5.1 – Introduction to Quadratic Functions Objectives: Define, identify, and graph quadratic functions. Multiply linear binomials to produce a quadratic expression. Standard: E. Use equations to represent curves.

I. Quadratic function is any function that can be written in the form f(x)= ax 2 + bx + c, where a ≠ 0.

Ex 2. Let f(x) = (2x – 5)(x - 2). Show that f represents a quadratic function. Identify a, b, and c when the function is written in the form f(x) = ax 2 + bx + c. FOIL  First – Outer – Inner – Last (2x – 5)(x – 2) = 2x 2 – 4x – 5x x 2 – 9x + 10 a = 2, b = -9, c = 10

II. The graph of a quadratic function is called a parabola. Each parabola has an axis of symmetry, a line that divides the parabola into two parts that are mirror images of each other. The vertex of a parabola is either the lowest point on the graph or the highest point on the graph. Axis of Symmetry Vertex

Example 1

Example 2 Ex 2. Identify whether f(x) = -2x 2 - 4x + 1 has a maximum value or a minimum value at the vertex. Then give the approximate coordinates of the vertex. First, graph the function: Next, find the maximum value of the parabola (2 nd, Trace): Finally, max(-1, 3).

III. Minimum and Maximum Values Let f(x) = ax 2 + bx + c, where a ≠ 0. The graph of f is a parabola. If a > 0, the parabola opens up and the vertex is the lowest point. The y- coordinate of the vertex is the minimum value of f. If a < 0, the parabola opens down and the vertex is the highest point. The y-coordinate of the vertex is the maximum value of f.

Ex 1. State whether the parabola opens up or down and whether the y-coordinate of the vertex is the minimum value or the maximum value of the function. Then check by graphing it in your Y = button on your calculator. Remember: F(X) means the same thing as Y! a. f(x) = x 2 + x – 6 b. g(x) = 5 + 4x – x 2 c. f(x) = 2x 2 - 5x + 2 d. g(x) = 7 - 6x - 2x 2 Opens up, has minimum value Opens down, has maximum value Opens up, has minimum value Opens down, has maximum value

Writing Activities 1. a. Give an example of a quadratic function that has a maximum value. How do you know that it has a maximum? 1. b. Give an example of a quadratic function that has a minimum value. How do you know that it has a minimum?