# 2 1 Linear and Quadratic Functions and Modeling

- Slides: 26

2. 1 Linear and Quadratic Functions and Modeling Copyright © 2011 Pearson, Inc.

What you’ll learn about n n n Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and 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. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 2

Polynomial Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 3

Polynomial Functions of No and Low Degree Name Zero Function Form f(x) = 0 Degree Undefined Constant Function f(x) = a (a ≠ 0) 0 Linear Function f(x) = ax + b (a ≠ 0) 1 Quadratic Function f(x) = ax 2 + bx + c (a ≠ 0) Copyright © 2011 Pearson, Inc. 2 Slide 2. 1 - 4

Example Finding an Equation of a Linear Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 5

Example Finding an Equation of a Linear Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 6

Average Rate of Change Copyright © 2011 Pearson, Inc. Slide 2. 1 - 7

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. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 8

Characterizing the Nature of a Linear Function Point of View Characterization Verbal polynomial of degree 1 Algebraic f(x) = mx + b (m≠ 0) Graphical slant line with slope m, y-intercept b Analytical function with constant nonzero rate of change m: f is increasing if m > 0, decreasing if m < 0; initial value of the function = f(0) = b Copyright © 2011 Pearson, Inc. Slide 2. 1 - 9

Properties of the Correlation Coefficient, r 1. – 1 ≤ r ≤ 1 2. When r > 0, there is a positive linear correlation. 3. When r < 0, there is a negative linear correlation. 4. When |r| ≈ 1, there is a strong linear correlation. 5. When |r| ≈ 0, there is weak or no linear correlation. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 10

Linear Correlation Copyright © 2011 Pearson, Inc. Slide 2. 1 - 11

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. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 12

Example Transforming the Squaring Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 13

Example Transforming the Squaring Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 14

The Graph of f(x)=ax 2 Copyright © 2011 Pearson, Inc. Slide 2. 1 - 15

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/(2 a) and k = c – ah 2. If a > 0, the parabola opens upward, and if a < 0, it opens downward. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 16

Example Finding the Vertex and Axis of a Quadratic Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 17

Example Finding the Vertex and Axis of a Quadratic Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 18

Example Using Algebra to Describe the Graph of a Quadratic Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 19

Example Using Algebra to Describe the Graph of a Quadratic Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 20

Example Using Algebra to Describe the Graph of a Quadratic Function Copyright © 2011 Pearson, Inc. Slide 2. 1 - 21

Example Using Algebra to Describe the Graph of a Quadratic Function The graph of f is a downwardopening parabola with vertex (3/2, 1) and axis of symmetry x = 3/2. The x-intercepts are at x = 1 and x = 2. Copyright © 2011 Pearson, Inc. Slide 2. 1 - 22

Characterizing the Nature of a Quadratic Function Point of View Verbal Characterization Algebraic f(x) = ax 2 + bx + c or f(x) = a(x – h)2 + k (a ≠ 0) Graphical parabola with vertex (h, k) and axis x = k; opens upward if a > 0, opens downward if a < 0; initial value = y-intercept = f(0) = c; polynomial of degree 2 Copyright © 2011 Pearson, Inc. Slide 2. 1 - 23

Vertical Free-Fall Motion Copyright © 2011 Pearson, Inc. Slide 2. 1 - 24

Quick Review Copyright © 2011 Pearson, Inc. Slide 2. 1 - 25

Quick Review Solutions Copyright © 2011 Pearson, Inc. Slide 2. 1 - 26

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