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ALGEBRA · EQUATIONS & LINES

Slope & Linear Functions

Slope measures steepness; the y-intercept tells you where the line starts.

15 min read 4 practice questions
A linear function has a constant rate of change. Every time xx increases by 11, yy changes by the same amount — the slope, written mm.
m=ΔyΔx=y2−y1x2−x1m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Slope is “rise over run”

The slope-intercept form y=mx+by = mx + b makes graphing fast: start at (0,b)(0, b) on the yy-axis, then use the slope to step to the next point.
Slope-intercept explorerInteractive
-8-6-4-22468-8-6-4-22468run 1rise 1

Drag the sliders to change the slope and intercept. The orange triangle shows rise over run for one step right.

y=1x+2y = 1x + 2

Crosses the x-axis at x = -2.00

Worked example · Line through two points

Find the equation of the line through (1,3)(1, 3) and (4,9)(4, 9).
  1. 1
    Slope: m=9−34−1=63=2m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2.
  2. 2
    Plug a point into y=2x+by = 2x + b: 3=2(1)+b3 = 2(1) + b, so b=1b = 1.
  3. 3
    The line is y=2x+1y = 2x + 1.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
What is the slope of the line through (2,5)(2, 5) and (6,17)(6, 17)?
2
What is the yy-intercept of y=−4x+7y = -4x + 7?
3
Which line is perpendicular to y=23x−1y = \frac{2}{3}x - 1?
4
The line y=3x+by = 3x + b passes through (2,4)(2, 4). Find bb.

Numeric answers accept decimals or fractions like 3/4.