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Functions & graphs
A function is a rule that turns one number into another. Plotting it gives you the picture of that rule.
Drag the two points
Slope (rise over run)
m = Δy / Δx = 4 / 6
m = 0.667
Line equation
y = 0.67x + 0.33
Drag the A and B points to any grid spot. The line snaps through both.
The dashed triangle shows rise (Δy) over run (Δx).
The line equation
The simplest function is a straight line: y = mx + b.
- m is the slope — how steep the line is. A slope of 2 means: every 1 step right, go 2 steps up.
- b is the y-intercept — where the line crosses the vertical axis.
Slope, in pictures
- Positive slope — line tilts up as you go right.
- Negative slope — line tilts down.
- Zero slope — flat horizontal line.
- Bigger slope (in size) — steeper line.
Finding slope from two points
Pick any two points on the line, say
(x₁, y₁) and (x₂, y₂). Then m = (y₂ − y₁) / (x₂ − x₁). Rise over run.Plotting a line
- Plot the y-intercept first — that's the dot on the y-axis.
- From there, use the slope to find a second point.
- Draw the line through them.
Quick check
For y = 2x + 1, what is y when x = 4?
Quick check
What is the slope of the line through (1, 2) and (4, 11)?