Algebra · KS3

Quadratic Graphs

Square the x and the graph stops being a line. It becomes a smooth U-shaped curve, the parabola: the path of every thrown ball, the shape of every satellite dish, and the first curve you'll plot from a table of values.

National curriculum KS3: recognise, sketch and produce graphs of quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane

1 · Why it curves

On a straight line, each step of 1 in x adds the same amount to y. For y = x² that stops being true: from 0 to 1 the y goes up by 1, from 1 to 2 it goes up by 3, from 2 to 3 by 5. The steps keep growing, so the graph gets steeper and steeper. Going left of zero, (−1)² = 1, (−2)² = 4: the same values again, mirrored. That mirror is what makes the U.

Plot it one point at a time. The table fills from the left; watch the differences in the bottom row.

Join with a smooth curve, not a ruler. The points are samples of a curve; a ruler between them gives a jagged shape that loses marks. The bottom of y = x² is at (0, 0), rounded, not pointed: draw through it smoothly. And it never goes below the x-axis, because a square is never negative.

2 · The negative-number trap

Most wrong quadratic graphs come from one calculator slip: typing −3² instead of (−3)². The first gives −9, the second +9. In a table of values, every negative x must be squared in brackets. Here's the same table done both ways. One of them is not a parabola.

With brackets: (−3)² = 9
Without: −3² read as −9
If your plotted points don't make a symmetric U, the first thing to check is the negative x-values.

3 · Symmetry and the turning point

Every parabola has a vertical line of symmetry. The point where the curve turns round, on that line, is the turning point (or vertex): the lowest point for a U, the highest for an upside-down one. Because of the symmetry, the table of values reads the same forwards and backwards from the turning point, which is a free check on your arithmetic.

Line of symmetry
Turning point
y-intercept
Crosses x-axis at

Where the curve crosses the x-axis, y = 0. Those crossing points are the roots, and they're the solutions of x² + bx + c = 0. At GCSE you'll solve that by factorising; for now, the graph shows you the answer.

4 · Moving and stretching the basic curve

All quadratic graphs are the same U, moved or stretched. Two simple changes to y = x²:

Turning point
Shape
Compared with x²

5 · Reading values off the curve

Once drawn, a graph answers questions in both directions. Given x, go up to the curve and across to read y. Given y, go across to the curve and down: there are usually two answers, one each side of the line of symmetry. That's new. A straight line gives one x for each y; a parabola can give two, one or none.

6 · Where parabolas appear

Throwing anything

A ball, a jet of water, a long-jumper: height against distance is an upside-down parabola, because gravity adds the same downward speed every second. The turning point is the highest point of the flight.

Area problems

A rectangle with perimeter 20 has sides x and 10 − x, so area = 10x − x². The graph peaks at x = 5: a square. Maximising anything with a quadratic rule means finding the turning point.

Satellite dishes and headlights

A parabola reflects every incoming parallel ray to one point, the focus. Dishes collect signals there; headlights put a bulb there and send light out in a parallel beam.

Braking distance

Double your speed and braking distance goes up four times, because kinetic energy is ½mv². The Highway Code's stopping distances follow a quadratic.

Computing

An algorithm that compares every item with every other (bubble sort) does roughly n² steps: its running time is a parabola in n. Doubling the data quadruples the wait. That's why n² algorithms are avoided on big inputs.

GCSE and beyond

Solving quadratics by factorising, completing the square (which finds the turning point exactly) and the formula are all about these curves. Differentiation finds the turning point as the place where the gradient is zero.

Worked examples

Draw y = x² − 2 for x from −3 to 3
Table, squaring in brackets: x = −3 → 9 − 2 = 7; −2 → 2; −1 → −1; 0 → −2; 1 → −1; 2 → 2; 3 → 7.
Symmetry check: 7, 2, −1, −2, −1, 2, 7 reads the same both ways ✓
Plot and join with a smooth curve. Turning point (0, −2); crosses the x-axis at about ±1.4 (that's √2).
Complete the table for y = x² + 2x − 3 and find the turning point
Table: x = −4 → 16 − 8 − 3 = 5; −3 → 0; −2 → −3; −1 → −4; 0 → −3; 1 → 0; 2 → 5.
Symmetry: the values −3, −4, −3 are centred on x = −1. Line of symmetry x = −1, turning point (−1, −4).
Roots: y = 0 at x = −3 and x = 1. (Later: (x + 3)(x − 1) = 0.)
Use the graph of y = x² − 4 to solve x² − 4 = 5
Draw the line y = 5 across the graph.
It meets the curve twice, at x = −3 and x = 3.
Check: (−3)² − 4 = 5 ✓ and 3² − 4 = 5 ✓. Two solutions, one either side of the symmetry line.
Exam-style: a ball's height is h = 20t − 5t² metres after t seconds. Sketch the graph and find the maximum height.
Shape: the t² coefficient is negative, so it's an upside-down U: a maximum.
Roots: h = 0 when t(20 − 5t) = 0, so t = 0 (launch) or t = 4 (landing).
Symmetry: the peak is halfway between the roots, at t = 2. h = 40 − 20 = 20 m.
Sketch: starts at origin, peaks at (2, 20), lands at (4, 0). Only t from 0 to 4 makes physical sense.

Quick check

What you've learned

A quadratic graph is a parabola, a symmetric U (or an upside-down U if the x² coefficient is negative), drawn as a smooth curve through points from a table of values in which every negative x is squared in brackets. It has a vertical line of symmetry, a turning point on that line, a y-intercept at the constant term, and crosses the x-axis at the roots, where the equation equals zero. Adding c slides the curve up or down; multiplying by a stretches or flips it. Reading across from a y-value can give two x-values, which is how graphs solve quadratic equations.