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

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y = x is a straight line.

Square every x to get $y=x^2$y = x². A negative x gives a positive y.

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

Coordinate Geometry #

How far up for each step across.

The gradient of a line is how far it rises for each 1 across

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The line crosses the y-axis at (0, 1).

From there it rises 2 for every 1 across, so its gradient is 2.

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

A point is on the line when its coordinates fit the equation: (4, 9) does, and (4, 8) does not.

It crosses the x-axis where y = 0: 2x + 1 = 0 gives $x=-\frac{1}{2}$x = −1/2.

Gradient from Two Points #

Rise over run, without the equation.

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

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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

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

Halfway across is (1 + 5) ÷ 2 = 3, and halfway up is (2 + 8) ÷ 2 = 5.

The midpoint averages the two endpoints, one coordinate at a time.

The Distance Between Two Points #

Pythagoras laid on the grid.

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

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How far is it from (1, 1) to (4, 5) in a straight line?

Going 3 across and 4 up makes a right triangle, with the straight line as its hypotenuse.

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

For any two points, square the run and the rise, add them, and take the square 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

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This triangle has corners (1, 1), (5, 2) and (3, 4), and no base or height is given.

Draw a 4 by 3 rectangle around it, area 12, and subtract the three corner triangles, 2, 2 and 3: the area is 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

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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.

So y = 2x + 1. Check it with the other point: 2 × 3 + 1 = 7.

On a graph, c is where the line crosses the y-axis: 5. Across 1 it goes down 2, so m = −2 and y = −2x + 5.

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

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Two lines that both rise 2 for every 1 across never meet: parallel lines have the same gradient.

Rotate one line by 90°: 2 up for 1 across becomes 1 down for 2 across, a gradient of −½.

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.