Quadratic function

Quadratic function formulas

The same quadratic function can have several forms. Each one highlights different information: coefficients, vertex, or roots.

General form

Vertex form

Factored form

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shape

direction and width of the arms

position

affects the vertex

start

Y-axis intercept

general form

y = ax² + bx + c

General form

f(x) = ax² + bx + c

This appears most often in exercises. Coefficient c immediately tells you where the graph crosses the Y-axis.

Vertex form

f(x) = a(x - p)² + q

The most convenient form for reading the vertex W = (p, q) and the axis of symmetry x = p.

Factored form

f(x) = a(x - x₁)(x - x₂)

This shows the roots x₁ and x₂. It is available when the function has real roots.

What do a, b, and c mean?

Coefficient a determines the direction of the arms and the width of the parabola. Coefficient b affects the position of the axis of symmetry, and c is the value of the function for x = 0.

For f(x) = 2x² - 4x + 1 we have a = 2, b = -4, and c = 1.

How do you get to the vertex?

From the general form, first calculate the x-coordinate of the vertex:p =-b2a. Then substitute p into the function to get q.

For x² - 2x - 3 we get p = 1 and q = -4, so W = (1, -4).

Mini example with three forms

The function f(x) = x² - 2x - 3 can also be written as f(x) = (x - 1)² - 4 and f(x) = (x + 1)(x - 3).

See how to draw the graph