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{{T}}
[[Category:English]]
[[Category:Linear Functions]]

{{BR}}
= 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 <math>y=mx+b</math> or <math>f(x)=mx+b</math>, where <math>m</math> is the slope and <math>b</math> 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 <math>f(x)=mx</math> that pass through the origin. In this course, '''linear function''' follows the common school-algebra convention and includes the form <math>f(x)=mx+b</math>.

Linear functions connect [[English:Algebra|algebra]], [[English:Function (mathematics)|functions]], [[English:Cartesian coordinate system|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.

[[File:Graph describing a linear function.svg|500px|frameless|center]]

The diagram above summarizes the central structure of <math>y=mx+b</math>. As you work through the course, keep asking two questions: '''What is the rate of change?''' and '''What is the starting value?'''

{{#ev:youtube|https://www.youtube.com/watch?v=MXV65i9g1Xg|500|center}}

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

{{BR}}
== Learning Goals ==

By the end of this aiMOOC, you should be able to:
# [[English:Slope|Slope and rate of change]]: Calculate and interpret slope from graphs, tables, equations, and pairs of points.
# [[English:Y-intercept|Y-intercept]]: Identify and interpret the starting value or initial value of a linear model.
# [[English:Linear equation|Linear equations]]: Write equations in slope-intercept form and convert information from other representations.
# [[English:Graph of a function|Graphing]]: Plot linear functions accurately and connect graphical features to algebraic meaning.
# [[English:Mathematical model|Modeling]]: Build and evaluate linear models for realistic situations and explain the limits of a model.
# [[English:Systems of linear equations|Systems]]: Recognize that the intersection of two lines represents a shared solution.

{{BR}}
= Core Ideas of Linear Functions =

{{BR}}
== Functions and Constant Rate of Change ==

A [[English:Function (mathematics)|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 <math>f(x)=2x+4</math>. If <math>x</math> increases by 1, then <math>f(x)</math> always increases by 2. If <math>x</math> increases by 3, then <math>f(x)</math> increases by 6. The constant rate of change is 2.

{| class="wikitable" style="margin:auto"
! x
! f(x)
|-
| -2
| 0
|-
| -1
| 2
|-
| 0
| 4
|-
| 1
| 6
|-
| 2
| 8
|}

The equal output differences show that the relationship is linear.

[[File:Graph of the function f(x) = 2x + 4.svg|450px|frameless|center]]

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

{{BR}}
== Slope: Measuring Steepness and Direction ==

The [[English:Slope|slope]] tells you how quickly a line rises or falls. For two distinct points <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>, with <math>x_1\neq x_2</math>,

<math>m=\frac{y_2-y_1}{x_2-x_1}</math>.

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

[[File:Diagram of gradient of linear function graph.svg|420px|frameless|center]]

The sign of the slope tells you the direction of the graph:
# [[English:Positive slope|Positive slope]]: If <math>m>0</math>, the line rises from left to right.
# [[English:Negative slope|Negative slope]]: If <math>m<0</math>, the line falls from left to right.
# [[English:Zero slope|Zero slope]]: If <math>m=0</math>, the line is horizontal.
# [[English:Undefined slope|Undefined slope]]: A vertical line has undefined slope and is not the graph of a function <math>y=f(x)</math>.

'''Worked example:''' The points <math>(2,1)</math> and <math>(6,9)</math> lie on a line. Its slope is

<math>m=\frac{9-1}{6-2}=\frac{8}{4}=2</math>.

This means that for every increase of 1 in <math>x</math>, <math>y</math> increases by 2.

{{#ev:youtube|https://www.youtube.com/watch?v=rpMu98yRk40|500|center}}

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.

{{BR}}
== The Y-Intercept: The Starting Value ==

In <math>y=mx+b</math>, the number <math>b</math> is the '''y-intercept'''. It is the output when <math>x=0</math>, so the graph crosses the y-axis at <math>(0,b)</math>.

For <math>y=3x-2</math>, the y-intercept is -2. This gives the point <math>(0,-2)</math>.

[[File:Y-intercept line slope.svg|420px|frameless|center]]

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 <math>x=0</math> makes sense in the situation.

{{BR}}
== Slope-Intercept Form ==

The form <math>y=mx+b</math> is especially useful because you can read the slope and y-intercept directly from the equation.

For the function <math>y=-\frac{1}{2}x+3</math>:
# The slope is <math>-\frac{1}{2}</math>.
# The y-intercept is 3.
# The graph passes through <math>(0,3)</math>.
# 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.

{{#ev:youtube|https://www.youtube.com/watch?v=IL3UCuXrUzE|500|center}}

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

{{BR}}
== Finding an Equation from Two Points ==

Suppose a line passes through <math>(1,4)</math> and <math>(5,12)</math>.

First calculate the slope:

<math>m=\frac{12-4}{5-1}=2</math>.

Next substitute one point into <math>y=mx+b</math>. Using <math>(1,4)</math> gives

<math>4=2\cdot1+b</math>,

so <math>b=2</math>. The equation is therefore

<math>y=2x+2</math>.

You can check your result by substituting the other point. If <math>x=5</math>, then <math>y=2\cdot5+2=12</math>, so the second point also fits.

{{BR}}
== Finding an Equation from a Table ==

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

Consider:

{| class="wikitable" style="margin:auto"
! x
! y
|-
| 0
| 5
|-
| 2
| 11
|-
| 4
| 17
|-
| 6
| 23
|}

Every increase of 2 in <math>x</math> produces an increase of 6 in <math>y</math>. Therefore,

<math>m=\frac{6}{2}=3</math>.

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

If the table does not include <math>x=0</math>, find the slope first, then substitute any known point into <math>y=mx+b</math> to solve for <math>b</math>.

{{BR}}
== X-Intercept and Zeros ==

The '''x-intercept''' is the point where a graph crosses the x-axis. At this point, <math>y=0</math>. The corresponding x-value is also called a [[English:Zero of a function|zero]] of the function.

For <math>y=2x-6</math>, set <math>y=0</math>:

<math>0=2x-6</math>,

so <math>x=3</math>. The x-intercept is <math>(3,0)</math>.

For a non-horizontal line <math>y=mx+b</math>, the x-intercept is <math>x=-\frac{b}{m}</math>.

{{BR}}
== Direct Proportion as a Special Case ==

A direct proportion has the form <math>y=kx</math>. It is a special linear function with y-intercept 0, so its graph passes through the origin <math>(0,0)</math>.

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

<math>C=2.5n</math>.

The constant of proportionality 2.5 is also the slope.

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

{{BR}}
= Connecting Representations =

{{BR}}
== Equation, Table, Graph, and Context ==

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

# [[English:Equation|Equation]]: A symbolic rule such as <math>y=1.5x+12</math>.
# [[English:Table (information)|Table]]: Pairs of input and output values.
# [[English:Graph of a function|Graph]]: A straight line in the coordinate plane.
# [[English:Mathematical model|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 <math>h</math> is the number of hours and <math>C</math> is the total cost, then

<math>C=1.5h+12</math>.

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.

{{BR}}
== Domain and Range in Context ==

The [[English:Domain of a function|domain]] is the set of allowed input values. The [[English:Range of a function|range]] is the set of resulting output values.

In pure algebra, <math>y=2x+3</math> 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.

{{BR}}
= Comparing Linear Functions =

{{BR}}
== 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:
# <math>y=4x+1</math> is steeper than <math>y=2x+1</math>.
# <math>y=-5x+2</math> decreases more steeply than <math>y=-x+2</math>.
# <math>y=3</math> 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.

{{BR}}
== Parallel and Perpendicular Lines ==

Distinct nonvertical [[English:Parallel (geometry)|parallel lines]] have the same slope but different y-intercepts. For example,

<math>y=2x+1</math> and <math>y=2x-4</math>

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 <math>-\frac{1}{3}</math>.

This relationship links linear functions with [[English:Analytic geometry|coordinate geometry]].

{{BR}}
== 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 [[English:Systems of linear equations|system of linear equations]].

[[File:Intersection point of two linear functions.svg|420px|frameless|center]]

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.

[[File:Linear Function Graph.svg|500px|frameless|center]]

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

{{BR}}
= Linear Models in the Real World =

{{BR}}
== 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 <math>t</math> is time in minutes and <math>V</math> is volume in liters, then

<math>V=120-8t</math>.

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.

{{BR}}
== 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.

{{BR}}
== Common Errors and How to Check Them ==

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

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

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

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

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

{{E}}
<br>

{{MC}}

'''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)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| 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
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! 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
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| 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?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Linear+Functions </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
A school-level linear function can often be written as { y = mx + b }. The symbol m represents the { slope }. The symbol b represents the { y-intercept }. A positive slope makes the graph { increasing }. A negative slope makes the graph { decreasing }. A direct proportion has a y-intercept of { zero }. The slope between two points is change in y divided by { change in x }. A linear model should be used only over a { reasonable domain }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Slope Photo Walk|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.
# [[English:Table Builder|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.
# [[English:Linear Function Poster|Linear Function Poster]]: Design a one-page poster that labels slope, y-intercept, x-intercept, increasing, decreasing, and horizontal lines with your own examples.
# [[English:Equation Story|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.

{{BR}}
=== Standard ===
# [[English:Video Explanation|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.
# [[English:Rate Interview|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.
# [[English:Measurement Experiment|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.
# [[English:Graphing Technology Comparison|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.

{{BR}}
=== Advanced ===
# [[English:Model Critique|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.
# [[English:Piecewise Linear Investigation|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.
# [[English:Systems Investigation|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.
# [[English:School Measurement Project|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.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Equation from Evidence|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.
# [[English:Error Analysis|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.
# [[English:Representation Transfer|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.
# [[English:Model Comparison|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.
# [[English:Model Reasonableness|Model Reasonableness]]: Evaluate a linear extrapolation beyond the original data range, identify assumptions that may fail, and propose a more defensible prediction range.
# [[English:New Context Challenge|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.

<br>
<br>

{{BR}}
= 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.

<br>
<br>

{{BR}}
= OERs on the Topic =

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

<iframe> https://en.m.wikipedia.org/wiki/Linear_function </iframe>

<br>

{{BR}}
= 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.

{| align=center
{{:D-Tab}}
'''[[English:Linear Functions|Linear Functions]]'''
# [[English:Function (mathematics)|Functions]]
# [[English:Linear equation|Linear equations]]
# [[English:Slope|Slope]]
# [[English:Y-intercept|Y-intercept]]
# [[English:Cartesian coordinate system|Coordinate plane]]
# [[English:Graph of a function|Function graphs]]
# [[English:Directly proportional|Direct proportion]]
# [[English:Systems of linear equations|Systems of linear equations]]
# [[English:Analytic geometry|Coordinate geometry]]
# [[English:Mathematical model|Mathematical modeling]]
|}

[[Category:English]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Functions]]
[[Category:Graphing]]
[[Category:Grades 9-10]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]

{{BR}}
= aiMOOC Projects =
[[Category:AI_MOOC]] [[Category:GPT aiMOOC]]
{{MT}}

English:Linear Functions - Versionsgeschichte

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