Graphs of Quadratic Functions Any quadratic function can be expressed in the form Where a, b, c are real numbers and the graph of any quadratic function is called a parabola. Graphing Quadratic Functions

Graphing Quadratic Functions With Equations in the Form T To graph 1) Determine whether the parabola opens upward or downward. If a > 0, it opens upward. It a < 0, it opens downward. 2) Determine the vertex of the parabola. The vertex is (h, k). 3) Find any x-intercepts by replacing f (x) with 0. Solve the resulting Quadratic equation for x. 4) Find the y-intercept by replacing x with 0. 5) Plot the intercepts and vertex and additional points as necessary. Connect these points with a smooth curve that is shaped like a cup. Graphing Quadratic Functions

EXAMPLE Graph the function SOLUTION 1) Determine how the parabola opens. Graphing Quadratic Functions 2) Find the vertex. The vertex of the parabola is at (h, k). 3) Find the x-intercepts. 4) Find the y-intercept. 5) Graph the parabola.

The Vertex of a Parabola Whose Equation is T Consider the parabola defined by the quadratic function The parabola’s vertex is Graphing Quadratic Functions

EXAMPLE Graph the function SOLUTION 1) Determine how the parabola opens. Graphing Quadratic Functions 2) Find the vertex. 3) Find the x-intercepts. 4) Find the y-intercept. 5) Graph the parabola.

Minimum and Maximum: Quadratic Functions Consider 1) If a > 0, then f has a minimum that occurs at This minimum value is 2) If a < 0, then f has a maximum that occurs at This maximum value is Minimums & Maximums

EXAMPLE A person standing close to the edge on the top of a 200-foot building throws a baseball vertically upward. The quadratic function models the ball’s height above the ground, s (t), in feet, t seconds after it was thrown. When will the ball reach its highest point? How high is it? How many seconds does it take until the ball finally hits the ground? Round to the nearest tenth of a second. Minimums & Maximums

11.3 Quadratic Functions & Their Graphs Reference: Blitzer’s Introductory and Intermediate Algebra