Quadratic Graphs and Coordinate Geometry

Quadratic Graphs #

Squaring bends a straight line into a curve.

Squaring the input bends a straight line into a curve

y = x is a straight line.

Square every x — that is $y=x^2$y = x². Negative inputs come back positive.

So the curve falls, turns, then climbs. That turning point is the vertex.

Coordinate Geometry #

How far up for each step across.

The steepness of a line is how far up it goes for each step across

Where the line crosses the y axis is where it starts out: here at 1.

From there, 1 step across and 2 steps up. That steepness is a gradient of 2.

So this line is y = 2x + 1: the 2 is the gradient and the 1 is the y-intercept.

Gradient from Two Points #

Rise over run, without the equation.

The gradient between two points is the rise divided by the run

You are given two points, (1, 2) and (4, 8), and no equation. How steep is the line?

Across from 1 to 4 is a run of 3. Up from 2 to 8 is a rise of 6.

Divide: 6 up over 3 across is 2 up for each 1 across. That is the gradient, m = 2.

The rule: subtract in the same order, numerator and denominator, then divide.

The Midpoint of a Segment #

Average the x-coordinates, then the y.

The midpoint of a segment averages the two x values and the two y values

From (1, 2) to (5, 8): which point sits exactly halfway between them?

Halfway across: x averages to (1 + 5) ÷ 2 = 3. Halfway up: y averages to 5.

The midpoint is the average of the ends — each coordinate separately.

The Distance Between Two Points #

Pythagoras laid on the grid.

The distance between two points is Pythagoras on their run and rise

How far is it from (1, 1) to (4, 5) in a straight line — not counting steps?

Walk 3 across and 4 up instead: a right triangle sits under the straight line.

Pythagoras on the legs: the distance squared is run² plus rise², so d = 5.

Any pair of points: square the run and the rise, add them, take the root.

The Area of a Polygon from Coordinates #

Cross-multiply the loop, halve the difference.

Cross-multiply down the corner list, cross-multiply up, and half the difference is the area

A triangle with corners (1, 1), (5, 2) and (3, 4). No base or height is given.

Draw a box around it: 4 × 3 = 12. Subtract the corner triangles 2, 2 and 3: area 5.

The shoelace method: half of (down-products − up-products) gives the same area, 5.

The Equation of a Line Through Two Points #

Gradient first, then anchor it.

Two points give the gradient, and then either point fixes c

Only one straight line runs through the two points (1, 3) and (3, 7).

Find the gradient first: rise 7 − 3 = 4 over run 3 − 1 = 2 gives m = 2.

So y = 2x + c for some c. Substitute either point: 3 = 2 + c, so c = 1.

y = 2x + 1 — and the point you did not use checks it: 2 × 3 + 1 = 7. It fits.

Gradients of Parallel and Perpendicular Lines #

Equally steep, or multiplying to minus one.

Parallel lines share a gradient, and perpendicular gradients multiply to minus one

Two lines that climb 2 for every 1 across stay the same distance apart forever.

Turn one a quarter turn: 2 up for 1 across becomes 1 down for 2 across.

When two lines cross at a right angle, their gradients multiply to −1: $m_1\times m_2=-1$m₁ × m₂ = −1.

Any pair works: $3\times \left(-\frac{1}{3}\right)=-1$3 × (−1/3) = −1, and those two lines cross at a right angle.