Linear graph explorer: y = mx + b

What the explorer shows

Every straight line that isn't vertical can be written as y = mx + b. The slope m says how steep the line is and which way it goes; the y-intercept b says where it crosses the y-axis. On this page, m steps through whole numbers and fractions from −3 to 3 (such as −2/3, 1/2 and 3/2), and b runs from −8 to 8. You can also drag the green dot on the y-axis to change b.

The slope triangle starts at the y-intercept: the blue side is the run and the red side is the rise. For m = 2/3, the line goes across 3 and up 2. Dots mark every point where the line passes through whole-number coordinates, and the table of values lists y for x from −2 to 3. The readout gives the equation and where the line crosses both axes.

Why explore lines this way

Changing one number at a time and watching the line separates the two ideas clearly: b slides the line up and down without turning it, and m turns it without moving the intercept. The slope triangle connects the number m to rise over run, and the table connects it to the pattern in the y-values. Match this line and Through two points then ask students to work backward from a graph to an equation.

Common mix-ups

• Swapping rise and run. A slope of 1/2 goes across 2 and up 1, so it is gentler than a slope of 2.
• Thinking a negative slope means the line is below the x-axis. A negative slope means the line goes down from left to right.
• Reading b as where the line crosses the x-axis. b is where it crosses the y-axis, at x = 0.
• Thinking a slope of 0 has no line. y = 3 is a flat line through (0, 3).

A 10-minute lesson on slope

• Set m = 1 and b = 0. Ask students to predict what m = 2 will look like, then show it.
• Step to m = 1/2 and compare the slope triangles.
• Change b only and ask what stays the same.
• Use the table to show that y changes by m for each step of 1 in x.
• Finish with two rounds of Match this line.

Where it fits by grade

• Grade 7: proportional relationships through the origin, y = kx, where k is the unit rate (7.RP.A.2).
• Grade 8: graph proportional relationships and interpret the unit rate as slope (8.EE.B.5), use similar triangles to explain why slope is the same between any two points and derive y = mx + b (8.EE.B.6), and interpret y = mx + b as a linear function (8.F.A.3).
• High school: graph linear functions and show intercepts (HSF-IF.C.7a).
• England writes y = mx + c and calls the slope the gradient. Key Stage 3 students rearrange linear equations into this form and interpret gradients and intercepts, and this continues at GCSE.

Learning more

Negative numbers, ratios and solving equations are skills on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Exam Prep includes GCSE Maths practice papers, and the AI tutor asks guiding questions when a graphing problem is tricky.