1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start- Up Day 10

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. What you’ll learn about Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Association, Correlation, and Linear Modeling Quadratic Functions and Their Graphs Applications of Quadratic Functions … and why Many business and economic problems are modeled by linear functions. Quadratic and higher-degree polynomial functions are used in science and manufacturing applications.

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2.1 Essential Question: What is a “rate of change”? What is the relationship between a quadratic function and its graph? How can we create linear/quadratic models to help us solve problems? Objective: Student will be able to recognize and graph linear and quadratic functions, and use these functions to model situations and solve problems. Home Learning: pg , 9, 13 ‐ 18, 21, 25 & 43

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Polynomial Function

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. No and Low Polynomial Functions NameFormDegree Zero Functionf(x) = 0 Undefined Constant Functionf(x) = a (a ≠ 0)0 Linear Functionf(x) = ax + b (a ≠ 0)1 Quadratic Functionf(x) = ax 2 + bx + c (a ≠ 0)2

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example 1: Which are Polynomial Functions?

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn: Try #1-#6

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Determining an Equation of a Linear Function

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution:

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Slope = Average Rate of Change

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Constant Rate of Change Theorem A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph.

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn: (#8) Write a linear equation that satisfies the given conditions & include a graph.

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Linear Correlation?

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Properties of the Correlation Coefficient, r 1. –1 ≤ r ≤ 1 2. When r > 0, there is a positive linear association. 3. When r < 0, there is a negative linear association. 4. When |r| ≈ 1, there is a strong linear association.. 5. When r ≈ 0, there is weak or no linear association.

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Regression Analysis 1. Enter and plot the data (scatter plot). 2. Find the regression model that fits the problem situation. 3. Superimpose the graph of the regression model on the scatter plot, and observe the fit. 4. Use the regression model to make the predictions called for in the problem.

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc.

17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Vertex Form of a Quadratic Equation Any quadratic function f(x) = ax 2 + bx + c, a ≠ 0, can be written in the vertex form f(x) = a(x – h) 2 + k The graph of f is a parabola with vertex (h, k) and axis x = h, where h = –b/(2a). If a > 0, the parabola opens upward, and if a < 0, it opens downward.

18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. The Graph of f(x)=ax 2

19 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Understanding Transformations of a Quadratic Function

20 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding the Vertex and Axis of a Quadratic Function

21 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

22 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Completing the Square to describe the graph of a Quadratic Function

23 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

24 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Can You Write the Equation?

25 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Characterizing the Nature of a Quadratic Function Point of Characterization View Verbalpolynomial of degree 2 Algebraic f(x) = ax 2 + bx + c or f(x) = a(x – h) 2 + k (a ≠ 0) Graphicalparabola with vertex (h, k) and axis x = h; opens upward if a > 0, opens downward if a < 0; initial value = y-intercept = f(0) = c;

26 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn: (#30) Rewrite the equation in vertex form. Identify the vertex and axis of the graph. (#34) Complete the Square to rewrite in vertex form. Describe the graph.

27 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Vertical Free-Fall Motion

28 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Let’s Try It: Vertical Free-Fall Motion