3.4Linear Functions
Every straight line tells a story — slope describes how fast it rises or falls, and the y-intercept tells you where it starts. Master y = mx + b and you can read, write, and graph any linear function.
Why This Matters
Linear functions model constant rates of change — from speed and distance to cost and revenue. They're the foundation of graphing in Algebra 2, Physics kinematics, and data analysis in statistics.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What do the slope and y-intercept of a linear function tell you about its graph, and how do you use them to write and graph equations?
Lesson Overview
A linear function is a function whose graph is a straight line. Every linear function can be written in slope-intercept form: y = mx + b, where m is the slope (the rate of change) and b is the y-intercept (where the line crosses the y-axis). The slope is calculated as rise over run: the vertical change divided by the horizontal change between any two points. A positive slope rises left to right; a negative slope falls; zero slope is horizontal; undefined slope is vertical.
Slope-Intercept Form — y = mx + b
m = slope (rate of change, steepness, direction)
b = y-intercept (where line crosses y-axis, value when x = 0)
Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)
= rise / run = Δy / Δx
Types of Slope
Positive Slope
rises left → right
Negative Slope
falls left → right
Zero Slope
horizontal line
Undefined Slope
vertical line
y = 2x − 1
Slope Triangle — Rise over Run
Rise = 6 — count 6 units up from (1, 1).
Run = 3 — count 3 units right to (4, 7).
m = rise / run = 6 / 3 = 2
Slope-Intercept Form: y = mx + b
y = mx + b
Slope (m): rise ÷ run — steepness and direction
Y-intercept (b): where the line crosses the y-axis
Example: y = 3x − 2 → m = 3, b = −2
Worked Examples
y = 2x + 1 (slope 2, y-intercept 1)
y = −x + 3 (slope −1, y-intercept 3)
Identify the slope and y-intercept of y = 2x + 3, then graph it.
The equation is in slope-intercept form y = mx + b.
Identify: m = 2, b = 3.
Plot the y-intercept: (0, 3).
Use slope m = 2 = 2/1: from (0, 3), go up 2, right 1 → (1, 5).
Draw the line through both points.
Check: x = 2 → y = 2(2) + 3 = 7. Point (2, 7) should be on the line ✓.
Identify the slope and y-intercept of y = −x + 5, then graph it.
Rewrite: y = −1x + 5. So m = −1, b = 5.
Plot y-intercept: (0, 5).
Slope = −1 = −1/1: from (0, 5), go down 1, right 1 → (1, 4).
Draw the line through (0, 5) and (1, 4).
Check: x = 3 → y = −3 + 5 = 2. Point (3, 2) should be on the line ✓.
Write the equation of the line with slope 3 and y-intercept −2.
Use slope-intercept form: y = mx + b.
Substitute m = 3 and b = −2.
y = 3x − 2
Check: x = 1 → y = 3(1) − 2 = 1. x = 0 → y = −2 (y-intercept ✓).
Write the equation of the line passing through (0, 4) and (2, 8).
m = (8 − 4) / (2 − 0) = 4 / 2 = 2
One point is (0, 4), so b = 4 (the y-intercept is given directly).
y = 2x + 4
Check: x = 2 → y = 2(2) + 4 = 8 ✓.
Write the equation from the table: x = 0, 1, 2, 3 and y = −2, 1, 4, 7.
As x increases by 1, y increases by 3 each time → m = 3.
When x = 0, y = −2. So b = −2.
y = 3x − 2
Check: x = 3 → y = 3(3) − 2 = 7 ✓.
Guided Practice
Guided Practice Video: Linear Functions
Watch the guided practice walkthrough for linear functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Identify the slope and y-intercept: y = 4x − 7
Hint: The equation is in y = mx + b form. m is the coefficient of x; b is the constant. m = 4, b = −7.
Graph y = −2x + 4. Identify two points on the line.
Hint: b = 4 → plot (0, 4). Slope = −2 = −2/1: from (0, 4) go down 2, right 1 → (1, 2). Draw the line.
Find the slope of the line through (1, 3) and (4, 9).
Hint: m = (9 − 3) / (4 − 1) = 6 / 3 = 2.
Write the equation of the line with slope −3 and y-intercept 5.
Hint: Substitute into y = mx + b: y = −3x + 5.
Write the equation from the table: x = 0, 1, 2, 3 and y = 5, 3, 1, −1.
Hint: Change in y per step: 3−5 = −2 (constant). So m = −2. When x = 0, y = 5, so b = 5. Equation: y = −2x + 5.
Table → Equation: Δy=3, Δx=1, b=1
Rise/run triangle: rise=9, run=3, m=3
Key Vocabulary
Linear Function
A function whose graph is a straight line. Can be written as y = mx + b.
Example: y = 3x − 2 is a linear function.
Slope-Intercept Form
y = mx + b, where m is the slope and b is the y-intercept.
Example: y = 2x + 5 has slope 2 and y-intercept 5.
Slope (m)
The rate of change of a line: rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁).
Example: Through (1,3) and (3,7): m = (7−3)/(3−1) = 2.
Y-Intercept (b)
The point where the line crosses the y-axis. The value of y when x = 0.
Example: In y = 4x − 7, the y-intercept is −7.
Rise
The vertical change between two points on a line (change in y).
Example: From (1,3) to (4,9): rise = 9 − 3 = 6.
Run
The horizontal change between two points on a line (change in x).
Example: From (1,3) to (4,9): run = 4 − 1 = 3.
Interactive Practice — 5 Questions
What is the slope of y = −3x + 7?
What is the y-intercept of y = 5x − 4?
What is the slope of the line through (2, 3) and (6, 11)?
Which equation has a slope of 0?
A table shows: x: 0, 1, 2, 3 → y: 4, 7, 10, 13. Which equation matches?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Identify the slope and y-intercept: y = 5x − 3
Find the slope through (2, 5) and (6, 13).
Write the equation: slope = 2, y-intercept = −5.
Write the equation from the table: x = 0, 1, 2, 3; y = 1, 4, 7, 10.
A taxi charges a $3 base fee plus $2 per mile. Write a linear function for the total cost C after m miles, and find the cost for 5 miles.
Common Mistakes
Plotting the y-intercept on the x-axis instead of the y-axis.
The y-intercept (b in y = mx + b) is where the line crosses the y-axis. Plot it at (0, b).
Using rise/run backwards — treating run as the numerator and rise as the denominator.
Slope = rise/run. Rise is the vertical change (up/down); run is the horizontal change (left/right).
Moving in the wrong direction for a negative slope — going up-right instead of down-right.
A negative slope means the line falls from left to right. From the y-intercept, go right and down.
Confusing slope-intercept form (y = mx + b) with standard form (Ax + By = C) and misidentifying m and b.
In y = mx + b, m is the coefficient of x (slope) and b is the constant (y-intercept). Rewrite in slope-intercept form first if needed.
Math Tips
Slope is a ratio, not a point: m = 2 means "for every 1 unit right, go 2 units up."
Negative slope trick: Write −3 as −3/1. Count 3 down, 1 right — OR 3 up, 1 left. Both give the same line.
Fraction slope: m = 1/2 means 1 up, 2 right. Always count run in the positive direction (right).
Parallel lines: Two lines with the same slope are parallel (they never intersect).
Table check: In a table, if x increases by 1 each row, the constant change in y equals the slope.