Linear Functions & Graphing
Slope, slope-intercept form, and how to write the equation of a line.
The idea
A linear function graphs as a straight line. Its slope m tells you how steep the line is — how much y changes for every 1-unit change in x. Its y-intercept b is where the line crosses the y-axis.
Given two points (x₁, y₁) and (x₂, y₂) on the line, the slope is:
Once you know the slope and one point, you can find b by substituting that point into y = mx + b and solving for b.
Practice problems
Find the slope of the line through (2, 3) and (6, 11).
Show answer
m = (11 − 3) / (6 − 2) = 8 / 4 = 2
Write the equation of the line with slope −3 that passes through (0, 5).
Show answer
Since (0, 5) is the y-intercept, b = 5. So y = −3x + 5.
A gym membership costs $20 to join, plus $15 per month. Write a linear function C(t) for the total cost after t months, then find C(6).
Show answer
C(t) = 15t + 20. C(6) = 15(6) + 20 = 90 + 20 = $110.
A line passes through (1, 4) and (3, 10). Find its equation.
Show answer
m = (10 − 4) / (3 − 1) = 6 / 2 = 3. Using point (1, 4): 4 = 3(1) + b → b = 1. So y = 3x + 1.