Copyright © Cengage Learning. All rights reserved. 4 Quadratic Functions

Copyright © Cengage Learning. All rights reserved. 4.1 Quadratic Functions and Parabolas

3 Objectives  Recognize a quadratic from its graph and equation.  Identify when a quadratic graph is increasing or decreasing.  Identify the vertex of a parabola and explain its meaning.

4 Introduction to Quadratics and Identifying the Vertex

5 Let us consider the functions T(m) = –1(m – 8) and B(t) = 28.7t 2 – 286.2t These functions are not linear, since they have a squared term and cannot be written in the general form of a line, Ax + By = C. These functions are instead called quadratic functions. Quadratic functions are most commonly represented by either the standard form or the vertex form.

6 Introduction to Quadratics and Identifying the Vertex

7 Parabolas can be described as a “U shape” or an “upside-down U shape.” A parabola will change from increasing to decreasing or from decreasing to increasing. The point where the parabola turns around and goes in the opposite direction is called the vertex. The vertex represents the point where the lowest or minimum value occurs on the graph if the graph opens upward and represents the highest or maximum value on the graph if the graph opens downward.

8 Introduction to Quadratics and Identifying the Vertex Examples of each of these types of graphs are shown below. Data that show this pattern can be modeled by using a quadratic function. Parabola opens upward. Vertex: (4, 1) Lowest point Minimum point Decreasing when x 4

9 Introduction to Quadratics and Identifying the Vertex Parabola opens downward. Vertex: (5, 4) Highest point Maximum point Increasing when x 5

10 Introduction to Quadratics and Identifying the Vertex

11 Recognizing Graphs of Quadratic Functions and Identifying the Vertex

12 Example 1 – Reading a quadratic graph Use the graph of f (x) to estimate the following. a. For what x-values is this curve increasing? Decreasing? Write your answer as inequalities. b. Vertex c. x-intercept(s) d. y-intercept e. f (5) = ? f. What x-value(s) will make f (x) = –2?

13 Example 1 – Solution a. Reading the graph from left to right, we see that the curve is increasing for x 2. b. This curve changes from increasing to decreasing when x = 2, so the vertex is (2, 3). c. The curve crosses the x-axis at x = 0 and x = 4, so (0, 0) and (4, 0) are the x-intercept(s).

14 Example 1 – Solution d. The curve crosses the y-axis at y = 0, so the y-intercept is (0, 0). e. When x = 5, the curve has an output of about y = –3.5, so f (5) = –3.5. f. The output of the function is y = –2 when the input is about x = –0.5 and x = –4.5. cont’d