3.1 Quadratic Functions and Models

A quadratic function is a function of the form:

Properties of the Graph of a Quadratic Function Parabola opens up if a > 0; the vertex is a minimum point. Parabola opens down if a < 0; the vertex is a maximum point.

Graphs of a quadratic function f(x) = ax2 + bx + c Vertex is highest point Axis of symmetry Axis of symmetry a > 0 a < 0 Opens up Opens down Vertex is lowest point

Steps for Graphing a Quadratic Function by Hand Determine the vertex. Determine the axis of symmetry. Determine the y-intercept, f(0). Determine how many x-intercepts the graph has. If there are no x-intercepts determine another point from the y-intercept using the axis of symmetry. Graph.

Since -3 < 0 the parabola opens down. Without graphing, locate the vertex and find the axis of symmetry of the following parabola. Does it open up or down? Vertex: Since -3 < 0 the parabola opens down.

Finding the vertex by completing the square:

(2,4) (0,0)

(0,0) (2, -12)

(2, 0) (4, -12)

Vertex (2, 13)

Determine whether the graph opens up or down. Find its vertex, axis of symmetry, y-intercept, x-intercept. x-coordinate of vertex: y-coordinate of vertex: Axis of symmetry:

There are two x-intercepts:

(0, 5) (-5.55, 0) (-0.45, 0) Vertex: (-3, -13)