2.2 Change in Linear and Exponential Functions

AP Precalculus · AP (Advanced Placement) · 4 min read · Updated: 7 Sept 2026

Welcome to 2.2: Change in Linear and Exponential Functions

In the previous chapter, we looked at patterns in sequences (lists of numbers). Now, we are stepping into the world of functions, where we look at how variables change together continuously. Understanding whether a situation is linear or exponential is one of the most important skills in AP Precalculus. It helps us predict everything from how fast a bank account grows to how quickly a medicine leaves the bloodstream.

Don't worry if these terms sound a bit technical! At their heart, linear and exponential functions are just two different ways of "growing" or "shrinking." One uses addition, and the other uses multiplication.

1. Linear Functions: The "Equal Addition" Rule

A function is linear if, for equal changes in the input $x$, there is a constant change in the output $y$. We call this constant change the common difference or the rate of change.

Think of it like walking: every step you take adds the exact same distance to your total. If you take 3 steps, you move 3 feet. If you take another 3 steps, you move another 3 feet. This is "additive" growth.

Key Characteristics of Linear Change:

The Pattern: $f(x+k) - f(x)$ is always the same value for a fixed interval $k$.
The Operation: You are adding (or subtracting) a fixed amount every time.
The Visual: On a graph, this creates a perfectly straight line.

Quick Example: Imagine a plant that grows exactly 2 inches every week.
Week 1: $5$ inches
Week 2: $7$ inches ($+2$)
Week 3: $9$ inches ($+2$)
Because we add the same amount each week, this is a linear function.

2. Exponential Functions: The "Equal Multiplication" Rule

A function is exponential if, for equal changes in the input $x$, the output $y$ changes by a constant factor (or a constant ratio). Instead of adding the same amount, we are multiplying by the same amount.

Think of this like a viral video. If one person shows it to two people, and those two show it to two more, the total views aren't just adding up; they are doubling.

Key Characteristics of Exponential Change:

The Pattern: The ratio $\frac{f(x+k)}{f(x)}$ is always the same value for a fixed interval $k$.
The Operation: You are multiplying by a fixed number (the base) every time.
The Visual: On a graph, this creates a curve that gets steeper and steeper (growth) or flattens out toward zero (decay).

Quick Example: Imagine a population of bacteria that doubles every hour.
Hour 1: $100$ bacteria
Hour 2: $200$ bacteria ($\times 2$)
Hour 3: $400$ bacteria ($\times 2$)
Because we multiply by the same factor each hour, this is an exponential function.

3. Comparing the Two (The "Cheat Sheet")

When you are looking at a table of data on the AP Exam, use this simple test to decide which function type you are seeing:

Feature	Linear Function	Exponential Function
Equal intervals of $x$ lead to...	Constant Differences in $y$	Constant Ratios in $y$
Math Relationship	$f(x+k) = f(x) + \text{constant}$	$f(x+k) = f(x) \cdot \text{constant}$
Key Word	"Per" or "Each" (e.g., \$5 per hour)	"Percent" or "Factor" (e.g., grows by 5%)

Note: For a refresher on how these relate to sequences, see Chapter 2.1 where we discussed Arithmetic (Linear) and Geometric (Exponential) patterns.

4. Identifying the Pattern: Step-by-Step

If you are given a table of values and asked to identify the function type, follow these steps:

Step 1: Check the $x$ values

Ensure the $x$ values are changing by the same amount (e.g., $x = 1, 2, 3...$ or $x = 10, 20, 30...$). If they aren't, you'll need to calculate the "rate" carefully!

Step 2: Test for Linear (Subtraction)

Subtract consecutive $y$ values: $y_2 - y_1$, $y_3 - y_2$, and so on.
If the results are all the same, it is Linear.

Step 3: Test for Exponential (Division)

Divide consecutive $y$ values: $\frac{y_2}{y_1}$, $\frac{y_3}{y_2}$, and so on.
If the results are all the same, it is Exponential.

Common Mistake: Don't just look at the first two numbers! Sometimes the first two might look like they could be either. Always check at least three points to confirm the pattern holds.

5. Why This Matters for the AP Exam

On the AP Precalculus exam, you will often be asked to justify your choice of a model. You cannot just say "it looks like a line." You must use the language of "change."

Correct Justification for Linear: "The function $f$ is linear because over equal intervals of $x$, the output $f(x)$ changes by a constant amount."
Correct Justification for Exponential: "The function $g$ is exponential because over equal intervals of $x$, the output $g(x)$ changes by a constant factor."

Did you know? Even if a constant factor is a fraction (like $1/2$), it is still exponential! This is often called "exponential decay." For example, if a radioactive substance loses half its mass every year, the ratio is a constant $0.5$.

Summary Key Takeaway:

To tell the difference between linear and exponential functions, look at how the outputs ($y$) change when the inputs ($x$) increase by a fixed step. If you add the same number, it’s linear. If you multiply by the same number, it’s exponential.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What defines a linear function in terms of its rate of change?

A function is linear if, for equal changes in the input $x$, there is a constant change (common difference) in the output $y$.

How is the pattern of change represented for a linear function over a fixed interval $k$?

$f(x+k) - f(x) = c$, where $c$ is a constant value.

What defines an exponential function in terms of its rate of change?

A function is exponential if, for equal changes in the input $x$, the output $y$ changes by a constant factor (or constant ratio).

How is the pattern of change represented for an exponential function over a fixed interval $k$?

The ratio $\frac{f(x+k)}{f(x)}$ is always the same constant value for a fixed interval $k$.

What is the "test" for identifying a linear function from a table of values?

Check for constant differences by subtracting consecutive $y$ values ($y_2 - y_1, y_3 - y_2$) for equal intervals of $x$.

What is the "test" for identifying an exponential function from a table of values?

Check for constant ratios by dividing consecutive $y$ values ($\frac{y_2}{y_1}, \frac{y_3}{y_2}$) for equal intervals of $x$.

How should you formally justify that a function is linear on the AP Exam?

State that the function is linear because over equal intervals of input, the output changes by a constant amount.

How should you formally justify that a function is exponential on the AP Exam?

State that the function is exponential because over equal intervals of input, the output changes by a constant factor.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practise This Topic

More AP Precalculus chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now

Feature	Linear Function	Exponential Function
Equal intervals of \(x\) lead to...	Constant Differences in \(y\)	Constant Ratios in \(y\)
Math Relationship	\(f(x+k) = f(x) + \text{constant}\)	\(f(x+k) = f(x) \cdot \text{constant}\)
Key Word	"Per" or "Each" (e.g., \$5 per hour)	"Percent" or "Factor" (e.g., grows by 5%)