1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.1Numbers, Data, and Problem Solving 1.2Visualization of Data 1.3Functions and Their Representations 1.4Types of Functions and Their Rates of Change Introduction to Functions and Graphs 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Numbers, Data and Problem Solving ♦Recognize common sets of numbers ♦Learn scientific notation and use it in applications ♦Apply problem solving strategies 1.1

3. Slide 1- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Natural Numbers and Integers Natural Numbers (or counting numbers)Natural Numbers (or counting numbers) are numbers in the set N = {1, 2, 3,...}. Integers are numbers in the setIntegers are numbers in the set I = {…  3,  2,  1, 0, 1, 2, 3,...}.

4. Slide 1- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Numbers Rational Numbers are real numbers which can beRational Numbers are real numbers which can be expressed as the ratio of two integers p/q where q  0 Note that: Every integer is a rational number.Every integer is a rational number. Rational numbers can be expressed as decimalsRational numbers can be expressed as decimals which either terminate (end) or repeat a sequence of digits.

5. Slide 1- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Irrational Numbers Irrational Numbers are real numbers which are notIrrational Numbers are real numbers which are not rational numbers. Irrational numbers Cannot be expressed as the ratio of two integers.Cannot be expressed as the ratio of two integers. Have a decimal representation which does notHave a decimal representation which does not terminate and does not repeat a sequence of digits.

6. Slide 1- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Classifying Real Numbers Classify each number as one or more of the following:Classify each number as one or more of the following: natural number, integer, rational number, irrational natural number, integer, rational number, irrational number. number.

7. Slide 1- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Scientific Notation A real number r is in scientific notationA real number r is in scientific notation when r is written as c x 10 n, where and n is an integer.

8. Slide 1- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Evaluate (5 x 10 6 ) (3 x 10  4 ), writing the result in scientific notation and in standard form. Examples of Evaluating Expressions Involving Scientific Notation

9. Slide 1- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Evaluate writing the answer in scientific notation and in standard form.

10. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Visualization of Data ♦ Learn to analyze one-variable data ♦ Find the domain and range of a relation ♦ Graph a relation in the xy-plane ♦ Calculate the distance between two points ♦ Find the midpoint of a line segment ♦ Learn to graph equations with a calculator (optional) 1.2

11. Slide 1- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Two-Variable Data: Relations A relation is a set of ordered pairs.A relation is a set of ordered pairs. If we denote the ordered pairs by (x, y)If we denote the ordered pairs by (x, y) The set of all x  values is the DOMAIN.The set of all x  values is the DOMAIN. The set of all y  values is the RANGE.The set of all y  values is the RANGE.Example The relation {(1, 2), (  2, 3), (  4,  4), (1,  2), (  3,0), (0,  3)}The relation {(1, 2), (  2, 3), (  4,  4), (1,  2), (  3,0), (0,  3)} Has domain D =Has domain D = And range R =And range R =

12. Slide 1- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The relation {(1, 2), (  2, 3), (  4,  4), (1,  2), (  3, 0), (0,  3)} has graph

13. Slide 1- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) in the xy-planeis (x 1, y 1 ) and (x 2, y 2 ) in the xy-plane is

14. Slide 1- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Using Distance Formula Use the distance formula to find the distance between the two points (  2, 4) and (1,  3).

15. Slide 1- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Midpoint Formula The midpoint of the segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) in the xy-planeis in the xy-plane is

16. Slide 1- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Using Midpoint Formula Use the midpoint formula to find the midpoint of the segment with endpoints (  2, 4) and (1,  3). Midpoint is:

17. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Functions and Their Representations ♦ Learn function notation ♦ Represent a function four different ways ♦ Define a function formally ♦ Identify the domain and range of a function ♦ Use calculators to represent functions (optional) ♦ Identify functions 1.3

18. Slide 1- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Idea Behind a Function Recall that a relation is a set of ordered pairs_______.Recall that a relation is a set of ordered pairs_______. If we think of values of x as being ________ and values of y as being _________, a function is a relation such thatIf we think of values of x as being ________ and values of y as being _________, a function is a relation such that for each input there is __________output.for each input there is __________output. This is symbolized by

19. Slide 1- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation y = f(x)y = f(x) Is pronounced “y is a function of x.”Is pronounced “y is a function of x.” Means that given a value of x (input), there is exactly one corresponding value of y (output).Means that given a value of x (input), there is exactly one corresponding value of y (output). x is called the ______________variable as it represents inputs, and y is called the ___________ variable as it represents outputs.x is called the ______________variable as it represents inputs, and y is called the ___________ variable as it represents outputs. Note that: f(x) is NOT f multiplied by x. f is NOT a variable, but the name of a function (the name of a relationship between variables).Note that: f(x) is NOT f multiplied by x. f is NOT a variable, but the name of a function (the name of a relationship between variables).

20. Slide 1- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domain and Range of a Function The set of all meaningful inputs is called the DOMAIN of the function.The set of all meaningful inputs is called the DOMAIN of the function. The set of corresponding outputs is called the RANGE of the function.The set of corresponding outputs is called the RANGE of the function. Formal Definition of a Function A function is a relation in which each element of the domain corresponds to exactly one element in the range.A function is a relation in which each element of the domain corresponds to exactly one element in the range.

21. Slide 1- 21 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Suppose a car travels at 70 miles per hour. Let y be the distance the car travels in x hours. Then y = 70 x.Suppose a car travels at 70 miles per hour. Let y be the distance the car travels in x hours. Then y = 70 x.

22. Slide 1- 22 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Input x 348 Output y 66  5 5 5 5 Given the following data, is y a function of x?

23. Slide 1- 23 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Undergraduate Classification at Tarleton State University (TSU) is a function of Hours Earned. We can write this in function notation as C = f(H).Undergraduate Classification at Tarleton State University (TSU) is a function of Hours Earned. We can write this in function notation as C = f(H). Why is C a function of H?Why is C a function of H?

24. Slide 1- 24 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) Classification of Students at SHUClassification of Students at SHU From Catalogue Evaluate f(20)Evaluate f(20) Evaluate f(30)Evaluate f(30) Evaluate f(0)Evaluate f(0) Evaluate f(61)Evaluate f(61) No student may be classified as a sophomore until after earning at least 30 semester hours. No student may be classified as a junior until after earning at least 60 hours. No student may be classified as a senior until after earning at least 90 hours.

25. Slide 1- 25 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What is the domain of f? What is the range of f?

26. Slide 1- 26 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Questions: Identifying Functions Referring to the previous example concerning TSU, is hours earned a function of classification? That is, is H = f(C)? Explain why or why not.Referring to the previous example concerning TSU, is hours earned a function of classification? That is, is H = f(C)? Explain why or why not. Is classification a function of years spent at TSU? Why or why not?Is classification a function of years spent at TSU? Why or why not?

27. Slide 1- 27 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Answers

28. Slide 1- 28 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Given x = y 2, is y a function of x?Given x = y 2, is y a function of x? Given x = y 2, is x a function of y?Given x = y 2, is x a function of y? Given y = x 2 2, is y a function of x?Given y = x 2  2, is y a function of x?

29. Slide 1- 29 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Five Ways to Represent a Function (page 31) VerballyVerbally NumericallyNumerically DiagrammaticlyDiagrammaticly SymbolicallySymbolically GraphicallyGraphically

30. Slide 1- 30 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) (Referring to previous TSU example) Verbal Representation.Verbal Representation. If you have less than 30 hours, you are a freshman.If you have less than 30 hours, you are a freshman. If you have 30 or more hours, but less than 60 hours, you are a sophomore.If you have 30 or more hours, but less than 60 hours, you are a sophomore. If you have 60 or more hours, but less than 90 hours, you are a junior.If you have 60 or more hours, but less than 90 hours, you are a junior. If you have 90 or more hours, you are a senior.If you have 90 or more hours, you are a senior.

31. Slide 1- 31 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) Numeric Representation

32. Slide 1- 32 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) Symbolic Representation

33. Slide 1- 33 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) Diagrammatic Representation

34. Slide 1- 34 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley C = f(H) Graphical Representation

35. Slide 1- 35 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Notes on Graphical Representation Vertical line test (p 39). To determine if a graph represents a function, simply visualize vertical lines in the xy-plane. If each vertical line intersects a graph at no more than one point, then it is the graph of a function.Vertical line test (p 39). To determine if a graph represents a function, simply visualize vertical lines in the xy-plane. If each vertical line intersects a graph at no more than one point, then it is the graph of a function. input output

36. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Types of Functions and Their Rates of Change ♦ Identify and use constant and linear functions ♦ Interpret slope as a rate of change ♦ Identify and use nonlinear functions ♦ Recognize linear and nonlinear data ♦ Use and interpret average rate of change ♦ Calculate the difference quotient 1.4

37. Slide 1- 37 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley f(x) = 2 Constant Function A function f represented by f(x) = b,A function f represented by f(x) = b, where b is a constant (fixed number), is a constant function. Examples: Note: Graph of a constant function is a horizontal line.

38. Slide 1- 38 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley f(x) = 2x + 3 Linear Function A function f represented by f(x) = ax + b,A function f represented by f(x) = ax + b, where a and b are constants, is a linear function. Examples: Note that a f(x) = 2 is both a linear function and a constant function. A constant function is a special case of a linear function.

39. Slide 1- 39 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rate of Change of a Linear Function Table of values for f(x) = 2x + 3.Table of values for f(x) = 2x + 3.

40. Slide 1- 40 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope of Line The slope m of the line passing through the points (x 1, y 1 ) and (x 2, y 2 ) isThe slope m of the line passing through the points (x 1, y 1 ) and (x 2, y 2 ) is

41. Slide 1- 41 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Calculation of Slope Find the slope of the line passing through theFind the slope of the line passing through the points (  2,  1) and (3, 9).

42. Slide 1- 42 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of a Nonlinear Function Table of values for f(x) = x 2Table of values for f(x) = x 2 xy 00 11 24

43. Slide 1- 43 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Average Rate of Change Let (x 1, y 1 ) and (x 2, y 2 ) be distinct points on theLet (x 1, y 1 ) and (x 2, y 2 ) be distinct points on the graph of a function f. The average rate of change of f from x 1 to x 2 is Note that the average rate of change of f from x 1 to x 2 is the slope of the line passing through (x 1, y 1 ) and (x 2, y 2 )

44. Slide 1- 44 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Difference Quotient The difference quotient of a function f is anThe difference quotient of a function f is an expression of the form where h is not 0. Note that a difference quotient is actually an average rate of change. Note that a difference quotient is actually an average rate of change.

45. Slide 1- 45 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example of Calculating a Difference Quotient Let f(x) = x 2 + 3x. Find the differenceLet f(x) = x 2 + 3x. Find the difference quotient of f and simplify the result.