Intercepts and Slope-Intercept Form

Where a line meets the axes

The y-intercept is the point where a line crosses the y-axis. On that axis, x is always 0 — so to find it, set x = 0 and solve. The x-intercept is where the line crosses the x-axis, where y is always 0 — so set y = 0 and solve.

Line: 2x + 3y = 12

y-intercept  (set x = 0):
  2(0) + 3y = 12  ->  3y = 12  ->  y = 4
  y-intercept is (0, 4)

x-intercept  (set y = 0):
  2x + 3(0) = 12  ->  2x = 12  ->  x = 6
  x-intercept is (6, 0)

Reading y = mx + b

The slope-intercept form is the friendliest way to write a linear equation: y = mx + b. Here m is the [[slope|slope]] and b is the y-coordinate of the [[y-intercept|y-intercept]] — the height where the line crosses the y-axis. The whole line is encoded in those two numbers.

y = 3x - 2
  slope m = 3        (rise 3, run 1)
  y-intercept b = -2 -> point (0, -2)

      2
y = - - x + 5
      3
  slope m = -2/3     (down 2 for every 3 right)
  y-intercept b = 5  -> point (0, 5)

Graphing from y = mx + b

1. Plot the y-intercept (0, b) first — that is your anchor point on the y-axis.
2. Write the slope as rise over run. From the anchor, count the rise up or down, then the run right.
3. Mark the second point you land on, then draw a straight line through both points.
4. Sanity-check: a positive slope should tilt up to the right; a negative slope should tilt down.

Graph y = (1/2)x + 1
  anchor: y-intercept (0, 1)
  slope 1/2 -> rise 1, run 2
    from (0,1): up 1, right 2 -> (2, 2)
    again:      up 1, right 2 -> (4, 3)
  draw the line through (0,1), (2,2), (4,3)