Recognizing Linear Functions — I can recognize a linear...

SECTION 1

Where Did Linear Functions Come From?

People have noticed straight-line patterns for thousands of years. Ancient farmers tracked how crops grew over time. Merchants calculated how the price of goods went up with each item they bought. These everyday observations led mathematicians to study linear relationships — patterns where things change at a steady, constant pace.

Over centuries, thinkers figured out how to turn those patterns into equations and graphs. The idea of a constant rate of change became one of the most useful tools in all of math. Let's see how that story unfolded.

~300 BC

Euclid and Straight Lines

The Greek mathematician Euclid wrote Elements, a famous book that described lines and their properties. He showed that a straight line is the shortest path between two points.

~850 AD

Al-Khwarizmi and Algebra

The Persian mathematician Al-Khwarizmi developed early algebra. He wrote methods for solving equations that describe constant-rate problems, like dividing goods equally.

1637

Descartes Invents Coordinate Graphing

René Descartes created the coordinate plane (the x-y grid you use today!). For the first time, people could draw equations as pictures. Linear equations became straight lines on a graph.

1800s

Functions Get a Name

Mathematicians began using the word "function" to describe a rule that turns an input into exactly one output. Linear functions — functions that graph as straight lines — became a key building block of modern math.

So here's the big question this lesson answers: How can you tell whether a set of data is linear just by looking at a table or a graph? The secret lies in one idea — the rate of change must stay the same every single time.

SECTION 2

Core Ideas You Need to Know

Before we dive into examples, let's nail down the key vocabulary and ideas. There are only a few concepts to learn, and once you have them, everything else clicks into place.

Function

A function is a rule that gives you exactly one output for every input. Think of it like a vending machine: you press one button (input), and you get one snack (output).

Linear Function

A linear function is a function whose graph is a perfectly straight line. The word "linear" actually comes from "line."

Rate of Change

The rate of change tells you how much the output (y) changes every time the input (x) goes up by a fixed amount. It answers: "How fast is this changing?"

Constant Rate of Change

If the rate of change is the same between every pair of points, we call it constant. A constant rate of change is the signature of a linear function.

✦ KEY TAKEAWAY

Imagine walking up a staircase where every step is exactly the same height. That's a constant rate of change. Now imagine a staircase where some steps are tall and some are short — that's a non-constant rate of change. Linear functions are the even staircases — perfectly steady, every single time.

SECTION 3

Seeing Linear Functions on a Graph

The fastest way to spot a linear function is to look at its graph. If the graph is a perfectly straight line, you have a linear function. If the graph curves, bends, or wiggles, it is not linear. The diagram below shows both types side by side so you can compare them.

On the left, the points form a straight line — that's linear. The gold dashed lines show that every time x goes up by 1, y goes up by 2. On the right, the points form a curve — that's non-linear because the rate of change keeps getting bigger.

💡 Quick Tip

Grab a ruler or the edge of a piece of paper. Hold it up to the graph. If all the plotted points lie along your straight edge, the function is linear!

SECTION 4

The Math Behind Constant Rate of Change

You don't always have a graph in front of you. Sometimes you only have a table of numbers. The good news is there's a simple formula to test whether the rate of change is constant.

RATE OF CHANGE

Rate of Change = (change in y) ÷ (change in x) = (y₂ − y₁) ÷ (x₂ − x₁)

Pick any two rows in a table. Subtract the y-values (output) and subtract the x-values (input). Divide the first result by the second. If you get the same number every time, the function is linear.

Let's break that formula into plain English. The top part, change in y, tells you how much the output went up or down. The bottom part, change in x, tells you how much the input went up. Together they answer: "For every 1 step in x, how many steps does y move?"

LINEAR FUNCTION FORM

y = mx + b

m = the constant rate of change (also called the slope). b = the y-value when x is 0 (called the y-intercept). Every linear function can be written in this form.

📌 Remember

The rate of change is the same thing as the slope. In this lesson, we use "rate of change" because it describes what the number actually means. You'll see "slope" used a lot in future classes.

SECTION 5

Spotting Linear Functions in a Table

Tables are one of the most common ways you'll see data in class. The trick is simple: calculate the rate of change between every consecutive pair of rows. If the answer is always the same, it's linear. If even one pair gives a different answer, it is not linear. Below are two tables. Let's compare them.

Table A shows y increasing by 3 every time — that's a constant rate of change, so it's linear. Table B shows y increasing by 1, then 2, then 4, then 8. The rate of change is not constant, so it's non-linear.

Notice that in Table A the x-values go up by 1 each time. When x increases by 1, you only need to look at the differences in y. If those differences are all the same, you're done — it's linear! When the x-values don't increase by 1, you need to use the full formula: (change in y) ÷ (change in x).

SECTION 6

Worked Example: Is This Table Linear?

A pet sitter charges customers based on the number of hours she works. Here is a table of her earnings. Is this a linear function?

Pet sitter earnings

Hours (x)	Earnings in $ (y)
1	12
2	24
3	36
5	60
8	96

Solution: Check Every Pair of Rows

Step 1 — Find the rate of change between Rows 1 and 2Change in y = 24 − 12 = 12. Change in x = 2 − 1 = 1. Rate of change = 12 ÷ 1 = 12.

Rate = 12

Step 2 — Find the rate of change between Rows 2 and 3Change in y = 36 − 24 = 12. Change in x = 3 − 2 = 1. Rate of change = 12 ÷ 1 = 12. Same as before!

Rate = 12 ✓

Step 3 — Find the rate of change between Rows 3 and 4 (watch the x gap!)Here x jumps from 3 to 5. That's a gap of 2, not 1. Change in y = 60 − 36 = 24. Change in x = 5 − 3 = 2. Rate of change = 24 ÷ 2 = 12. Still 12!

Rate = 12 ✓

Step 4 — Find the rate of change between Rows 4 and 5Change in y = 96 − 60 = 36. Change in x = 8 − 5 = 3. Rate of change = 36 ÷ 3 = 12.

Rate = 12 ✓

Step 5 — Draw your conclusionEvery pair of rows gave us a rate of change of 12. Because the rate is constant, this is a linear function. The pet sitter earns $12 for every hour she works.

Linear function — constant rate of change = 12

SECTION 7

Linear vs. Non-Linear — Side by Side

It helps to see the differences between linear and non-linear functions all in one place. The table below summarizes the key features you should look for.

Feature	Linear Function	Non-Linear Function
Graph shape	Straight line	Curve, zigzag, or other shape
Rate of change	Always the same (constant)	Changes from pair to pair
Equation form	y = mx + b	Has x², x³, √x, or other non-linear terms
Table test	Differences in y (when x goes up by 1) are equal	Differences in y are not equal
Real-world example	Earning $10 per hour	A ball bouncing lower each time

✦ KEY TAKEAWAY

Think of a linear function like a car driving on cruise control — it covers the same distance every minute. A non-linear function is like a car speeding up or slowing down — the distance covered changes each minute. Checking for a constant rate of change is like checking whether cruise control is on.

SECTION 8

What Comes Next? Slope and Beyond

You've just learned how to recognize a linear function. In future math classes, you'll take this further. The constant rate of change you've been calculating has a more famous name: slope. You'll learn to write equations, graph lines from equations, and solve real-world problems using linear models. The table below shows how today's ideas connect to what's ahead.

What You Know Now	What You'll Learn Next
Constant rate of change	Slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁)
Recognize linear from a table	Write the equation y = mx + b from a table
Recognize linear from a graph	Graph a line using slope and y-intercept
Linear vs. non-linear	Quadratic, exponential, and other function families

Everything in algebra builds on this foundation. If you can spot a constant rate of change, you already understand the most important feature of linear functions. Nice work!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In your own words, what does it mean for a function to have a "constant rate of change"? Give an everyday example.

PROBLEM 2 — BASIC CALCULATION

A table shows x = 0, 1, 2, 3 and y = 4, 7, 10, 13. Is this a linear function? Show your work.

PROBLEM 3 — INTERMEDIATE

A table shows x = 2, 4, 6, 8 and y = 10, 16, 22, 28. Notice that x goes up by 2 each time, not 1. Is this linear? Calculate the rate of change.

PROBLEM 4 — APPLIED

A phone plan costs $20 per month plus $0.10 per text message. Your friend says the total monthly cost is a linear function of the number of texts sent. Do you agree? Explain using the idea of constant rate of change.

PROBLEM 5 — CRITICAL THINKING

A table shows x = 1, 2, 3, 4, 5 and y = 3, 6, 12, 24, 48. Marcus says it's linear because the y-values are always being multiplied by 2. Is Marcus correct? Explain your reasoning using the rate of change.

SUMMARY

Lesson Summary

A linear function is a function whose graph forms a straight line. The key feature that makes it linear is a constant rate of change. This means every time the input (x) increases by the same amount, the output (y) also increases (or decreases) by the same amount. You can check this in a table by computing (change in y) ÷ (change in x) between consecutive rows. If that number is the same for every pair, the function is linear.

On a graph, a linear function looks like a straight line — no curves or bends. The constant rate of change is also called the slope, and every linear function can be written in the form y = mx + b, where m is the slope and b is the y-intercept. Remember: linear functions add the same amount each time, while non-linear patterns (like exponential functions) multiply by the same amount each time.