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Lines and graphs — Formula Sheet

4 formulas from Math Foundations, each with what it means, when to use it and the trap it sets.

Slope between two points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

What it says: How much the line climbs for each step it takes to the right.
When to reach for it: Two points, and you need the steepness or the equation of the line through them.
Watch out: Both differences must be taken in the same order. Subtracting y one way and x the other flips the sign.

m = \frac{7 - 1}{4 - 2} = 3

This same ratio, with the two points brought together, becomes the derivative.

Equation of a line

y = m x + c or y - y_{1} = m (x - x_{1})

What it says: Slope m, and either the height at x = 0 or one point the line is known to pass through.
When to reach for it: Writing down a line from a graph, or from a slope and a point.
Watch out: c is the y-value where x = 0, not where the line crosses the x-axis.

y - 1 = 3 (x - 2) \Rightarrow y = 3 x - 5

The point-slope form is usually faster; expand only if you need c.

Multiplying an inequality by a negative

a < b ⟹ - a > - b

What it says: Multiplying or dividing by a negative reverses the direction of the sign.
When to reach for it: Any inequality where you divide through by a negative coefficient.
Watch out: This is the single most common inequality error, and it is silent — the answer just comes out backwards.

- 2 x < 6 \Rightarrow x > - 3

Dividing by −2 flipped < into >.

Function composition

(f \circ g) (x) = f (g (x))

What it says: Run g first, then feed its output into f.
When to reach for it: Whenever one expression sits inside another — which is exactly what the chain rule later differentiates.
Watch out: Order matters: f(g(x)) and g(f(x)) are usually different functions.

f (x) = x^{2}, g (x) = x + 3 \Rightarrow f (g (x)) = (x + 3)^{2}

The inner function is the one written closest to x.

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