (when the base is greater than one) or always decreasing (when the base is between zero and one). Unlike polynomials, they never have extremas, so there are no maximums, minimums, or inflection points.

Since exponential functions do not have any inflection points, they will also always be concave up or concave down. A positive exponential function will always be concave up, and a negative exponential function will always be concave down.

Let’s look at an example!

The graph of the function $f (x) = 4^{x}$ is shown below. Since the base $b$ is greater than $1$ , this function represents exponential growth. As you can see, the graph is always increasing. There is no maximum, no minimum, and no inflection point anywhere on the graph. Since it is also positive, the function is always concave up throughout.

Figure 2.3.2: A function representing exponential growth. Made using desmos.com. — Figure 2.3.2: A function representing exponential growth. Made using **desmos.com**.

Transformations of Exponential Functions

Just like linear and quadratic functions, exponential functions can also be shifted, stretched, or reflected. These changes are called transformations, and they do not change the fact that the function is exponential.

Let’s start with our base function:

$f (x) = 2^{x}$

This is our parent exponential function. Now, let’s look at the transformations one at a time.

1. Vertical Shifts (Up or Down)

Adding or subtracting a number outside the function moves the graph up or down.
Example: $f (x) = 2^{x} + 3$

This graph looks like the base $f (x) = 2^{x}$ , but it is shifted upward by $3$ units.

2. Horizontal Shifts (Left or Right)

Adding or subtracting a number inside the exponent moves the graph left or right.
Example: $f (x) = 2^{x - 4}$

This graph has the same shape as $2^{x}$ , but it is shifted $4$ units to the right.

3. Vertical Stretches and Compressions

Multiplying by a constant outside the function makes the graph stretch or compress vertically.
Example (Stretch): $f (x) = 5 \cdot 2^{x}$

The factor of $5$ makes the graph rise more steeply, which is a vertical stretch.

Example (Compression): $f (x) = \frac{1}{4} \cdot 2^{x}$

The factor of $\frac{1}{4}$ reduces every output to one-fourth of its original value, making the graph flatter against the $x$ -axis. This is a vertical compression.

4. Reflections

If the coefficient in front is negative, the graph is reflected across the $x$ -axis.
Example: $f (x) = - 2^{x}$

The negative sign flips the graph upside down, but the exponential shape is still there.

5. Horizontal Stretches and Compressions

Multiplying the exponent by a constant changes how the function grows horizontally. The effect is the reciprocal of that number:
Example 1: $f (x) = 2^{2 x}$

The factor of $2$ in the exponent compresses the graph by $\frac{1}{2}$

Example 2: $f (x) = 2^{\frac{1}{2} x}$

The factor of $\frac{1}{2}$ in the exponent stretches the graph by $2$ .

Let’s see how a transformation looks on a graph!

Let’s start with the parent function $f (x) = 2^{x}$ graphed. This function goes through the point $(0, 1)$ and rises quickly as $x$ increases.

Figure 2.3.3: An exponential function. Made using **desmos.com**.

Now, let’s apply a reflection. Consider $f (x) = - 2^{x}$ . The negative sign in front flips the entire graph across the $x$ -axis. Instead of rising upward, the reflected graph now points downward, but it keeps the same exponential shape.

Figure 2.3.4: An exponential function reflected over the x-axis. Made using desmos.com. — Figure 2.3.4: An exponential function reflected over the x-axis. Made using **desmos.com**.

End Behavior of Exponential Functions

You have definitely been introduced to limit notation by now, and exponential functions are another place where this idea is very useful. The end behavior of an exponential function describes what happens to the outputs as $x$ becomes very large or very small. This behavior depends on the base of the function.

If the base is greater than $1$ :

As $x$ gets larger, the function increases without bound.
As $x$ gets smaller, the function gets closer and closer to zero.

If the base is between $0$ and $1$ :

As $x$ gets larger, the function gets closer and closer to zero.
As $x$ gets smaller, the function increases without bound.

However, keep in mind that if the base is less than $0$ , everything reverses:

As $x$ gets larger, the function gets closer and closer to zero.
As $x$ gets smaller, the function increases without bound.

If you forget this, these trends can also be explained with exponential growth and decay. An exponential function will increase as $x$ gets larger when the base is greater than one because as you multiply a number by a constant that is greater than one, the number will increase and vice versa.

Quick Check: Growth

For $f (x) = 3^{x}$

Figure 2.3.5: A graph of f(x)=3^x. Made using desmos.com. — Figure 2.3.5: A graph of f(x)=3^x. Made using **desmos.com**.

As you can see in the graph of $f (x) = 3^{x}$ , the curve keeps rising higher and higher as $x$ increases. We can express this as:

$x \to \infty lim f (x) = \infty$

On the other side, as $x$ moves further into the negatives, the graph gets closer and closer to zero. That can be written as:

$x \to \infty lim f (x) = 0$

Quick Check: Decay

For $f (x) = (\frac{1}{4})^{x}$

Figure 2.3.6: A graph of f(x)=0.25^x. Made using

desmos.com. — Figure 2.3.6: A graph of f(x)=0.25^x. Made using **desmos.com**.

But if we look at negative values of $x$ , the graph grows larger and larger without bound. That can be written as:

$x \to - \infty lim f (x) = \infty$

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