1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.1 Linear and Quadratic Functions and Modeling

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6 What you’ll learn about 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 to model some manufacturing applications.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Polynomial Function

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 8 Polynomial Functions of No and Low Degree

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Example Finding an Equation of a Linear Function

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10 Average Rate of Change

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11 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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12 Characterizing the Nature of a Linear Function Point of ViewCharacterization Verbalpolynomial of degree 1 Algebraic f(x) = mx + b (m≠0) Graphicalslant line with slope m and y-intercept b Analyticalfunction 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

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 13 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.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 14 Linear Correlation

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 15 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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 16 Example Transforming the Squaring Function

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 17 The Graph of f(x)=ax 2

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 18 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) and k = c – ah 2. If a > 0, the parabola opens upward, and if a < 0, it opens downward.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19 Example Finding the Vertex and Axis of a Quadratic Function

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 20 Characterizing the Nature of a Quadratic Function Point of View Characterization

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 21 Vertical Free-Fall Motion

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 22 Free Fall Example A machine throws a baseball straight up with an initial velocity of 48 fps from a height of 3.5ft. A.Find an equation that models the height of the ball t seconds after it is thrown.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 23 Free Fall Example B. What is the maximum height the ball reaches? How many seconds does it take to reach that height?

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 24 Homework Review Section 2.1 Page 182, Exercises: 1 – 77 (EOO)

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.2 Power Functions and Modeling

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 26 Quick Review

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 27 Quick Review Solutions

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 28 What you’ll learn about Power Functions and Variation Monomial Functions and Their Graphs Graphs of Power Functions Modeling with Power Functions … and why Power functions specify the proportional relationships of geometry, chemistry, and physics.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 29 Power Function Any function that can be written in the form f (x) = k·x a, where k and a are nonzero constants, is a power function. The constant a is the power, and the k is the constant of variation, or constant of proportion. We say f (x) varies as the a th power of x, or f (x) is proportional to the a th power of x.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 30 Example Analyzing Power Functions

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 31 Monomial Function Any function that can be written as f(x) = k or f(x) = k·x n, where k is a constant and n is a positive integer, is a monomial function.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 32 Example Graphing Monomial Functions

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 33 Graphs of Power Functions For any power function f(x) = k·x a, one of the following three things happens when x < 0. f is undefined for x < 0. f is an even function. f is an odd function.

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 34 Graphs of Power Functions

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 35 Example, Keeping Warm Scientist have determined that the pulse rate r of mammals is a power function their body weight w. MammalWeight (kg)Pulse rate (bpm) Rat0.2420 Guinea pig0.3300 Rabbit2205 Small dog5120 Large dog3085 Sheep5070 Human7072

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 36 Example, Keeping Warm a. Draw a scatter plot of the data. b. Find the power regression model

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 37 Example, Keeping Warm c. Superimpose the regression curve on the scatter. d. Predict the pulse rate of a 450-kg horse. Compare to the 38 bpm reported by Clark.