The shape of a quadratic

A quadratic is an expression whose highest power of $x$ is $x^2$, like $x^2 - x - 6$ or $2x^2 + 3x - 5$. We usually write it as

where $a$, $b$ and $c$ are numbers and $a$ is not 0. Before solving quadratics, it helps to see one. Every quadratic draws the same kind of curve, called a parabola.

See it for yourself

Work through the challenges on the board. Try to predict which slider to move before you touch it.

What each number does

• $a$ decides the shape. If $a$ is positive the curve opens upwards like a cup, so it has a lowest point. If $a$ is negative it opens downwards, with a highest point. The bigger $a$ is (ignoring the sign), the narrower the curve.
• $c$ is where the curve cuts the $y$-axis. Put $x = 0$ into $y = ax^2 + bx + c$ and everything disappears except $c$. So the curve always passes through $(0, c)$.
• $b$ slides the curve sideways (and up or down with it). You’ll see exactly how in lesson 3.

Roots: where the curve meets the x-axis

On the $x$-axis, $y = 0$. So the points where the curve meets the $x$-axis are the values of $x$ that make $ax^2 + bx + c = 0$. These are called the roots of the equation (or its solutions).

Between its two roots the curve stays on one side of the $x$-axis. For $y = x^2 - x - 6$ (a cup shape), the curve is below the axis between $x = -2$ and $x = 3$, so $y < 0$ there, and above it outside. That is how a quadratic inequality is solved: find the roots, then read which side you need (more in inequalities↺).

The vertex and the line of symmetry

A parabola is symmetrical. The line down its middle is the line of symmetry, and the curve turns round on it at the vertex (its lowest or highest point). When there are two roots, the line of symmetry is exactly halfway between them.

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