L3 · Slope-intercept form y = mx + b; the two intercepts

Learning goal

“I can write a linear function in slope-intercept form y = mx + b, read the slope and the y-intercept straight from the equation, find the x-intercept, and state both intercepts as ordered pairs. I can also write y = b for a horizontal line and x = a for a vertical line.”

Slope-intercept form — the equation y = mx + b, where m is the slope and b is the y-intercept.

y-intercept — where the graph crosses the y-axis; as an ordered pair it is (0, b).

x-intercept — where the graph crosses the x-axis; it is the x-value where y = 0, written as an ordered pair (x, 0).

Constant (b) — the starting value of the function: the output when the input is 0.

Worked example

A graph is described: the line crosses the y-axis at (0, -2) and passes through (3, 4). Step 1: The y-intercept is b = -2, so the equation will end in "- 2". Step 2: Slope from the two points: rise = 4 - (-2) = 6, run = 3 - 0 = 3, m = 6/3 = 2. Step 3: The equation is y = 2x - 2. Step 4: Find the x-intercept by setting y = 0: 0 = 2x - 2, so 2x = 2 and x = 1. The x-intercept, as an ordered pair, is (1, 0). Say it plainly: m tells you how the line climbs; b tells you where it starts on the y-axis. Two facts, and the whole line is fixed.

Check yourself — explain it

In y = 2x - 2, the -2 at the end is the y-intercept because...

Paired problem — example still visible

A line crosses the y-axis at (0, 1) and passes through (2, 5). Write its equation. -> Slope (5 - 1)/(2 - 0) = 2, b = 1: y = 2x + 1.

Faded completion — last step is yours

"The line y = 3x + b passes through (1, 7). Substituting gives 7 = 3(1) + b, so b = ___" -> 4

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