Slope-Intercept Form and Graphing a Line

In this lesson you will: Read the slope and the y-intercept straight off f(x)=mx+b; Graph a line from slope-intercept form in two moves; Rewrite an equation in slope-intercept form when it does not start there.

Two numbers, two jobs

Write a linear function as f(x)=mx+b. The number b is f(0) — the output when the input is zero, which is where the graph meets the y-axis. The number m is the slope, the rate at which the output changes.

One number places the line; the other tilts it. Change b and the line slides up or down without tilting. Change m and the line pivots about its y-intercept.

Graphing in two moves

Plot (0,b). From there, use the slope as a set of directions: up m for every 1 across, or, if the slope is a fraction, up the numerator for every denominator across. Mark the second point and draw the line through both.

Two points are enough to fix a line, but plot a third. It costs five seconds and it catches a miscount immediately — three points that do not line up mean one of them is wrong.

When the equation is not in that form yet

Until y is alone on one side, the numbers you can see are not the slope and the intercept. Solve for y first, then read them off.

That single habit removes most sign errors in this lesson. In 4x+2y=10 the coefficient of x is 4, but the slope of the graph is -2.

Common mistake — Walking the slope sideways first

To graph y=3x+2, a student plots (0,2), counts 3 to the right and then 1 up, landing on (3,3).

The reasoning is that 3 is \"the number next to x\", so 3 must be the horizontal move. It feels right because the 3 is written first.

Slope is rise over run, and the whole number 3 is really 3/1: up 3 for every 1 across. The correct second point is (1,5). The student's line is far too shallow and passes through nothing the equation predicts.

How to catch it: Substitute your second point into the equation. Here (3,3) gives 3(3)+2=11, not 3, so the point is not on the line. One substitution per graph is enough to catch this every time.