Graphing a line from its equation

TL;DR:

• In y = mx + b, b is where the line crosses the y axis and m is the slope.

• Start at the intercept and use the slope as rise over run to plot more points.

• Parallel lines share a slope. Perpendicular slopes multiply to −1.

An equation like y = −2x + 4 packs a whole picture into a few symbols. Two numbers in it tell you where to start and which way to go. Once you can read them, a line takes less than a minute to draw, and you can check it with a single test point.

Reading y = mx + b

In the slope-intercept form y = mx + b, the number b is the y-intercept: the point (0, b) where the line crosses the vertical axis. The number m is the slope, rise over run, as in slope as a rate of change. A positive slope climbs to the right and a negative slope falls.

Worked example: y = −2x + 4

1. Read the intercept: b = 4, so plot (0, 4).

2. Read the slope: m = −2, which is −2/1. From (0, 4), go right 1 and down 2 to reach (1, 2).

3. Repeat: right 1, down 2 again gives (2, 0). This is the x-intercept.

4. You can go backwards too: left 1, up 2 gives (−1, 6).

5. Draw a straight line through the points, extending with arrows at both ends.

Check every row: −2(−1) + 4 = 6, −2(0) + 4 = 4, −2(1) + 4 = 2, −2(2) + 4 = 0, −2(3) + 4 = −2. All match.

Check with a test point

Pick a point you did not use for plotting and see whether it lies on the line. Try x = 5: y = −10 + 4 = −6. Is (5, −6) on your drawing? Follow the slope: from (3, −2), right 1 and down 2 gives (4, −4), then (5, −6). Yes. A point that does not fit the equation is off the line. For example, (1, 3) gives −2 + 4 = 2 ≠ 3, so it lies above the line.

Two quick intercepts

Another way to graph is to find where the line meets each axis. Set x = 0 for the y-intercept: y = 4. Set y = 0 for the x-intercept: 0 = −2x + 4, so x = 2. Plot (0, 4) and (2, 0) and join them. That uses only two points, so a third point is a worthwhile check. The same line can be written in standard form as 2x + y = 4.

Parallel and perpendicular lines

• Parallel lines have the same slope and different intercepts. A line parallel to y = −2x + 4 through (0, 1) is y = −2x + 1.

• Perpendicular lines have slopes whose product is −1, the negative reciprocal. A line perpendicular to slope −2 has slope 1/2, since −2 × 1/2 = −1. Through (0, 4): y = ½x + 4.

Test perpendicularity with a point: the two lines meet at (0, 4). From there, the first goes right 1 and down 2, and the second goes right 2 and up 1. Those two moves form a right angle, as the vectors (1, −2) and (2, 1) have a dot product 1×2 + (−2)×1 = 0.

Try three more yourself

Graph y = ½x − 3. The intercept is (0, −3). The slope is 1/2, so go right 2 and up 1: (2, −2), (4, −1), (6, 0). Check: ½(6) − 3 = 0.

Build an equation from two points, (1, 5) and (3, 9). The slope is (9 − 5)/(3 − 1) = 4/2 = 2. Substitute a point to find b: 5 = 2(1) + b, so b = 3. The line is y = 2x + 3. Check the other point: 2(3) + 3 = 9.

Convert 3x + 2y = 12 to slope-intercept form: 2y = −3x + 12, so y = −1.5x + 6. The intercepts are (0, 6) and, setting y to 0, 3x = 12, so (4, 0). Check: −1.5(4) + 6 = 0.

Special cases

• A horizontal line y = 3 has slope 0.

• A vertical line x = 2 has an undefined slope and is not a function of x.

• A line through the origin has b = 0, like y = 3x.

If you are given a graph rather than an equation, work in reverse. Read the y-intercept where the line crosses the vertical axis, pick two clear grid points to find the rise and run, and write y = mx + b. Then check your equation against a third point on the picture. This works both ways, so you can move freely between an equation and its line.

How to check your answer

• Intercept. Does your line cross the y axis at b?

• Direction. Negative slope should fall left to right.

• Third point. Plug a new x into the equation and see if the point lies on the drawn line.

• Steepness. A slope of magnitude 2 is steeper than a slope of 1.

Common mistakes

• Plotting rise and run in the wrong order, using right 2 and down 1.

• Ignoring the sign of the slope.

• Misreading y = −2x + 4 as intercept −2.

• Joining points with a curve.

• Using the x-intercept as the b value.

Where you will use it next

Boundary lines like this one are the first step in graphing inequalities, as in inequalities and why the sign flips. Directions and steps along a line become arrows in vectors: adding forces and velocities.

Frequently asked questions

What if the equation is not in slope-intercept form?

Solve it for y first. For example, 2x + y = 4 becomes y = −2x + 4.

How many points do I need to draw a line?

Two define a line. A third acts as a check.