Quadratic functions: move a, b and c and watch the parabola respond

How to use this simulation

1. Start by just watching: the screen shows y = x² − 2x − 3. The parabola crosses the x-axis at −1 and 3, and those two green dots are the roots.
2. Drag the a slider below zero and the parabola flips to open downwards. Set a to exactly 0 and the curve straightens into a line.
3. Raise c to lift the whole curve, and watch the discriminant D in the readings: D = 0 at the moment it touches the x-axis, and negative once it floats above it.
4. Grab the orange vertex with your finger or mouse and move it, or pick “Vertex form” and use the h and k sliders. The axis of symmetry follows.
5. Keep “Show completing the square” on: five steps appear one by one below the graph, using your own numbers, and end in vertex form.
6. Pick “Factored form”, set the roots α and β directly, and check the sum −b/a and product c/a in the readings.

Throw a ball, watch a fountain: the same curve everywhere

Toss a ball up and a little forward. It slows as it rises, pauses for an instant at the top, and falls back down. Trace its path and you get a smooth upside-down U. Mathematicians call that shape a parabola, and the equation behind it is a quadratic.

The arc of water from a drinking fountain, the dish of a satellite antenna, the mirror inside a torch and the curve of some bridges all belong to the same family. Each one follows a rule of the form y = ax² + bx + c.

Shopkeepers use this maths without knowing it. Price something too low and the profit is thin; price it too high and nobody buys. Somewhere in the middle is the price that earns the most. A graph of profit against price is a downward parabola, and the best price is its vertex.

On this page you will change a, b and c in the simulation and see for yourself how the parabola moves, when it cuts the x-axis twice, when it just touches, and when it misses completely. Then you will learn three ways to solve a quadratic equation and work through a set of exam-style problems.

Starting from zero: what is a quadratic?

The degree of a term is how many times the variable is multiplied by itself: x has degree 1, x² has degree 2, x³ has degree 3. An equation whose highest power of the variable is 2 is called a quadratic equation. “Quadratic” comes from the Latin quadratus, a square, because the key term is x squared.

The standard form of a quadratic equation in one variable is ax² + bx + c = 0. Here a is the coefficient of x², b is the coefficient of x and c is the constant term. They are real numbers, and one condition is essential: a ≠ 0. If a were 0 the x² term would vanish and you would be left with bx + c = 0, which is a linear equation.

b or c can be zero without any problem. x² − 9 = 0 (where b = 0) and x² − 4x = 0 (where c = 0) are both quadratics. Only a is not allowed to be zero. Slide a to 0 in the simulation and the parabola straightens into a line, with a label saying it is no longer quadratic.

Keep the difference between an equation and a function clear. ax² + bx + c = 0 is an equation: it asks “which values of x make the left-hand side zero?” y = ax² + bx + c is a function: it gives one y for every x. The graph of the function is a parabola, and the x-values where it meets the x-axis (where y = 0) are exactly the solutions of the equation.

A root, zero or solution is a number that makes the equation true when you put it in place of x. A quadratic equation has at most two roots. Sometimes there are two different roots, sometimes the two roots are equal, and sometimes there are no real roots at all. The discriminant, coming up shortly, tells you which.

Key terms

Definition questions come straight from this list, so it is worth learning the words precisely.

Method 1: solving by factoring

Factoring is usually the fastest method when it works. It rests on one simple fact, the zero product property: if two numbers multiply to zero, at least one of them must be zero. 3 × 0 = 0 and 0 × 7 = 0, but 3 × 7 is never 0.

So if you can write ax² + bx + c as a product of two linear factors, such as (x − p)(x − q) = 0, then either x − p = 0 or x − q = 0. That gives x = p or x = q. One quadratic has become two tiny linear equations.

When a = 1, look for two numbers that multiply to c and add to b, then split the middle term bx using them. When a ≠ 1, the two numbers must multiply to a × c and add to b (the “ac method”), and you group the four terms in pairs.

Factoring has limits. If the roots are awkward fractions or irrational numbers such as 2 + √3, you will never find neat integer factors. That is when completing the square or the formula takes over. It is also why the simulation’s factored form only works when there are real roots.

Method 2: completing the square

Completing the square means rearranging the equation so that one side is a perfect square (x + p)². Why bother? Because once you have (x + p)² = r, you take square roots to get x + p = ±√r, and x is one step away.

The trick comes from the identity (x + p)² = x² + 2px + p². The coefficient of x is 2p and the last term is p². So if you take half the coefficient of x and square it, adding that number turns the expression into a perfect square. x² + 6x plus (6/2)² = 9 gives x² + 6x + 9 = (x + 3)².

The steps are always the same: (1) divide the whole equation by a so that x² has coefficient 1; (2) move the constant to the right-hand side; (3) add (half the x coefficient)² to both sides; (4) write the left side as (x + p)²; (5) take square roots, remembering the ± sign.

This method works for every quadratic, whether or not it factors. Even more important, it is where the quadratic formula comes from: run exactly these steps on the general equation ax² + bx + c = 0 and the formula drops out. Below the simulation the same steps run on whatever numbers you choose.

Method 3: the quadratic formula, derived

Below, completing the square is applied to the general equation ax² + bx + c = 0. Each line follows from the one before, so read it as an argument rather than something to memorise. (In India this is often called Sridharacharya’s formula, after the mathematician who described the method more than a thousand years ago.)

In the first step we divide by a, which is allowed because a ≠ 0. Next c/a moves to the right. Then (b/2a)² is added to both sides, since half of the x coefficient b/a is b/2a. Now the left side is a perfect square, and the right side is put over the common denominator 4a².

Taking the square root of both sides brings in ±, because every positive number has two square roots (the square roots of 9 are +3 and −3). Moving b/2a across finishes the job. Notice what is under the root: b² − 4ac, the discriminant. If it is negative the square root is not a real number, so there are no real roots.

The discriminant: the nature of the roots without solving

The expression under the square root, b² − 4ac, is called the discriminant and written D (or Δ). It discriminates between the possible kinds of answer: from its sign alone you know how many real roots there are and what they are like, before you solve anything.

If D > 0, √D is a positive real number, so the + and − give two different real roots, and the parabola cuts the x-axis at two points. If D is also a perfect square (16, 25, …) and a, b, c are rational, the roots are rational and the quadratic factors nicely. If not, the roots are irrational.

If D = 0, √D = 0 and both signs give the same answer, x = −b/2a. We say there are two equal real roots (a repeated root). On the graph the parabola touches the x-axis at its vertex without crossing it.

If D < 0, you would need the square root of a negative number, which is not real. So there are no real roots and the parabola never meets the x-axis. Later, once you meet complex numbers, you will say the roots form a complex conjugate pair.

Sum and product of the roots

If α and β are the roots of ax² + bx + c = 0, adding the two formula answers cancels the ± parts and leaves α + β = −b/a. Multiplying them uses (p + q)(p − q) = p² − q² and gives αβ = c/a. These two relations (sometimes called Vieta’s formulas) appear constantly in exams.

Their power is that you can answer questions about the roots without finding them. For example α² + β² = (α + β)² − 2αβ, and 1/α + 1/β = (α + β)/αβ. With irrational roots this saves a lot of messy arithmetic.

They also work in reverse. If you know the roots, the equation is x² − (sum of roots)x + (product of roots) = 0, because multiplying out (x − α)(x − β) gives exactly x² − (α + β)x + αβ.

The graph: parabola, vertex and axis of symmetry

The graph of y = ax² + bx + c is always a parabola. If a > 0 it opens upwards like a U, the vertex is the lowest point and the function has a minimum value. If a < 0 it opens downwards, the vertex is the highest point and the function has a maximum value.

The bigger |a| is, the narrower and steeper the parabola; the smaller |a| is, the wider and flatter. In the simulation, bring a down from 3 to 0.5 and watch the curve spread out.

The vertex is (h, k) with h = −b/2a and k = c − b²/4a (or just substitute x = h to find y). The vertical line x = h through the vertex is the axis of symmetry: fold the graph along it and the two arms match exactly. So when there are two roots they sit the same distance either side of the axis, and h is their average.

Writing the function as y = a(x − h)² + k, found by completing the square, is called vertex form. Because (x − h)² is never negative, when a > 0 the smallest value of y happens at x = h and equals k. It is the quickest form for reading the vertex at a glance.

The effect of b is the surprising one: changing b moves the vertex along a curved path rather than sliding the whole parabola sideways. And c is always the y-intercept, because putting x = 0 leaves y = c. On the starting graph that is (0, −3).

Try this in the simulation

Each experiment takes a minute or two. Predict what will happen first, then check.

Worked examples (every number computed)

Each problem is laid out the way an examiner likes to see it. The first one is the simulation’s starting graph, so you can compare with the readings on screen.

Common mistakes

These cost marks every year. Knowing them in advance is half the battle.

• Getting the sign of −b wrong: if b = −2 then −b = +2, not −2.
• Forgetting that −4ac becomes positive when a and c have opposite signs.
• Taking only the positive square root and losing one of the two roots.
• Dividing only the square-root part by 2a instead of the whole numerator.
• Dividing both sides of x² = 5x by x, which throws away the root x = 0. Write x(x − 5) = 0 instead.
• Reading off a, b and c before rearranging the equation into ax² + bx + c = 0.
• Keeping a root that makes no sense in a word problem, such as a negative length or time.
• Writing the vertex x-coordinate as b/2a instead of −b/2a.

Quadratics in real life

Parabolas and quadratic equations are hiding all around you:

• The path of a thrown ball, a football kick or a jet of water is a parabola if you ignore air resistance.
• Satellite dishes and torch reflectors are parabolic because a parabola gathers parallel rays at a single point, the focus.
• Businesses find maximum profit or minimum cost at the vertex of a quadratic model.
• Builders and designers find side lengths from an area, which leads straight to a quadratic.
• A car’s braking distance grows with the square of its speed, so doubling your speed roughly quadruples the distance.
• The cables of a suspension bridge hang in a shape very close to a parabola.

Exam corner

Most exam systems test quadratics in three ways: solving (by factoring, completing the square or the formula), analysing (discriminant, vertex, axis, intercepts) and modelling (area, projectile and profit problems). Check which methods your syllabus expects you to show: some papers insist on completing the square for particular questions.

For graph sketching, collect the key features before drawing: the direction of opening from the sign of a, the y-intercept (0, c), the roots if D ≥ 0, and the vertex. Plot these, draw a smooth symmetric curve, and label everything.

For multiple-choice questions, remember the shortcuts: sum of roots −b/a, product c/a; D = 0 means equal roots; and if a and c have opposite signs then D is automatically positive, so there are certainly two real roots.

One-screen revision summary

Everything you need the night before the exam:

• Standard form: ax² + bx + c = 0, with a ≠ 0.
• Three methods: factoring, completing the square, the quadratic formula.
• Formula: x = (−b ± √(b² − 4ac)) / 2a.
• Discriminant D = b² − 4ac: > 0 two real roots, = 0 equal roots, < 0 no real roots.
• α + β = −b/a, αβ = c/a; the equation from its roots is x² − (α + β)x + αβ = 0.
• The graph is a parabola; a > 0 opens up, a < 0 opens down.
• Vertex (−b/2a, c − b²/4a); axis of symmetry x = −b/2a; y-intercept (0, c).
• Vertex form y = a(x − h)² + k.

Frequently asked questions