Quadratic Functions and Modeling

Properties of Quadratic Functions

Parabolas

The graph of a quadratic function is a parabola.
A quadratic function is a function, where aa, bb, and cc are real numbers and a0a\neq0, written in the form:
f(x)=ax2+bx+cf(x)=ax^2+bx+c
The shape of the graph of a quadratic function is a parabola. Any object thrown into the air is a projectile. When a projectile, such as a baseball, is thrown into the air at an angle, it follows a path in the shape of a parabola. So a quadratic function can be used to model the path of this type of projectile.
A projectile thrown into the air at an angle moves upward and then downward along a curved path. If the effect of air resistance is ignored, this path has the shape of a parabola.

Properties of Parabolas

Every parabola has a vertex and an axis of symmetry.
A parabola can open upward or downward. The vertex is the lowest point of a parabola that opens upward or the highest point of a parabola that opens downward. The variable aa is the leading coefficient in a quadratic function of the form:
f(x)=ax2+bx+cf(x)=ax^2+bx+c
If the leading coefficient is positive, then the parabola opens upward. If the leading coefficient is negative, then the parabola opens downward. An axis of symmetry is a line that divides a graph into two halves that are mirror images. For the graph of a quadratic function, the axis of symmetry is a vertical line that passes through the vertex of the parabola.
If a parabola opens upward, then the vertex is the lowest point and the quadratic function has a minimum value. If a parabola opens downward, then the vertex is the highest point and the quadratic function has a maximum value.