Quadratic Function

The equation of a quadratic function is:

y = ax² + bx +c

Its graph is a parabola.

Graphical Representation of the Parabola

A parabola can be built from these points:

1. Vertex

Finding the Vertex

The axis of symmetry passes through the vertex of the parabola.

The equation of the axis of symmetry is:

Axis of Symmetry

2. x-intercepts

For the intercept with the x-axis, the second coordinate is always zero:

ax² + bx +c = 0

To find the x-intecepts, solve the resultant quadratic equation:

Two intercept points: (x1, 0) (x2, 0) if b² − 4ac > 0

One intercept point: (x1, 0) if b² − 4ac = 0

No intercept points if b² − 4ac < 0

3. y-intercept

For the intercept with the y-axis, the first coordinate is always zero:

f(0) = a · 0² + b · 0 + c = c        (0, c)


Example

Graph the quadratic function y = x² − 4x + 3.

1. Vertex

xv = − (−4)/2 = 2     yv = 2² − 4 · 2 + 3 = −1       

 V(2, −1)

2. x-intercepts

x² - 4x + 3 = 0

ecuación       

(3, 0)      (1, 0)

3. y-intercept

(0, 3)

Quadratic Function