1. Chapter 1 Graphs and Functions
Copyright © 2014, Pearson Education, Inc.

2. Section 1.1 Graphs of Equations
Find the distance between two points.
Find the midpoint of a line segment.
Sketch a graph by plotting points.
Find the intercepts of a graph.
Find the symmetries in a graph.
Find the equation of a circle.

3. THE DISTANCE FORMULA IN THE COORDINATE PLANE
Recall that the Pythagorean Theorem states that in a right triangle with hypotenuse of length c and the legs of lengths a and b,

4. THE DISTANCE FORMULA IN THE COORDINATE PLANE
Let P(x1, y1) and Q(x2, y2) be any two points in the coordinate plane. Then the distance between P and Q, denoted d(P,Q), is given by the distance formula:

5. THE DISTANCE FORMULA IN THE COORDINATE PLANE

6. Example: Finding the Distance between Points
Find the distance between the points P(–2, 5) and Q(3, – 4). Let (x1, y1) = (–2, 5) and (x2, y2) = (3, – 4).

7. THE MIDPOINT FORMULA
The coordinates of the midpoint M(x, y) of the line segment joining P(x1, y1) and Q(x2, y2) are given by

8. Example: Find the Midpoint of a Line Segment
Find the midpoint of the line segment joining the points P(–3, 6) and Q(1, 4). Let (x1, y1) = (–3, 6) and (x2, y2) = (1, 4).

9. Definitions
Symmetry means that one portion of the graph is a mirror image of another portion. The mirror line is called the axis of symmetry or line of symmetry.

10. TESTS FOR SYMMETRY
A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph,
the point (–x, y) is also on the graph.

11. TESTS FOR SYMMETRY
2. A graph is symmetric with respect to the x-axis if, for every point (x, y) on the graph,
the point (x, –y) is also on the graph.

12. TESTS FOR SYMMETRY
3. A graph is symmetric with respect to the origin if, for every point (x, y) on the graph, the point (–x, –y)
is also on the graph.

13. Example: Checking for Symmetry
Determine whether the graph of the equation is symmetric with respect to the y-axis. Replace x with –x in the original equation.
When we replace x with –x in the equation, we obtain the original equation. The graph is symmetric with respect to the y-axis.

14. Example: Sketching a Graph Using Symmetry
OBJECTIVE Use symmetry to sketch the graph of an equation. Step 1 Test for all three symmetries. About the x-axis: Replace y with y. About the y-axis: Replace x with x. About the origins: Replace x with x and y with y.
Use symmetry to sketch the graph of y = 4x  x3.

15. Example: Sketching a Graph Using Symmetry
OBJECTIVE Use symmetry to sketch the graph of an equation. Step 2 Make a table of values using any symmetries found in Step 1.
2. Origin symmetry: If (x, y) is on the graph, so is (x, y). Use only positive x-values in the table. x
0.5 1
1.5 2
2.5
y = 4x  x3
1.875 3
2.625
5.625

16. Example: Sketching a Graph Using Symmetry
OBJECTIVE Use symmetry to sketch the graph of an equation. Step 3 Plot the points from the table and draw a smooth curve through them.

17. Example: Sketching a Graph Using Symmetry
OBJECTIVE Use symmetry to sketch the graph of an equation. Step 4 Extend the portion of the graph found in Step 3 using symmetry.

18. CIRCLE
A circle is a set of points in a Cartesian coordinate plane that are at a fixed distance r from a specified point (h, k). The fixed distance r is called the radius of the circle, and the specified point (h, k) is called the center of the circle.

19. CIRCLE
The graph of a circle with center (h, k) and radius r.

20. CIRCLE
The equation of a circle with center (h, k) and radius r is This equation is also called the standard form of an equation of a circle with radius r and center (h, k).

21. Example: Finding the Equation of a Circle
Find the standard form of the equation of the circle with center (–3, 4) and radius 7.

22. Example: Graphing a Circle
Graph each equation. Center: (0, 0) Radius: 1 Called the unit circle

23. Example: Graphing a Circle (cont)
Graph each equation. Center: (–2, 3) Radius: 5

24. GENERAL FORM OF THE EQUATION OF A CIRCLE
The general form of the equation of a circle is

25. Example: Converting the General Form to Standard Form
Find the center and radius of the circle with equation Complete the squares on both x and y.
Center: (3, – 4) Radius:

26. Section 1.2 Lines Find the slope of a line.
Write the point-slope form of the equation of a line.
Write the slope-intercept form of the equation of a line.
Recognize the equations of horizontal and vertical lines.
Recognize the general form of the equation of a line.
Find equations of parallel and perpendicular lines.

27. Definitions
Since the graphs of first degree equations in two variables are straight lines, these equations are called linear equations. We measure the “steepness” of a line by a number called its slope.

28. Definitions
The rise is the change in y-coordinates between the points and the run is the corresponding change in the x-coordinates.

29. SLOPE OF A LINE
The slope of a nonvertical line that passes through the points P(x1, y1) and Q(x2, y2) is denoted by m and is defined by The slope of a vertical line is undefined.

30. Example: Finding and Interpreting the Slope of a Line
Sketch the graph of the line that passes through the points P(1, –1) and Q(3, 3). Find and interpret the slope of the line. The graph of the line passing through the points P(1, –1) and Q(3, 3) is sketched here.

31. Example: Finding and Interpreting the Slope of a Line (cont)
The slope m of the line through P(1, –1) and Q(3, 3) is given by

32. Example: Finding and Interpreting the Slope of a Line (cont)
Interpretation: A slope of 2 means that the value of y increases two units for every one unit increase in the value of x.

33. MAIN FACTS ABOUT SLOPES LINES
1. Scanning graphs from left to right, lines with positive slopes rise and lines with negative slopes fall. 2. The greater the absolute value of the slope, the steeper the line. 3. The slope of a vertical line is undefined. 4. The slope of a horizontal line is 0.

34. POINT–SLOPE FORM OF THE EQUATION OF A LINE
If a line has slope m and passes through the point (x1, y1), then the point-slope form of an equation of the line is

35. Example: Finding an Equation of a Line with Given Point and Slope
Find the point-slope form of the equation of the line passing through the point (1, –2) and with slope m = 3. Then solve for y. We have x1 = 1, y1 = –2, and m = 3.

36. Example: Finding an Equation of a Line Passing Through Two Given Points
Find the point-slope form of the equation of the line l passing through the points (–2, 1) and (3, 7). Then solve for y. First, find the slope.

37. Example: Finding an Equation of a Line Passing Through Two Given Points (cont)
We have x1 = 3, y1 = 7.

38. Example: Finding an Equation of a Line with a Given Slope and y-intercept
Find the point-slope form of the equation of the line with slope m and y-intercept b. Then solve for y. The line passes through (0, b).

39. SLOPE–INTERCEPT FORM OF THE EQUATION OF A LINE
The slope-intercept form of the equation of the line with slope m and y-intercept b is

40. Example: Graph Using the Slope and y-intercept
Graph the line whose equation is Plot the y-intercept (0, 2). Start from the y-intercept (0, 2) and rise 2 and run 3 to locate a second point (3, 4).

41. GRAPHS OF HORIZONTAL AND VERTICAL LINES
For any constant k, the graph of the equation y = k is a horizontal line with slope 0. The graph of the equation x = k is a vertical line with undefined slope.

42. Example: Recognizing Horizontal and Vertical Lines
Discuss the graph of each equation in the xy-plane. a. y = 2 b. x = 4 a. The equation y = 2 may be considered as an equation in two variables x and y by writing 0 ∙ x + y = 2. Any ordered pair of the form (x, 2) is a solution of the equation. The graph of y = 2 is a line parallel to the x-axis and 2 units above it with a slope of 0.

43. Example: Recognizing Horizontal and Vertical Lines
Discuss the graph of each equation in the xy-plane. a. y = 2 b. x = 4 b. The equation may be written as x + 0 ∙ y = 4. Any ordered pair of the form (4, y) is a solution of the equation. The graph of x = 4 is a line parallel to the y-axis and 4 units to the right of it. The slope of a vertical line is undefined.

44. GENERAL FORM OF THE EQUATION OF A LINE
The graph of every linear equation ax + by + c = 0, where a, b, and c are constants and not both a and b are zero, is a line. The equation ax + by + c = 0 is called the general form of the equation of a line.

45. Example: Graphing a Linear Equation using Intercepts
Find the slope, y-intercept, and x-intercept of the line with equation 3x – 4y +12 = 0. Then sketch the graph.
The slope is
The y-intercept is 3.
Set y = 0 and solve for x: 3x + 12 = 0.
The x-intercept is – 4.

46. Example: Graphing a Linear Equation using Intercepts (cont)
Find the slope, y-intercept, and x-intercept of the line with equation 3x – 4y +12 = 0. Then sketch the graph. Slope = 3/4. y-intercept = (0, 3) x-intercept = (–4, 0)

47. PARALLEL AND PERPENDICULAR LINES
Let l1 and l2 be two distinct lines with slopes m1 and m2, respectively. Then l1 is parallel to l2 if and only if m1 = m2. l1 is perpendicular l2 to if and only if m1∙ m2 = –1. Any two vertical lines are parallel, and any horizontal line is perpendicular to any vertical line.

48. Example: Finding Equations of Parallel and Perpendicular Lines
Let l: ax + by + c = 0. Find the equation of each line through the point (x1, y1): (a) l1 parallel to l (b) l2 perpendicular to l Step 1 Find slope m of l.
Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8): (a) l1 parallel to l (b) l2 perpendicular to l 1. l :

49. Example: Finding Equations of Parallel and Perpendicular Lines
Let l: ax + by + c = 0. Find the equation of each line through the point (x1, y1):
(a) l1 parallel to l
(b) l2 perpendicular to l
Step 2 Write slope of l1 and l2.
The slope m1 of l1 is m.
The slope m2 of l2 is
Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8):
(a) l1 parallel to l
(b) l2 perpendicular to l 2.

50. Example: Finding Equations of Parallel and Perpendicular Lines
Let l: ax + by + c = 0. Find the equation of each line through the point (x1, y1):
(a) l1 parallel to l
(b) l2 perpendicular to l
Step 3 Write the equations of l1 and l2.
Use point-slope form to write equations of l1 and l2.
Simplify to write equations in the requested form.
Let l: 2x  3y + 6 = 0. Find the equation of each line through the point (2, 8):
(a) l1 parallel to l
(b) l2 perpendicular to l 3.

51. Section Functions
SECTION 1.1
Use functional notation and find function values.
Find the domain of a function.
Identify the graph of a function.
Find the average rate of change of a function.
Solve applied problems by using functions.

52. Definitions
A special relationship such as y = 10x in which to each element x in one set there corresponds a unique element y in another set is called a function. y is sometimes referred to as the dependent variable and x as the independent variable.

53. DEFINTION OF A FUNCTION
A function from a set X to a set Y is a relation in which each element of X corresponds to one and only one element of Y. The set X is the domain of the function. The set of those elements of Y that correspond (are assigned) to the elements of X is the range of the function.

54. Function Notation
For each x in the domain of f, there corresponds a unique y in its range. The number y is denoted by f (x) read as “f of x” or “f at x”. We call f(x) the value of f at the number x and say that f assigns the f (x) value to x.

55. Functions
Functions can be defined by: Ordered pairs Tables and graphs Equations

56. Example: Functions Defined by Ordered Pairs
Determine whether each relation defines a function. a. r = {(–1, 2), (1, 3), (5, 2), (–1, –3)} b. s = {(–2, 1), (0, 2), (2, 1), (–1, –1)} a. The domain of r is {–1, 1, 5}. The range of r is {2, 3, –3}. It is NOT a function because two ordered pairs have the same first component.

57. Example: Functions Defined by Ordered Pairs (cont)
Determine whether each relation defines a function. a. r = {(–1, 2), (1, 3), (5, 2), (–1, –3)} b. s = {(–2, 1), (0, 2), (2, 1), (–1, –1)} b. The domain of s is {–2, –1, 0, 2}. The range of s is {–3, 1, 2}. The relation is a function because no two ordered pairs have the same first component.

58. Example: Determining Whether an Equation Defines a Function
Determine whether y is a function of x for each equation. a. 6x2 – x = 12 b. y2 – x2 = 4 a. One value of y corresponds to each value of x so it defines y as a function of x.

59. Example: Determining Whether an Equation Defines a Function (cont)
Determine whether y is a function of x for each equation. a. 6x2 – x = 12 b. y2 – x2 = 4 b. Two values of y correspond to each value of x so y is not a function of x.

60. Example: Evaluating a Function
Let g be the function defined by the equation y = x2 – 6x + 8. Evaluate each function value. a. g(3) b. g(–2) c. d. g(a + 2) e. g(x + h)

61. Example: Evaluating a Function (cont)
Let g be the function defined by the equation y = x2 – 6x + 8.

62. AGREEMENT ON DOMAIN
If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.

63. Example: Finding the Domain of a Function
Find the domain of each function. a. f is not defined when the denominator is 0.
Domain: {x|x ≠ –1 and x ≠ 1}

64. Example: Finding the Domain of a Function (cont)
Find the domain of each function. The square root of a negative number is not a real number and is excluded from the domain. Domain: {x|x ≥ 0}, [0, ∞)

65. Example: Finding the Domain of a Function (cont)
Find the domain of each function. The square root of a negative number is not a real number, so x – 1 ≥ 0 and since therefore denominator ≠ 0, x > 1. Domain: {x|x > 1}, or (1, ∞)

66. Example: Finding the Domain of a Function (cont)
Find the domain of each function. Any real number substituted for t yields a unique real number. Domain: {t|t is a real number}, or (–∞, ∞)

67. VERTICAL LINE TEST
If no vertical line intersects the graph of a relation at more than one point, then the graph is the graph of a function.
Graph does not represent a function

68. Example: Test for Functions
Determine which graphs in the figures are graphs of functions. a.
Not a function
Does not pass the vertical line test since a vertical line can be drawn through the two points farthest to the left

69. Example: Test for Functions
Determine which graphs in the figures are graphs of functions. b.
Not a function Does not pass the vertical line test.

70. Example: Test for Functions
Determine which graphs in the figures are graphs of functions. c.
Is a function
Does pass the vertical line test

71. Example: Test for Functions
Determine which graphs in the figures are graphs of functions. d.
Is a function
Does pass the vertical line test

72. Example: Examining the Graph of a Function
Let f(x) = x2 – 2x – 3.
Is the point (1, –3) on the graph of f ?
Find all values of x such that (x, 5) is on the graph of f.
Find all y-intercepts of the graph of f.
Find all x-intercepts of the graph of f.
a. Check whether (1, –3) satisfies the equation.
(1, –3) is not on the graph of f. ?

73. Example: Examining the Graph of a Function
Let f(x) = x2 – 2x – 3. b. Find all values of x such that (x, 5) is on the graph of f. Substitute 5 for y and solve for x. (–2, 5) and (4, 5) are on the graph of f.

74. Example: Examining the Graph of a Function
Let f(x) = x2 – 2x – 3. c. Find all y-intercepts of the graph of f. Substitute 0 for x and solve for y. The only y-intercept is –3.

75. Example: Examining the Graph of a Function
Let f(x) = x2 – 2x – 3.
d. Find all x-intercepts of the graph of f.
Substitute 0 for y and solve for x.
The x-intercepts of the graph of f are –1 and 3.

76. THE AVERAGE RATE OF CHANGE OF A FUNCTION
Let (a, f (a)) and (b, f (b)) be points on the graph of a function f. Then the average rate of change of f (x) as x changes from a to b is defined by

77. Example: Finding the Average Rate of Change
OBJECTIVE Find the average rate of change of a function f as x changes from a to b.
Step 1 Find f (a) and f (b).
Step 2 Use the values from Step 1 in the definition of average rate of change.
EXAMPLE Find the average rate of change of f (x) = 2  3x2 as x changes from x = 1 to x =

78. Example: Finding the Average Rate of Change
Find the average rate of change of f (x) = 2x2  3 as x changes from x = c to x = c + h, h  0.

79. DIFFERENCE QUOTIENT
For a function f , the quantity is called the difference quotient.

80. Example: Evaluating and Simplifying a Difference Quotient
Let f(x) = 2x2 – 3x + 5. Find and simplify First, we find f(x + h)

81. Example: Evaluating and Simplifying a Difference Quotient (cont)

82. Section 1.4 A Library of Functions
Define linear functions.
Discuss important properties of functions.
Evaluate and graph piecewise functions.
Graph basic functions.

83. LINEAR FUNCTIONS
Let m and b be real numbers. The function f (x) = mx + b is called a linear function. If m = 0, the function f (x) = b is called a constant function. If m = 1 and b = 0, the resulting function f (x) = x is called the identity function.

84. GRAPH OF f (x) = mx + b
The graph of a linear function is a nonvertical line with slope m and y-intercept b.

85. GRAPH OF f (x) = mx + b

86. INCREASING, DECREASING, AND CONSTANT FUNCTIONS
Let f be a function, and let x1 and x2 be any two numbers in an open interval (a, b) contained in the domain of f . The symbols a and b may represent real numbers, –∞, or ∞. Then

87. INCREASING, DECREASING, AND CONSTANT FUNCTIONS
f is an increasing function on (a, b) if x1 < x2 implies
f (x1) <  f (x2).

88. INCREASING, DECREASING, AND CONSTANT FUNCTIONS
(ii) f is a decreasing function on (a, b) if x1 < x2 implies f (x1) > f (x2).

89. INCREASING, DECREASING, AND CONSTANT FUNCTIONS
(iii) f is a constant on (a, b) if x1 < x2 implies f (x1) = f (x2).

90. Example: Tracking the Behavior of a Function
From the graph of g, find the intervals over which g is increasing, decreasing, or is constant.
increasing on the interval (–∞, –2)
constant on the interval (–2, 3)
decreasing on the interval (3, ∞)

91. DEFINITION OF RELATIVE MAXIMUM AND RELATIVE MINIMUM
If a is in the domain of a function f, we say that the value f (a) is a relative minimum of f if there is an interval (x1, x2) containing a such that f (a) ≤ f (x) for every x in the interval (x1, x2). We say that the value f (a) is a relative maximum of f if there is an interval (x1, x2) containing a such that f (a) ≥ f (x) for every x in the interval (x1, x2).

92. RELATIVE MAXIMUM AND RELATIVE MINIMUM

93. EVEN FUNCTION
A function f, is called an even function if, for each x in the domain of f, –x is also in the domain of f and f (–x) = f (x). The graph of an even function is symmetric with respect to the y-axis.

94. ODD FUNCTION
A function f, is an odd function if, for each x in the domain of f, –x is also in the domain of f and f (–x) = – f (x). The graph of an odd function is symmetric with respect to the origin.

95. EVALUATING A PIECEWISE FUNCTION
To evaluate F(a) for piecewise function F. Step 1 Determine which line of the function applies to the number a. Step 2 Evaluate F(a) using the line chosen in Step 1.

96. Example: Evaluating a Piecewise Function
OBJECTIVE Evaluate F(a) for piecewise function F. Step 1 Determine which line of the function applies to the number a. Step 2 Evaluate F(a) using the line chosen in Step 1.
Let Find F(0) and F(2). 1. Let a = 0. Because a < 1, use the first line, F(x) = x2. Let a = 2. Because a > 1, use the second line, F(x) = 2x + 1.

97. Example: Evaluating a Piecewise Function
OBJECTIVE Evaluate F(a) for piecewise function F. Step 2 Evaluate F(a) using the line chosen in Step 1.
Let Find F(0) and F(2). 2. F(0) = 02 = 0 F(2) = 2(2) + 1 = 5

98. Example: Graphing a Piecewise Function
Let Sketch the graph of y = F(x). In the definition of F the formula changes at x =1. We call such numbers the transition points of the formula. For the function F the only transition point is 1.

99. Example: Graphing a Piecewise Function
Graph the function separately over the open intervals determined by the transition points and then graph the function at the transition points themselves. For the function y = F(x), the formula for F specifies that we graph the equation y = –2x + 1 on the interval (–∞, 1)

100. Example: Graphing a Piecewise Function
Graph the function separately over the open intervals determined by the transition points and then graph the function at the transition points themselves. For the function y = F(x), the formula for F specifies that we graph the equation y = –2x + 1 on the interval (–∞, 1). Next, we graph the equation y = 3x + 1 on the interval (1, ∞) and at the transition point 1.

101. Example: Graphing a Piecewise Function
Graphs:

102. BASIC FUNCTIONS
The following are some of the common functions of algebra, along with their properties, and should be included in a library of basic functions.

103. Constant Function f (x) = c
Domain: (–∞, ∞) Range: {c} Constant on (–∞, ∞) Even function (y–axis symmetry)

104. Identity Function f (x) = x
Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry)

105. Squaring Function f (x) = x2
Domain: (–∞, ∞) Range: [0, ∞) Decreasing on (–∞, 0) Increasing on (0, ∞) Even function (y–axis symmetry)

106. Cubing Function f (x) = x3
Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry)

107. Absolute Value Function
Domain: (–∞, ∞) Range: [0, ∞) Decreasing on (–∞, 0) Increasing on (0, ∞) Even function (y–axis symmetry)

108. Square Root Function
Domain: [0, ∞) Range: [0, ∞) Increasing on (0, ∞) Neither even nor odd (no symmetry)

109. Cube Root Function
Domain: (–∞, ∞) Range: (–∞, ∞) Increasing on (–∞, ∞) Odd function (origin symmetry)

110. Reciprocal Function Domain: (–∞, 0) U (0, ∞) Range: (–∞, 0) U (0, ∞)
Decreasing on (–∞, 0) U (0, ∞)
Odd function (origin symmetry)

111. Section 1.5 Transformations of Functions
Define transformations of graphs.
Use vertical or horizontal shifts to graph functions.
Use reflections to graph functions.
Use stretching or compressing to graph functions.

112. TRANSFORMATIONS
If a new function is formed by performing certain operations on a given function f , then the graph of the new function is called a transformation of the graph of f.

113. Example: Graphing Vertical Shifts
Let f(x) = |x|, g(x) = |x| + 2, and h(x) = |x| – 3 . Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f. Make a table of values and graph the equations.

114. Example: Graphing Vertical Shifts
Let f(x) = |x|, g(x) = |x| + 2, and h(x) = |x| – 3 . Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f. Make a table of values and graph the equations.

115. Example: Graphing Vertical Shifts (cont)
y = |x|
y = |x| + 2 –5 5 7 –3 3 –1 1 2
The graph is shifted 2 units up.

116. Example: Graphing Vertical Shifts (cont)
y = |x|
y = |x| – 3 –5 5 2 –3 3 –1 1 –2
The graph is shifted 3 units down

117. VERTICAL SHIFT
Let c > 0. The graph of y = f (x) + c is the graph of y = f (x) shifted c units up, and the graph of y = f (x) – c is the graph of y = f (x) shifted c units down.

118. Example: Writing Functions for Horizontal Shifts
Let f(x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2. A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide. Describe how the graphs of g and h relate to the graph of f.

119. Example: Writing Functions for Horizontal Shifts (cont)

120. Example: Writing Functions for Horizontal Shifts (cont)

121. Example: Writing Functions for Horizontal Shifts
All three functions are squaring functions. a. g is obtained by replacing x with x – 2 in f . For each point (x, y) on the graph of f , there will be a corresponding point (x + 2, y) on the graph of g. The graph of g is the graph of f shifted 2 units to the right.
The x-intercept of f is 0.
The x-intercept of g is 2.

122. Example: Writing Functions for Horizontal Shifts
b. h is obtained by replacing x with x + 3 in f . For each point (x, y) on the graph of f , there will be a corresponding point (x – 3, y) on the graph of h. The graph of h is the graph of f shifted 3 units to the left.
The x-intercept of f is 0.
The x-intercept of h is –3.

123. HORIZONTAL SHIFT
Let c > 0. The graph of y = f (x – c) is the graph of y = f (x) shifted c units to the right, and the graph of y = f (x + c) is the graph of y = f (x) shifted c units to the left.

124. Example: Graphing Combined Vertical and Horizontal Shifts
Objective: Sketch the graph of g(x) = f(x – c) + d, where f is a function whose graph is known. Step 1: Identify the graph the known function f. Step 2: Identify the constants d and c.
Example: Sketch the graph of the function 1. Choose 2. c = –2 and d = –3

125. Example: Graphing Combined Vertical and Horizontal Shifts
Step 3: For c > 0: i. graph y = f(x – c) by shifting the graph f horizontally c units to the right. ii. graph y = f(x + c) by shifting the graph f horizontally c units to the left.
3. Since c = 0 – 2 < 0, the graph is shifted horizontally two units to the left. (see blue graph on next slide)

126. Example: Graphing Combined Vertical and Horizontal Shifts
Step 4: For d > 0: i. graph y = f(x – c) + d by shifting the graph vertically up d units. ii. graph y = f(x – c) + d by shifting the graph vertically down d units.
4. Shift the graph three units down.

127. REFLECTION IN THE x–AXIS
The graph of y = – f (x) is a reflection of the graph of y = f (x) in the x–axis. If a point (x, y) is on the graph of y = f (x), then the point (x, –y) is on the graph of y = – f (x).

128. REFLECTION IN THE x–AXIS

129. REFLECTION IN THE y–AXIS
The graph of y = f (–x) is a reflection of the graph of y = f (x) in the y–axis. If a point (x, y) is on the graph of y = f (x), then the point (–x, y) is on the graph of y = f (–x).

130. REFLECTION IN THE y–AXIS

131. Example: Combining Transformations
Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|. Step 1 Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|.

132. Example: Combining Transformations
Step 2 Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

133. Example: Combining Transformations
Step 3 Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

134. Example: Stretching or Compressing a Function Vertically
Let f(x) = |x|, g(x) = 2|x|, and h(x) = Sketch the graphs of f, g, and h on the same coordinate plane and describe how the graphs of g and h are related to the graph of f. x –2 –1 1 2
f(x)
g(x) 4
h(x)
1/2

135. Example: Stretching or Compressing a Function Vertically (cont)

136. Example: Stretching or Compressing a Function Vertically (cont)
The graph of y = 2|x| is the graph of y = |x| vertically stretched (expanded) by multiplying each of its y–coordinates by 2. The graph of is the graph of y = |x| vertically compressed (shrunk) by multiplying each of its y–coordinates by 1/2.

137. VERTICAL STRETCHING OR COMPRESSING
The graph of y = af(x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is 1. A vertical stretch away from the x-axis if a > 1; 2. A vertical compression toward the x-axis if 0 < a < 1. If a < 0, first graph y = |a|f(x) by stretching or compressing the graph of y = f(x) vertically. Then reflect the resulting graph about the x-axis.

138. HHORIZONTAL STRETCHING OR COMPRESSING
The graph of y = f(bx) is obtained from the graph of y = f(x) by multiplying the x-coordinate of each point on the graph of y = f(x) by 1/b and leaving the y-coordinate unchanged. The result is
A horizontal stretch away from the y-axis if
0 < b < 1;
2. A horizontal compression toward the y-axis if b > 1.
If b < 0, first graph f(|b|x) by stretching or compressing the graph of y = f(x) horizontally. Then reflect the resulting graph about the y-axis.

139. Example: Stretching or Compressing a Function Horizontally
Using the graph of a function y = f (x) in the figure, whose equation is not given, sketch each of the following graphs.

140. Example: Stretching or Compressing a Function Horizontally (cont)
a. Stretch the graph of y = f (x) horizontally by a factor of 2. Each point (x, y) transforms to (2x, y).

141. Example: Stretching or Compressing a Function Horizontally (cont)
b. Compress the graph of y = f (x) horizontally by a factor of 1/2. Each point (x, y) transforms to (1/2x, y).

142. Example: Stretching or Compressing a Function Horizontally (cont)
c. Reflect the graph of y = f (2x) in the y–axis. Each point (x, y) transforms to (–x, y).

143. Example: Combining Transformations
Sketch the graph of the function f (x) = 3 – 2(x – 1)2. Step 1 y = x2 Identify basic function.
Step 2 y = (x – 1)2 Shift right 1.

144. Example: Combining Transformations
Step 3 y = 2(x – 1)2
Stretch vertically by a factor of 2.
Step 4 y = –2(x – 1)2 Reflect about x–axis.

145. Example: Combining Transformations
Step 5 y = 3 – 2(x – 1)2 Shift three units up.

146. Section 1.6 Combining Functions; Composite Functions
Form composite functions.
Find the domain of a composite function.
Decompose a function.
Apply composition to practical problems.

147. COMPOSITION OF FUNCTIONS
If f and g are two functions, the composition of function f with function g is written as
and is defined by the equation
where the domain of
values x in the domain of g for which g(x) is in the domain of f.
consists of those

148. COMPOSITION OF FUNCTIONS

149. Example: Evaluating a Composite Function
Let Find each of the following.

150. Example: Evaluating a Composite Function (cont)
Let

151. Example: Finding Composite Functions
Let Find each composite function.

152. Example: Finding Composite Functions (cont)
Let

153. Example: Finding Composite Functions (cont)
Let

154. Example: Finding the Domain of a Composite Function
Domain is (–∞, 0) U (0, ∞).

155. Example: Finding the Domain of a Composite Function (cont)
Domain is (–∞, –1) U (–1, ∞).

156. Example: Decomposing a Function
Step 1 Define g(x) as any expression in the defining equation for H. Let g(x) = 2x2 + 1.

157. Example: Decomposing a Function (cont)
Step 2 Replace the letter H with f and replace the expression chosen for g(x) with x. Step 3 Now we have

158. Example: Calculating the Area of an Oil Spill from a Tanker
Oil is spilled from a tanker into the Pacific Ocean and the area of the oil spill is a perfect circle.
The radius of this oil slick increases at the rate of 2 miles per hour.
Express the area of the oil slick as a function of time.
Calculate the area covered by the oil slick in 6 hours.
The area of the oil slick is a function its radius.

159. Example: Calculating the Area of an Oil Spill from a Tanker (cont)
The radius is a function time: increasing 2 mph a. The area is a composite function b. Substitute t = 6. The area of the oil slick is 144π square miles or about 452 square miles.

160. Section 1.7 Inverse Functions
Define an inverse function.
Find the inverse function.

161. DEFINITION OF A ONE-TO-ONE FUNCTION
A function is called a one-to-one function if each y-value in its range corresponds to only one x-value in its domain.

162. A ONE-TO-ONE FUNCTION
Each y-value in the range corresponds to only one x-value in the domain.

163. NOT A ONE-TO-ONE FUNCTION
The y-value y2 in the range corresponds to two x-values, x2 and x3, in the domain.

164. NOT A FUNCTION
The x-value x2 in the domain corresponds to two y-values, y2 and y3, in the range.

165. HORIZONTAL- LINE TEST
A function f is one-to-one if no horizontal line intersects the graph of f in more than one point.

166. Example: Using the Horizontal-Line Test
Use the horizontal-line test to determine which of the following functions are one-to-one.
No horizontal line intersects the graph of f in more than one point, therefore the function f is one-to-one.

167. Example: Using the Horizontal-Line Test (cont)
There are many horizontal lines that intersect the graph of f in more than one point, therefore the function f is not one-to-one.

168. Example: Using the Horizontal-Line Test (cont)
No horizontal line intersects the graph of f in more than one point, therefore the function f is one-to-one.

169. DEFINITION OF f –1 FOR A ONE-TO-ONE FUNCTION f
Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f –1.
If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f –1, and we write x = f –1(y).
We have y = f (x) if and only if f –1(y) = x.

170. Example: Relating the Values of a Function and Its Inverse
Assume that f is a one-to-one function.
If f (3) = 5, find f –1(5).
If f –1(–1) = 7, find f (7).
We know that y = f (x) if and only if f –1(y) = x.
Let x = 3 and y = 5. Now 5 = f (3) if and only if f –1(5) = 3. Thus, f –1(5) = 3.
Let y = –1 and x = 7. Now, f –1(–1) = 7 if and only if f (7) = –1. Thus, f (7) = –1.

171. INVERSE FUNCTION PROPERTY
Let f denote a one-to-one function. Then Further, if g is any function such that (for the values of x in these equations)
for every x in the domain of f –1. 1.
for every x in the domain of f . 2.

172. Example: Verifying Inverse Functions
Verify that the following pairs of functions are inverses of each other: Form the composition of f and g.

173. Example: Verifying Inverse Functions (cont)
Now form the composition of g and f.
Since
we conclude that
f and g are inverses of each other.

174. SYMMETRY PROPERTY OF THE GRAPHS OF f AND f –1
The graph of the function f and the graph of f –1 are symmetric with respect to the line y = x.

175. Example: Finding the Graph of f –1 from the Graph of f
The graph of the function f is shown. Sketch the graph of the f –1.

176. PROCEDURE FOR FINDING f –1
Step 1 Replace f (x) by y in the equation for f (x). Step 2 Interchange x and y. Step 3 Solve the equation in Step 2 for y. Step 4 Replace y with f –1(x).

177. Example: Finding the Inverse Function
Find the inverse of the one-to-one function
Step 1
Step 2

178. Example: Finding the Inverse Function (cont)
Find the inverse of the one-to-one function
Step 3

179. Example: Finding the Inverse Function (cont)
Find the inverse of the one-to-one function
Step 4

180. Example: Finding the Domain and Range of a One-to-One Function
Find the domain and the range of the function Domain of f is the set of all real numbers x such that x ≠ 2. In interval notation that is (–∞, 2) U (2, –∞). Range of f is the domain of f –1. Range of f is (–∞, 1) U (1, –∞).

181. Example: Finding an Inverse Function
Find the inverse of G(x) = x2 – 1, x ≥ 0. Step 1 y = x2 – 1, x ≥ 0 Step 2 x = y2 – 1, y ≥ 0 Step 3 Step 4
Since y ≥ 0, reject

182. Example: Finding an Inverse Function (cont)
Here are the graphs of G and G–1