Grade 8–9 · Free math lesson
Linear Functions
After this lesson, you will be able to draw and read a straight-line graph using its slope and y-intercept.
Practice Linear Functions freeSkills in this unit
Pick a skill to try three free questions on just that.
- Slope from two points
- Slope-intercept form
- Graphing lines from equations
- Writing a line's equation from a graph or table
- Point-slope and standard form
- Parallel and perpendicular lines
- Function notation and evaluating functions
- Rate of change in word problems
- Comparing linear functions across representations
What Is a Linear Function?
A linear function makes a perfectly straight line on a graph. It follows the rule y = mx + b, where x is your input and y is your output.
Example: y = 2x + 1. Plug in x = 3: y = 2(3) + 1 = 7. So the point (3, 7) is on the line.
Remember: If the equation has no x² or x³, it's linear — it stays straight!
Slope: The Steepness
The slope (m) tells you how steep the line is — how much y goes up or down for every 1 step right. Slope = rise ÷ run.
Example: In y = 3x + 1, the slope is 3. Every time x increases by 1, y increases by 3.
Remember: Rise over Run — always divide the up-move by the right-move!
Y-Intercept: The Starting Point
The y-intercept (b) is where the line crosses the y-axis. It's the value of y when x = 0.
Example: In y = 2x + 4, set x = 0: y = 4. So the line starts at the point (0, 4) on the y-axis.
Remember: b is where the line BEGINS — think 'b for beginning'!
Worked example
Graph the function y = 2x + 3.
- 1Find the y-intercept: set x = 0, so y = 2(0) + 3 = 3. Plot the point (0, 3).
- 2Use the slope m = 2: from (0, 3), go UP 2 and RIGHT 1 to reach (1, 5). Plot that point.
- 3Draw a straight line through (0, 3) and (1, 5) and extend it both ways.
Answer: A straight line crossing the y-axis at 3 and rising 2 units for every 1 unit to the right.
The #1 mistake to avoid
Students swap slope and y-intercept. Remember: in y = mx + b, the number next to x is ALWAYS the slope, and the lone number is ALWAYS the y-intercept.
Key takeaways
- A linear function forms a straight line and is written as y = mx + b.
- The slope (m) tells you how steep the line is — rise divided by run.
- The y-intercept (b) is where the line crosses the y-axis when x = 0.
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