Quadratic functions

What it is

A quadratic function is a function f(x) = ax² + bx + c, where a, b and c are fixed numbers and a ≠ 0. Its graph is a parabola. With a = 0 the x² term disappears and what is left is a linear function; that is why the definition requires a ≠ 0.

The three numbers steer the parabola in different ways:

• a decides which way the parabola opens: upwards when a > 0, with a lowest point, and downwards when a < 0, with a highest point. The larger |a| is, the narrower the parabola.
• c is f(0), where the parabola crosses the y-axis.
• b, together with a, decides where the parabola turns.

Three properties are asked about most: the axis of symmetry, the vertex and the zeros. Take f(x) = x² − 4x + 3, with a = 1, b = −4 and c = 3.

• The axis of symmetry is the vertical line the parabola is mirror-symmetric about: x = −b/(2a) = 4/2 = 2. The reason is on the axis of symmetry page.
• The vertex lies on the axis of symmetry: f(2) = 4 − 8 + 3 = −1, so (2, −1). Since a > 0 it is a lowest point.
• The zeros are the values of x where f(x) = 0, where the graph crosses the x-axis. x² − 4x + 3 = 0 gives x = 2 ± √(4 − 3) = 2 ± 1, so x = 1 and x = 3. They lie equally far from the axis of symmetry, one step each way.

With a < 0 everything turns over: f(x) = −x² + 6x − 5 has the highest point (3, 4) and the zeros 1 and 5.

The same function can be written in three ways, and each one shows something:

• f(x) = x² − 4x + 3 shows where it crosses the y-axis, c = 3.
• f(x) = (x − 2)² − 1, the vertex form, shows the vertex (2, −1).
• f(x) = (x − 1)(x − 3) shows the zeros 1 and 3.

The number of zeros is two, one or none, and the discriminant decides which. f(x) = x² − 4x + 3 has two. f(x) = x² has one, x = 0, because the lowest point lies on the x-axis. f(x) = x² + 1 has none, because the lowest point (0, 1) lies above the x-axis.

Not quadratic functions:

• f(x) = 2x + 1. Here a = 0, and the graph is a straight line.
• f(x) = x³ − x. The highest power is 3, so it is a cubic function.
• x = y². A parabola lying on its side, but not a function of x: x = 4 gives both y = 2 and y = −2.

Why it matters

Quadratic functions appear wherever something has a largest or smallest value. Without air resistance and with g ≈ 10 m/s², the height of a ball thrown straight up at 20 m/s is h(t) = −5t² + 20t metres after t seconds. The axis of symmetry t = −20/(2 · (−5)) = 2 gives the greatest height h(2) = −20 + 40 = 20 m, and the zeros t = 0 and t = 4 are the throw and the landing. A rectangle with perimeter 20 has sides x and 10 − x and area A(x) = x(10 − x) = −x² + 10x, which is largest, 25, when x = 5.

Which tool fits depends on the question:

• The zeros come from the quadratic formula or the pq formula, and the equation solver shows the steps.
• The vertex comes from the axis of symmetry, or from completing the square to vertex form.
• The factored form is tied to the factor theorem: x − 1 and x − 3 are factors because 1 and 3 are zeros.

Change a, b and c with the sliders in the graphing calculator and watch the parabola turn, widen and move.

Common misconceptions

• “A parabola has two zeros.” It can have two, one or none, like x² − 4x + 3, x² and x² + 1. The discriminant decides.
• “c is the lowest point of the parabola.” c is f(0), where it crosses the y-axis. For x² − 4x + 3, c = 3 but the lowest value is −1. The two coincide only when b = 0.
• “Axis of symmetry, vertex and zero are three names for one thing.” The axis of symmetry is a vertical line, x = 2. The vertex is a point on it, (2, −1). The zeros are the values of x where y = 0, here 1 and 3.
• “If I double x, y doubles.” That holds for f(x) = kx, where y is proportional to x. For f(x) = x², y becomes four times as large: f(3) = 9 and f(6) = 36.

Common questions