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Linear Functions

Introduction

A linear function describes a relationship in which the output changes at a constant rate as the input changes. In Grades 9–10, you will usually work with functions written in the form $y = m x + b$ or $f (x) = m x + b$ , where $m$ is the slope and $b$ is the y-intercept. The graph is a straight line.

In some higher-level mathematics, the word "linear" is used more narrowly for functions of the form $f (x) = m x$ that pass through the origin. In this course, linear function follows the common school-algebra convention and includes the form $f (x) = m x + b$ .

Linear functions connect algebra, functions, coordinate graphs, rates, patterns, equations, and real-world modeling. You will move among equations, tables, graphs, and verbal descriptions and learn how each representation expresses the same relationship.

The diagram above summarizes the central structure of $y = m x + b$ . As you work through the course, keep asking two questions: What is the rate of change? and What is the starting value?

This Math Antics video introduces the basic idea of linear functions and provides a useful visual overview before you begin the detailed sections below.

Learning Goals

By the end of this aiMOOC, you should be able to:

Slope and rate of change: Calculate and interpret slope from graphs, tables, equations, and pairs of points.
Y-intercept: Identify and interpret the starting value or initial value of a linear model.
Linear equations: Write equations in slope-intercept form and convert information from other representations.
Graphing: Plot linear functions accurately and connect graphical features to algebraic meaning.
Modeling: Build and evaluate linear models for realistic situations and explain the limits of a model.
Systems: Recognize that the intersection of two lines represents a shared solution.

Core Ideas of Linear Functions

Functions and Constant Rate of Change

A function assigns exactly one output to each allowed input. For a linear function, equal changes in the input produce equal changes in the output. This property is called a constant rate of change.

For example, consider $f (x) = 2 x + 4$ . If $x$ increases by 1, then $f (x)$ always increases by 2. If $x$ increases by 3, then $f (x)$ increases by 6. The constant rate of change is 2.

x	f(x)
-2	0
-1	2
0	4
1	6
2	8

The equal output differences show that the relationship is linear.

A straight-line graph is the visual signature of a linear relationship over its domain.

Slope: Measuring Steepness and Direction

The slope tells you how quickly a line rises or falls. For two distinct points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , with $x_{1} \neq x_{2}$ ,

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

You can remember this as change in y divided by change in x, or rise over run.

The sign of the slope tells you the direction of the graph:

Positive slope: If $m > 0$ , the line rises from left to right.
Negative slope: If $m < 0$ , the line falls from left to right.
Zero slope: If $m = 0$ , the line is horizontal.
Undefined slope: A vertical line has undefined slope and is not the graph of a function $y = f (x)$ .

Worked example: The points $(2, 1)$ and $(6, 9)$ lie on a line. Its slope is

$m = \frac{9 - 1}{6 - 2} = \frac{8}{4} = 2$ .

This means that for every increase of 1 in $x$ , $y$ increases by 2.

This Math Antics video reviews slope as a geometric and numerical measure. Pause the video when examples appear and calculate each slope before the explanation continues.

The Y-Intercept: The Starting Value

In $y = m x + b$ , the number $b$ is the y-intercept. It is the output when $x = 0$ , so the graph crosses the y-axis at $(0, b)$ .

For $y = 3 x - 2$ , the y-intercept is -2. This gives the point $(0, - 2)$ .

In a real-world model, the y-intercept often represents an initial amount, fixed fee, starting distance, opening balance, or another value that exists before the changing part begins. However, you should always check whether $x = 0$ makes sense in the situation.

Slope-Intercept Form

The form $y = m x + b$ is especially useful because you can read the slope and y-intercept directly from the equation.

For the function $y = - \frac{1}{2} x + 3$ :

The slope is $- \frac{1}{2}$ .
The y-intercept is 3.
The graph passes through $(0, 3)$ .
From that point, moving 2 units right means moving 1 unit down.

To graph the function, plot the y-intercept first, then use the slope to find another point, and draw the straight line through the points.

This Khan Academy video focuses on slope-intercept form. Use it to compare the algebraic symbols $m$ and $b$ with the geometric features of the graph.

Finding an Equation from Two Points

Suppose a line passes through $(1, 4)$ and $(5, 12)$ .

First calculate the slope:

$m = \frac{12 - 4}{5 - 1} = 2$ .

Next substitute one point into $y = m x + b$ . Using $(1, 4)$ gives

$4 = 2 \cdot 1 + b$ ,

so $b = 2$ . The equation is therefore

$y = 2 x + 2$ .

You can check your result by substituting the other point. If $x = 5$ , then $y = 2 \cdot 5 + 2 = 12$ , so the second point also fits.

Finding an Equation from a Table

A table represents a linear function when the rate of change is constant for equal input steps.

Consider:

x	y
0	5
2	11
4	17
6	23

Every increase of 2 in $x$ produces an increase of 6 in $y$ . Therefore,

$m = \frac{6}{2} = 3$ .

Because the table contains the point $(0, 5)$ , the y-intercept is 5. The equation is $y = 3 x + 5$ .

If the table does not include $x = 0$ , find the slope first, then substitute any known point into $y = m x + b$ to solve for $b$ .

X-Intercept and Zeros

The x-intercept is the point where a graph crosses the x-axis. At this point, $y = 0$ . The corresponding x-value is also called a zero of the function.

For $y = 2 x - 6$ , set $y = 0$ :

$0 = 2 x - 6$ ,

so $x = 3$ . The x-intercept is $(3, 0)$ .

For a non-horizontal line $y = m x + b$ , the x-intercept is $x = - \frac{b}{m}$ .

Direct Proportion as a Special Case

A direct proportion has the form $y = k x$ . It is a special linear function with y-intercept 0, so its graph passes through the origin $(0, 0)$ .

For example, if one notebook costs $2.50 and there is no fixed fee, the total cost $C$ for $n$ notebooks is

$C = 2.5 n$ .

The constant of proportionality 2.5 is also the slope.

A relationship such as $y = 2.5 x + 4$ is linear but not directly proportional because it has a nonzero starting value.

Connecting Representations

Equation, Table, Graph, and Context

A strong understanding of linear functions means being able to translate among four representations:

Equation: A symbolic rule such as $y = 1.5 x + 12$ .
Table: Pairs of input and output values.
Graph: A straight line in the coordinate plane.
Context: A real situation in which slope and intercept have units and meaning.

Suppose a bike rental costs a fixed $12 plus $1.50 per hour. If $h$ is the number of hours and $C$ is the total cost, then

$C = 1.5 h + 12$ .

Here, the slope 1.5 means $1.50 per hour, while the y-intercept 12 means a $12 fixed starting fee.

When interpreting a model, always include units. A slope without units may hide the real meaning of the relationship.

Domain and Range in Context

The domain is the set of allowed input values. The range is the set of resulting output values.

In pure algebra, $y = 2 x + 3$ can often use all real numbers as inputs. In a real situation, the domain may be restricted. For example, a model for the number of tickets sold cannot sensibly use a negative number of tickets, and a model based on hourly measurements might use only nonnegative values within the observed time period.

A graph can look like an infinite line mathematically but represent only a limited segment in a practical context.

Comparing Linear Functions

Steeper, Increasing, and Decreasing Lines

When two nonvertical lines are drawn with the same axis scales, the line with the larger absolute value of slope is steeper.

For example:

$y = 4 x + 1$ is steeper than $y = 2 x + 1$ .
$y = - 5 x + 2$ decreases more steeply than $y = - x + 2$ .
$y = 3$ has slope 0 and is horizontal.

Do not compare steepness reliably from visual appearance alone if the graph uses different scales on the axes. Calculate the slope when accuracy matters.

Parallel and Perpendicular Lines

Distinct nonvertical parallel lines have the same slope but different y-intercepts. For example,

$y = 2 x + 1$ and $y = 2 x - 4$

are parallel.

For nonvertical and nonhorizontal perpendicular lines, the slopes are negative reciprocals. A line with slope 3 is perpendicular to a line with slope $- \frac{1}{3}$ .

This relationship links linear functions with coordinate geometry.

Intersections and Systems

Two linear functions may intersect at one point, never intersect because they are parallel, or coincide because they represent the same line. An intersection point satisfies both equations at the same time, so it is a solution of a system of linear equations.

Fehler beim Erstellen des Vorschaubildes:

The intersection of two graphs can represent a meaningful comparison. For example, if two phone plans have different starting fees and different rates per gigabyte, their intersection can represent the usage level at which the two plans cost the same.

Graphical solutions are useful for estimation and visualization, while algebraic methods can give exact coordinates.

Linear Models in the Real World

Building a Model

You can build a linear model when a situation has an approximately constant rate of change. A useful process is:

Identify the input and output variables and their units.
Determine the rate of change and use it as the slope.
Determine the output when the input is zero and use it as the intercept.
Write the equation.
Check the equation against known values.
Decide what input values make sense in context.

For example, a tank contains 120 liters of water and drains at 8 liters per minute. If $t$ is time in minutes and $V$ is volume in liters, then

$V = 120 - 8 t$ .

The slope -8 means the volume decreases by 8 liters per minute. The intercept 120 is the starting volume. The practical domain ends when the tank becomes empty, so the model should not be extended indefinitely.

Interpolation and Extrapolation

Interpolation means predicting within the range of observed or known input values. Extrapolation means predicting beyond that range.

Even when a line fits known data well, extrapolation can be risky because the real relationship may change. A constant rate that is reasonable for ten minutes may not remain reasonable for ten years.

When using a linear model, ask:

Does a constant rate make sense?
Is the prediction within a reasonable input range?
Are there physical, economic, or social limits that the equation ignores?
Are the data exact, measured, or approximate?

These questions help you use mathematical models responsibly.

Common Errors and How to Check Them

A correct method is easier to trust when you also know how to detect mistakes.

Slope: Keep the order of subtraction consistent in the numerator and denominator.
Y-intercept: Do not confuse the y-intercept with the x-intercept.
Graphing: Check that the plotted y-intercept is at $x = 0$ .
Rate of change: Include units when the problem has a real context.
Model validation: Substitute a known point into the equation to verify it.
Graph scale: Read axis intervals carefully before estimating coordinates.

A fast check for $y = m x + b$ is to ask whether the graph crosses the y-axis at $b$ and changes vertically by $m$ units for every horizontal change of 1 unit.

Interactive Tasks

Quiz: Test Your Knowledge

In the equation y = mx + b, what does m represent? (Slope) (!Y-intercept) (!X-intercept) (!Domain)

What is the y-intercept of y = -3x + 5? (5) (!-3) (!3) (!-5)

What is the slope through the points 2,1 and 6,9? (2) (!4) (!8) (!One half)

Which equation has slope 4 and y-intercept -2? (y = 4x - 2) (!y = -2x + 4) (!y = 4x + 2) (!y = -4x - 2)

What happens to a line from left to right when its slope is negative? (It decreases) (!It increases) (!It stays horizontal) (!It becomes vertical)

Which equation represents a direct proportion? (y = 3x) (!y = 3x + 1) (!y = x - 3) (!y = 3)

A table has x-values 0, 1, 2 and y-values 3, 5, 7. What is the slope? (2) (!3) (!5) (!7)

Which line is parallel to y = 3x + 1? (y = 3x - 7) (!y = -3x + 1) (!y = one third x + 1) (!y = -one third x - 7)

What is the x-intercept of y = 2x - 6? (3) (!-3) (!2) (!6)

In the cost model C = 1.5x + 12, what does 12 most naturally represent? (A fixed starting cost) (!The cost per unit) (!The slope of the graph) (!The maximum possible cost)

Memory Game

Slope	Change in output divided by change in input
Y-intercept	Output value when the input equals zero
X-intercept	Input value where the output equals zero
Direct proportion	Relationship whose graph passes through the origin
Parallel lines	Distinct lines that have equal slopes
Domain	Set of allowed input values

Drag and Drop

Match the correct terms.	Topic
y = 2x + 3	Positive slope and y-intercept 3
y = -x + 4	Negative slope and y-intercept 4
y = 5	Horizontal line with zero slope
y = 3x	Direct proportional relationship through the origin
y = one half x - 2	Slope one half and y-intercept negative two

...

Crossword Puzzle

Slope	What quantity measures change in y divided by change in x?
Intercept	What word names a point where a graph crosses an axis?
Origin	What is the point where both coordinate axes meet?
Increasing	What describes a line with positive slope from left to right?
Parallel	What describes distinct lines with equal slopes?
Function	What relation assigns exactly one output to each allowed input?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Slope Photo Walk: Find and photograph two safe examples of straight sloping objects or paths, sketch coordinate axes over each image, and describe which example appears to have the larger absolute slope.
Table Builder: Create a value table for y = 2x - 3 using at least six input values, plot the points, and explain how the constant difference in outputs reveals the slope.
Linear Function Poster: Design a one-page poster that labels slope, y-intercept, x-intercept, increasing, decreasing, and horizontal lines with your own examples.
Equation Story: Write a short real-world story that can be modeled by y = 4x + 10 and explain what both numbers mean in your situation.

Standard

Video Explanation: Record a two-minute teaching video that shows how to find the equation of a line from two points and includes a substitution check.
Rate Interview: Interview an adult about a job or activity that uses a constant rate, such as pay per hour, distance per time, or cost per item, and translate one example into a linear equation.
Measurement Experiment: Collect at least six measurements from a simple repeated process, such as total mass after adding equal objects, graph the data, and decide whether a linear model is reasonable.
Graphing Technology Comparison: Graph the same three linear functions by hand and with a graphing tool, compare the results, and write a short reflection on what technology helps you notice.

Advanced

Model Critique: Find a real data set that looks approximately linear, create a linear model, interpret its slope and intercept, and explain where extrapolation would become unreliable.
Piecewise Linear Investigation: Create a situation in which the rate changes once, represent it with two connected linear pieces, and explain why one single linear function would not model the whole situation well.
Systems Investigation: Invent two real-world cost models with different slopes and intercepts, graph them, find their intersection algebraically, and explain the practical meaning of that point.
School Measurement Project: Measure a safe linear relationship around your school or home, such as distance traveled at steady walking intervals, document your method, build a model, and evaluate sources of measurement error.

Learning Assessment

Equation from Evidence: Given two points and a short context, derive a linear equation, justify the slope calculation, interpret the intercept, and verify the equation using both points.
Error Analysis: Examine an incorrect solution in which the slope signs or subtraction order have been mishandled, identify the exact error, and repair the reasoning without restarting the whole problem.
Representation Transfer: Convert one linear relationship from a verbal description to an equation, then to a table and graph, and explain how the same slope and intercept appear in every representation.
Model Comparison: Compare two linear models for the same situation, decide which has the greater starting value and greater rate of change, and determine when one model overtakes the other.
Model Reasonableness: Evaluate a linear extrapolation beyond the original data range, identify assumptions that may fail, and propose a more defensible prediction range.
New Context Challenge: Create and solve a linear-function problem from a new context, include units throughout, and explain why a linear model is suitable rather than merely stating the equation.

Evidence of Learning

Knowledge: You can explain slope, y-intercept, x-intercept, constant rate of change, domain, range, direct proportion, and the meaning of a linear model.
Skills: You can calculate slope, write and graph equations, translate among representations, solve for intercepts, compare lines, and check solutions.
Reasoning: You can justify why a relationship is or is not linear, interpret parameters with units, and identify limitations of extrapolation.
Products: You can produce accurate graphs, tables, equations, written explanations, data-based models, posters, or short instructional videos.
Transfer: You can recognize and use linear relationships in unfamiliar mathematical, scientific, financial, technical, or everyday contexts.

OERs on the Topic

The English Wikipedia article below offers an additional reference for the mathematical idea of a linear function and related terminology.

Linked Learning Areas

Linear functions connect algebraic rules with geometric representations and real-world rates. The most important linked areas are functions, equations, coordinate geometry, proportional reasoning, systems, and mathematical modeling.

Linear Functions

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