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Daily Math Minute

Linear Functions

Slope-Intercept Form

Graphing and writing equations in slope-intercept form.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Slope of a Line

Two Numbers That Describe an Entire Line

A line has infinitely many points on it. Before reading on, think about this: could just two numbers — a starting height and a constant rate of change — be enough to pin down every single one of those infinite points, with nothing left ambiguous?

Definition — Slope-Intercept Form

A linear equation written as y = mx + b, where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis, at x = 0).

This form works because it directly encodes a starting point and a rule for moving away from it. Setting x = 0 gives y = b immediately, confirming b is where the line starts on the y-axis. From there, every one-unit step to the right changes y by exactly m, since that's the definition of slope — so knowing m and b really does determine every other point on the line, one step at a time.

Worked Example — Graphing from Slope-Intercept Form

Graph y = 2x − 3. Start at the y-intercept, (0, −3). The slope, 2, means rise 2 and run 1 — from (0, −3), move right 1 and up 2 to reach (1, −1), then repeat to reach (2, 1), and so on.

Worked Example — Writing an Equation from a Graph

A line crosses the y-axis at (0, 5) and passes through (2, 1). Write its equation. The y-intercept is b = 5. The slope is (1 − 5)/(2 − 0) = −4/2 = −2. The equation is y = −2x + 5.

Worked Example — Rewriting an Equation into Slope-Intercept Form

Rewrite 3x + 2y = 12 in slope-intercept form. Isolate y: 2y = −3x + 12, then divide by 2: y = −(3/2)x + 6. The slope is −3/2 and the y-intercept is 6.

Graph Visualizer

Domain & range
2
Evaluate a point
  • x^2 = 0

Tip

Rewriting any linear equation into slope-intercept form first — even one that doesn't start that way — makes both its slope and its graph immediately readable, without any extra calculation.

Common Mistakes

  • Mixing up which number in y = mx + b is the slope and which is the intercept, especially after rearranging an equation.

    In y = mx + b, m always multiplies x (the slope), and b always stands alone as the constant (the y-intercept) — check which number is attached to x before naming either one.

  • Forgetting to divide every term by the coefficient of y when isolating it, such as rewriting 2y = −3x + 12 as y = −3x + 6 instead of y = −(3/2)x + 6.

    Every term on both sides must be divided by the same coefficient, not just the constant term — divide −3x and 12 by 2 together, giving −(3/2)x + 6.

Key Takeaways

  • Slope-intercept form, y = mx + b, encodes a line's starting height (b) and constant rate of change (m).
  • Graphing from this form starts at the y-intercept and uses the slope to step to additional points.
  • Any linear equation can be rearranged into slope-intercept form to reveal its slope and intercept directly.

Summary

Slope-intercept form fully describes a line using just two numbers, anchored at the y-axis. The next lesson writes a line's equation from a point that isn't necessarily the y-intercept at all.

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