289. Slope & Y-Intercept Unlocked · Decode y = mx + c visually

Try to break it

Drag C up or down to change the y-intercept c — the line slides without rotating. Drag M up or down to change the slope m (M sits one unit to the right of C, so the vertical distance from C to M is exactly the rise per unit run). Together c and m pin down a unique line: every y = mx + c is one C and one M.

How you build it

Plot a line from its slope and y-intercept.

• Point tool: plot C on the y-axis at (0, 1). This is the y-intercept c = 1 — where the line meets the y-axis.
• Point tool: from C go 1 right and 2 up, to (1, 3). Run = 1, rise = 2, so the slope m = rise/run = 2.
• Line tool: click C, then M. The line is y = 2x + 1 — slope m = 2, y-intercept c = 1.

The proof, step by step

Prove that in y = mx + c, m is the slope and c is the y-intercept.

• The y-intercept c is the constant term in y = mx + c. It represents the point where the line crosses the y-axis (x = 0).
• The slope m is the ratio of vertical change to horizontal change (rise over run). In our diagram, moving M horizontally by 1 unit changes the line vertically by m units.

Worked example

A line has a slope of -2 and a y-intercept of 4. Which equation represents this line?

Comparing with y = mx + c, m = -2 and c = 4. Thus, the equation is y = -2x + 4.

• y = -2x + 4 — correct
• y = 4x - 2
• y = -2x - 4
• y = 2x + 4