Quadratic Graphs

Throw a ball to a friend and watch its flight: it rises, levels off, and comes down along a smooth, perfectly symmetric arc. A fountain's jet traces the same arc. So does water from a hosepipe, a basketball at the top of a free throw, and — famously — the golden arches outside a certain burger restaurant. Satellite dishes and car headlights are built from the very same curve, spun around its axis.

That shape is the parabola, and here is the remarkable thing: every one of them is the graph of a quadratic. One family of equations,

y = ax^2 + bx + c,

draws them all — only the width, position, and direction change. Better still, the traffic runs both ways: every feature of the curve can be read straight from the algebra. Where it crosses the axes, where it turns, which way it opens, how steep it is — the equation tells you everything, if you know where to look. This page is the guided tour.

Worked example: plot y = x^2 - 4x + 3 from a table

The most honest way to meet a graph is to build it point by point. Pick some x-values, substitute each into y = x^2 - 4x + 3, and record the y that comes out. For x = -1, say: y = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8.

x	-1	0	1	2	3	4	5
y	8	3	0	-1	0	3	8

Before you even plot it, the table is whispering secrets. Read the y row out loud: 8, 3, 0, -1, 0, 3, 8 — the values march down to a low point and then climb back up through exactly the same numbers. That mirror pattern is the parabola's symmetry, visible in raw arithmetic before a single point is drawn. Now step through the figure below and watch each feature appear.

The roots — where it crosses the x-axis

The points where the parabola meets the x-axis are the roots. On the axis y = 0, so the roots are exactly the solutions of ax^2 + bx + c = 0 — solving a quadratic and reading its graph are the same job in two costumes. A factorised quadratic shows the roots at a glance:

y = (x - r_1)(x - r_2) \quad\Rightarrow\quad \text{roots at } x = r_1 \text{ and } x = r_2.

Check it against our worked example: x^2 - 4x + 3 = (x - 1)(x - 3), so the roots should be x = 1 and x = 3 — and that is precisely where the table produced y = 0 and the curve cut the axis. The algebra and the picture agree, as they must.

Not every parabola gets two crossings. A parabola can have two roots (it cuts clean through), one root (it just kisses the axis at its turning point — a repeated root, like y = (x-2)^2), or none (it floats entirely clear of the axis, like y = x^2 + 1, whose lowest point is already at height 1).

The y-intercept — read it for free

Where does the curve cross the y-axis? On that axis x = 0, and substituting x = 0 into y = ax^2 + bx + c wipes out the first two terms:

y = a(0)^2 + b(0) + c = c.

So the y-intercept is simply c — no working required. The graph of y = x^2 - 4x + 3 crosses at (0, 3); the graph of y = 2x^2 + x - 7 crosses at (0, -7). It is the cheapest piece of information on the whole curve, and examiners love asking for it precisely because so many people forget it is free.

The turning point and the line of symmetry

A parabola has a single turning point — its vertex — the lowest point when it opens up, or the highest point when it opens down. A vertical line of symmetry runs straight through the vertex, so the two halves of the curve are perfect mirror images (that's the 8, 3, 0, -1, 0, 3, 8 pattern from the table).

Symmetry is not just pretty — it is a tool. When there are two roots, the line of symmetry sits exactly halfway between them, so its position is just the average of the roots:

x = \frac{r_1 + r_2}{2}.

For y = x^2 - 4x + 3 the roots are 1 and 3, so the line of symmetry is x = \tfrac{1+3}{2} = 2. And once you know the vertex's x-coordinate, one substitution gives its height: y = 2^2 - 4(2) + 3 = -1. Turning point: (2, -1). Two lines of working, the hardest point on the curve — that's symmetry paying its way.

Which way does it open? Flip it upside down

The sign of a — the coefficient of x^2 — decides the direction:

a > 0: the parabola opens up — a valley, with a minimum at the vertex.
a < 0: the parabola opens down — a hill, with a maximum at the vertex.

The size of a matters too: the larger |a| is, the narrower and steeper the curve (compare y = 3x^2 with y = \tfrac{1}{3}x^2 — triple the y-values, a much skinnier valley).

Everything we learned about valleys works for hills. Take y = -x^2 + 2x + 3. Pull out the minus sign and factorise: -(x^2 - 2x - 3) = -(x - 3)(x + 1), so the roots are x = 3 and x = -1. Symmetry sits midway: x = \tfrac{3 + (-1)}{2} = 1, and substituting gives y = -1 + 2 + 3 = 4. The y-intercept is c = 3. So: a hill peaking at (1, 4), feet on the axis at -1 and 3 — exactly what the graph shows.

Sketching without a table

Here is the skill GCSE really wants: a sketch — the right shape in the right place, key points labelled — with no table of values at all. Everything you need is in the algebra. The recipe:

Direction: look at the sign of a — valley or hill?
y-intercept: read off c. Free.
Roots: factorise and solve y = 0; mark the crossings.
Turning point: average the roots for the line of symmetry, substitute for the height.
Draw: one smooth symmetric curve through the lot.

Try it on y = x^2 - 2x - 8. It's a valley (a = 1 > 0). The y-intercept is -8. Factorising, (x - 4)(x + 2) gives roots at x = 4 and x = -2. Midway: x = \tfrac{4 + (-2)}{2} = 1, and y = 1 - 2 - 8 = -9, so the vertex is (1, -9). Five facts, four lines of working, and the sketch draws itself — no table in sight.

Three traps catch parabola-sketchers again and again:

Forgetting the symmetry. The turning point sits exactly midway between the roots — not "roughly in the middle", not at one of the roots. If your roots are 1 and 3 but your vertex isn't at x = 2, something has gone wrong. Use the symmetry: it is the fastest checking device you own.
Working out the y-intercept the hard way (or wrongly). It is just c — substitute x = 0 and see. No factorising, no formula. Read it for free.
Mixing up what a and c do. A steeper, narrower parabola comes from a bigger a (bigger |a|, to be precise). Changing c does not change the shape at all — it only slides the whole curve up or down. Test it on the sliders below: crank c and watch the curve keep its exact width while it rises.

A parabola has a superpower no other curve shares. Inside every parabola there is a special point called the focus, and the curve is shaped so that every ray arriving parallel to its axis of symmetry bounces off the surface and passes through that one point. A satellite's signal comes from so far away that its rays are effectively parallel — so a dish shaped like a spun parabola gathers a metre-wide sheet of faint signal and concentrates all of it onto a receiver the size of your thumb, mounted at the focus.

Run the trick backwards and you get a headlight: put a bulb at the focus, and every ray it throws at the mirror reflects out in a perfectly parallel beam. Radio telescopes, solar cookers, whispering dishes in science museums — same shape, same theorem.

And the arc of a thrown ball? Around 1600, Galileo worked out why it is a parabola too: the ball's sideways motion ticks along steadily while gravity pulls it down with distance growing like t^2 — one coordinate linear, the other quadratic, and a parabola is exactly what that combination draws. Four hundred years later, every basketball free throw and every fountain still signs its flight with the graph on this page.

Try it live

Here is the whole family y = ax^2 + bx + c under your fingers. Slide the controls and watch the parabola redraw. Some experiments worth running:

Drag a through zero: the valley flattens toward a straight line, then flips into a hill.
Push a to \pm 2: the curve narrows — steeper walls, same family.
Slide c up and down: the shape is untouched; the curve just rides the lift. Watch the roots merge into one, then vanish as the curve lifts clear of the axis.
Move b: the vertex drifts along a curved path — the subtlest of the three, shifting the parabola sideways and vertically at once.

Khan Academy graphs a parabola from its roots and vertex here: