1. College Algebra
Exam 3 Material

2. Ordered Pair Consists of: Examples of different ordered pairs:
Two real numbers,
Listed in a specific order,
Separated by a comma,
Enclosed within parentheses
Examples of different ordered pairs:
(7, 2)
(2, 7)
(3, -4)

3. Coordinates of an Ordered Pair
The first number within an ordered pair is called the “x-coordinate”
The second number within an ordered pair is called the “y-coordinate”
Example, given the ordered pair (5, -2)
The x-coordinate is: 5
The y-coordinate is: -2

4. Ordered Pairs & Interval Notation
(2, 7) can have two meanings:
It can be an ordered pair.
It can be interval notation and have the meaning of all real numbers between 2 and 7, but not including 2 and 7.
Context will tell the difference, just like in the meaning of certain words.
English Example, “bow”:
ribbon decoration on a package
an instrument for shooting an arrow

5. Rectangular Coordinate System
Consists of:
Horizontal number line called “x-axis”
Vertical number line called “y-axis”
Intersecting at the zero points of each axis in a point , designated as the ordered pair (0, 0), and called the “origin”
Forms a rectangular grid that can be used to show ordered pairs as points

6. Graphing Ordered Pair as a Point
The x-coordinate gives horizontal directions from the origin
The y-coordinate gives vertical directions from the origin
When both directions have been followed, the point is located.
Example, graph: (-2, 3)

7. Rectangular Coordinate System
Axes divide plane into four quadrants numbered counterclockwise from upper right
The signs of the coordinates of a point are determined by its quadrant
I :
II :
III :
IV :

8. Distance Formula
The “distance” between two points (x1, y1) and (x2, y2) in a rectangular coordinate system is given by the formula:
Example: Distance between (2, -1) and (-3, 4)?
x1 = 2, y1 = -1, x2 = -3, y2 = 4

9. Midpoint Formula
The “midpoint” of two points is a point exactly half way between the two points
The formula for finding the coordinates of the midpoint of two points (x1, y1) and (x2, y2) is:
Example: Midpoint of (2, -1) and (-3, 4)?

10. Homework Problems Section: 2.1 Page: 192 Problems: All: 5 – 14
MyMathLab Assignment 35 for practice

11. Ordered Pairs as Solutions to Equations in Two Variables
An equation in two variables, “x” and “y,” requires a pair of numbers, one for “x” and one for “y”, to form a solution
Example: Given the equation:
2x + y = 5
One solution consists of the pair:
x = 2 and y = 1
If it is agreed that “x” will be listed first and “y” second, this solution can be shown as the ordered pair:
(2, 1)

12. Ordered Pairs as Solutions to Equations in Two Variables
Given: 2x + y = 5
Complete the missing coordinate to form ordered pair solutions:
( 1, __ )
( 3, __ )
( __, 5 )
( __, -3)
How many ordered pair solutions can be found?

13. Graphs of Equations in Two Variables
A “graph” of an equation in two variables is a “picture” in a rectangular coordinate system of all solutions to the equation
A graph can be constructed by “point plotting,” finding enough (x, y) pairs to establish a pattern of points and then connecting the points with a smooth curve
If the pattern indicates that the curve should extend beyond the graph grid, an arrow is placed on the curve where it exits the grid

14. Graph: 2x + y = 5
Find (x, y) pairs by picking a number for “x” and solving for “y”, or vice versa
Show solutions as (x, y) pairs
Plot each point
When pattern is seen
connect with smooth curve
with arrows at both ends
Solutions:

15. Graph:
Find and plot solutions (using a “T-chart” helps):

16. Homework Problems Section: 2.1 Page: 194 Problems: All: 33 – 44
MyMathLab Assignment 36 for practice

17. Graphs vs. Form of Equation
Experience will teach us that the general shape of a graph depends on the form of the equation in two variables
Examples related to last section:
Equations of form: Ax + By = C will always have a straight line graph
Equation of form: y = |mx + b| will always have a V-shaped graph

18. Center Radius Equation of Circle
Equation of the form:
Will always have a graph that is a circle with center:
and radius:

19. Center Radius Equation
Considering:
What would be the equation of a circle with center (-1, 2) and radius 3?

20. Center Radius Equation
Given each of the following center radius equations, find the center and the radius:

21. Homework Problems Section: 2.1 Page: 194 Problems: All: 45 – 52
MyMathLab Assignment 37 for practice

22. General Equation of a Circle
The equation will represent a real circle, if, when put in “center-radius form”,
Example:
Don’t know yet, if it is a “real” circle, until written:

23. Converting General Equation to Center-Radius Equation
“Complete the Square” twice, once on “x” and once on “y” (keep both sides balanced)
Example:
The “general equation” is a “real” circle since:
Center and radius:

24. Homework Problems Section: 2.1 Page: 195 Problems: All: 57 – 64
MyMathLab Assignment 38 for practice
MyMathLab Quiz 2.1 due for a grade on the date of our next class meeting

25. Relation Relation – any set of ordered pairs
Example: M = {(-3,2), (1,0), (4,-5)}
Domain of a Relation – set of first members (“x coordinates”)
Example: Domain M = {-3, 1, 4}
Range of a Relation – set of all second members (“y coordinates”)
Example: Range M = {2, 0, -5}

26. Equations in Two Variables
Considered to be relations because solutions form a set of ordered pairs
Example:
There are an infinite number of ordered pair solutions, but some are:

27. Domain of Equations in Two Variables
Set of all x’s for which y is a real number
May help to find domain by first solving for “y”
Example:
Easy to see that x can be anything except -1:
Domain =

28. Range of Equations in Two Variables
Set of all y’s for which x is a real number
May help to find range by first solving for “x”
Example:
Easy to see that y can be anything except 3:
Range =

29. Function
A special relation in which each x coordinate is paired with exactly one y coordinate
Example – only one of these is a function:
R = {(2,1), (3,-5), (2,3)}
Not a Function
S = {(3,2), (1,2), (-5,3)}
FUNCTION

30. Homework Problems Section: 2.2 Page: 209 Problems: All: 5 – 15
MyMathLab Assignment 39 for practice

31. Dependent & Independent Variables in Functions
Since every x is paired with exactly one y, we say that “y depends on x”
For a function defined by an equation in two variables, x is the “independent variable” and y is the “dependent variable”

32. Functions Defined by Equations in Two Variables
To determine if an equation in two variables is a function, solve for y and consider whether one x can give more than one y – if not, it is a function
Example – only one of these is a function:
Solve each equation for y and analyze:

33. Example Continued

34. Function Notation
Functions represented by equations in two variables are traditionally solved for y because doing so shows how y depends on x
Example – each of these is a different function:

35. Function Notation Continued
When working with functions it is also traditional to replace y with the symbol f(x) or some variation using a letter other than f. This gives different functions different names:
Previous Example Written in “Function Notation”

36. Function Notation Continued
Function notation “f(x)” is read as “f of x” and means “the value of y for the given x”
Example:
If:
Then:
This means that for this f function, when x is -2, y is -7

37. Graphs of Relations and Functions
Can be accomplished by point plotting methods already discussed
The graph of a relation will be a function if, and only if, every vertical line intersects the graph in at most one point (VERTICAL LINE TEST)

38. Example of Vertical Line Test Which graph represents a function?
Above: passes vertical line test – Function
Below: fails vertical line test – Not a Function

39. Practice Using Function Notation
Given the functions f, g and h defined as follows, and remembering that f(2) means “the value of y in the f function when x is 2”, find the value of each function for the value of “x” shown: If
If g = {(1,3),(-2,4),(2,5)} find g(-2)
g(-2) = 4

40. Practice Using Function Notation
Given the graph of y = h(x),
find h(-3)
h(-3) = 1

41. Determining if Equation in Two Variables is a Function
Solve the equation for y.
If one x gives exactly one y, it is a function.
If it is a function, it may be written in function notation by replacing y with f(x).

42. Example
Determine if the following equation defines a function, if so, write in function notation:
This is a function since one x gives one y

43. Increasing, Decreasing and Constant Functions
A function is increasing over some interval of its domain if its graph goes up as x values go from left to right within the interval
A function is decreasing over some interval of its domain if its graph goes down as x values go from left to right within the interval
A function is constant over some interval of its domain if its graph is flat as x values go from left to right within the interval

44. Example Show the “interval of the domain” where the given function,
is increasing:
is decreasing:
is constant:

45. Homework Problems Section: 2.2 Page: 210
Problems: Odd: 17 – 37, 41 – 63,
MyMathLab Assignment 40 for practice
MyMathLab Quiz 2.2 due for a grade on the date of our next class meeting

46. Linear Functions Any function that can be written in the form:
Example: f(x) = 3x - 1
This is called the slope intercept form of a linear function (reason for name – explained later).
The graph of a linear function will always be a non-vertical straight line

47. Graphing Linear Functions
Find any two ordered pairs that are solutions (chose two different x’s and find corresponding y’s)
Plot the two points in a rectangular coordinate system.
Draw a straight line connecting the two points with arrows on both ends.

48. Graphing a Linear Function
Graph f(x) = 3x – 1
Choose two values of “x” and calculate corresponding “y” values:
f(0) = 3(0) – 1 = -1 [ordered pair: (0,-1)]
f(2) = 3(2) – 1 = 5 [ordered pair: (2, 5)]

49. Linear Functions With Horizontal Line Graphs
A linear function of the form:
can be simplified to:
This means y will always be the same (y = b) no matter the value of “x”
The graph will be a horizontal line through all the points that have a “y” value of “b”

50. Example of Linear Function With Horizontal Line Graph
Graph f(x) = -3
(Note: this function says no matter what value “x” is, the “y” value will be -3.)

51. Homework Problems Section: 2.3 Page: 221 Problems: All: 7 – 18
MyMathLab Assignment 41 for practice

52. Linear Relation A first degree polynomial equation in two variables
Examples:
Standard form of a linear relation:
First example is in Standard Form (A=2, B=-3, C=6)
Rest can be put in standard form, if desired
Last example is in standard form of a linear function

53. Facts About Linear Relations
Every linear relation,
is a linear function, can (by solving for y) be written in the form:
except when B = 0 (y term is missing)

54. Example
Write the linear relation in the standard form of a linear function:

55. Facts About Linear Relations
If a linear relation,
has B = 0 (the y term is missing), the relation is not a linear function, and it’s equation takes the form:
This equation has as solutions all the ordered pairs that have an x value of n (y can be any number) and its graph will be a vertical line.

56. Example of Linear Relation with Vertical Line Graph
Consider the linear relation:
It can be written in the form:
Two solutions are:
Fails vertical line test -
Not a linear function

57. Homework Problems Section: 2.3 Page: 221 Problems: All: 19 – 28
MyMathLab Assignment 42 for practice

58. Slope of a Line
The slope, m, of a line is defined to be the ratio of the amount of vertical movement (rise) to the amount of horizontal movement (run) required to get from one point to another

59. Slope of a Line Formula
Given two points on a line, (x1, y1) and (x2, y2), the slope, m, can be calculated from the formula:
For a horizontal line y1 = y2 , so the slope of a horizontal line is always: ___
For a vertical line x1 = x2 , so the slope of a horizontal line is always: _________

60. Calculating Slope of a Line
Find two points on the line and substitute into the formula
Example: Find the slope of:
If x = 0, then y = ,
If y = 0, then x = ,
rise = 2, run = 3

61. Homework Problems Section: 2.3 Page: 222 Problems: All: 35 – 42
MyMathLab Assignment 43 for practice

62. Graphing a Line Given a Point and a Slope
Given a point and a slope, m:
Plot the point
From the point, rise and run, according to the value of m, to plot a second point
Connect the two points with a straight line with arrows at both ends

63. Example: Graph line through (-2, 1) with slope -2/3

64. Calculating Slope and y-intercept by “Solving for y”
When a linear relation in two variables is solved for y it takes the form:
where “m” is the slope, and “b” is the y-intercept
In a previous example we used the slope formula to calculate slope: Calculating Slope of a Line
A new approach is to find slope by solving for y:
What is “m”?
What is “b”?

65. Graphing a Line in Slope-Intercept Form
Write the equation in slope-intercept form
Identify the y-intercept, b, and slope, m
Plot the y-intercept on the graph
From y-intercept, “rise” and “run” to another point according to the ratio, m
Connect the points with a straight line with arrows on each end

66. Example of Graphing a Line Written in Slope Intercept Form
y intercept is: 1, slope is:
Plot y intercept
From there rise -2, run 3 and plot
Connect points with a line

67. Homework Problems Section: 2.3 Page: 222 Problems: All: 45 – 58
MyMathLab Assignment 44 for practice
MyMathLab Quiz 2.3 is due for a grade on the date of our next class meeting

68. Equations with Line Graphs

69. Point-Slope Equation of a Line
Given a point (x1, y1) and a slope, m, the equation of the line through that point with that slope is:
y – y1 = m(x – x1)

70. Using Point Slope Equation
Write the slope intercept equation of the line through (5, -1) with slope, m = -4/3

71. Homework Problems Section: 2.4 Page: 236 Problems: All: 1 – 22
MyMathLab Assignment 45 for practice

72. Parallel and Perpendicular Lines
Two lines are “parallel” if they have the same slope, but different y-intercepts
Two lines are “perpendicular” if they have slopes that are “negative reciprocals”
Examples of slopes that are negative reciprocals:

73. Application Problem Involving Perpendicular Lines
Write the slope intercept equation of a line that goes through the point (-1, 5) that is perpendicular to the line: x + 2y = 4
First find slope of the line: x + 2y = 4 by solving for y: y = (-1/2)x + 2
This line has slope, m = -1/2
The perpendicular slope, m = 2
Now use point-slope equation to write equation of line through (-1, 5) with slope 2:
y – 5 = 2(x + 1)
y = 2x + 7

74. Homework Problems Section: 2.4 Page: 237 Problems: All: 27 – 42
MyMathLab Assignment 46 for practice
MyMathLab Quiz 2.4 is due for a grade on the date of our next class meeting

75. Basic Function Graphs The most basic functions include:
Identity function:
Squaring function:
Cubing function:
Square root function:
Cube root function:
Absolute value function:
Greatest integer function:

76. Identify Function:
Note: This is the slope intercept form of the linear function f(x) = mx + b, with m = 1 and b = 0

77. Squaring Function:
Note: This is a special case of the “quadratic function” that will be discussed later

78. Cubing Function:

79. Square Root Function:
Note: This is the first of the basic functions to have a restricted domain:

80. Cube Root Function:
Does this function have a restricted domain like the last one?

81. Absolute Value Function:

82. Greatest Integer Function:
Note: This function pairs every real number x with the “greatest integer less than or equal to x”
There is no restriction on the domain, but the range will be the set of all integers

83. Homework Problems Section: 2.5 Page: 249
Problems: All: 7 – 14, 16, 33 – 35
MyMathLab Assignment 47 for practice

84. Continuity of Functions
A function is “continuous over some interval of its domain” if, over that interval, its graph can be drawn from one end to the other, using a pencil, without lifting the pencil from the paper
The points where a function is not continuous are called “points of discontinuity”

85. Give domain and intervals of domain where these are continuous:

86. This function is continuous over only short intervals of the domain
This function is continuous over only short intervals of the domain. Give domain and the points of discontinuity.

87. Describe domain and the intervals of the domain where the function is continuous:

88. Homework Problems Section: 2.5 Page: 249 Problems: All: 1 – 6, 15
MyMathLab Assignment 48 for practice

89. Piecewise Defined Functions
These are functions that are defined by different rules for different intervals of their domain
Example:
To evaluate the function for specific values of “x”, use the formula for the interval that contains “x”

90. Graphing Piecewise Defined Functions
Plot enough points in each interval of the domain to establish the pattern in each interval

91. Homework Problems Section: 2.5 Page: 250 Problems: Odd: 17 – 31
MyMathLab Assignment 49 for practice
MyMathLab Quiz 2.5 is due for a grade on the date of our next class meeting

92. Transformations of Graphs
Given a basic function graph, or any other, specific changes in the definition of the function lead to very similar graphs as follows:
Given the graph of y = f(x), and assuming that h and v are positive, each of the following alterations to the function modifies the original graph as shown:
y = - f(x)
Reflects original graph over x axis
y = af(x)
Vertically stretches or squeezes original graph by a factor of a

93. Transformations of Graphs
y = f(x – h)
Horizontally translates the graph h units to the right
y = f(x + h)
Horizontally translates the graph h units to the left

94. Transformations of Graphs
y = f(x) – v
Vertically translates the graph v units downward
y = f(x) + v
Vertically translates the graph v units upward

95. Transformations of Graphs
y = af(x – h) + v
Has the combination of effects previously discussed – moves original graph horizontally right h units, shifts vertically upward v units and vertically stretches or squeezes by a factor of a

96. Example of Graph Transformations

97. Example of Graph Transformations
How does this transform the graph?
Should reflect graph across the x axis:

98. Example of Graph Transformations
How does this transform the graph?
Vertically stretches by a factor of 2:

99. Example of Graph Transformations
How does this transform the graph?
Vertically squeezes by a factor of ½ :

100. Example of Graph Transformations
How does this transform the graph?
Moves it horizontally one unit to the right:

101. Example of Graph Transformations
How does this transform the graph?
Translates it vertically down 4 units:

102. Example of Graph Transformations
How does this transform the graph?
Horizontally translates 1 unit left
Vertically translates 3 units down
Vertically stretches by 2:

103. Homework Problems Section: 2.6 Page: 264
Problems: All: 1 – 3, Odd: 5 – 11, – 47
MyMathLab Assignment 50 for practice

104. Even Functions
A function f(x) is “even” if f(-x) = f(x) for every x in the domain
Test to determine if a function is even:
If substituting “–x” for “x” makes f(-x) = f(x), the function is “even”
Even functions have graphs that are symmetric with respect to the y-axis
every line perpendicular to the y-axis that intersects the graph at a distance of “d” from the y-axis will also intersect the graph at a distance of “d” on the other side of the y-axis

105. Determine if f(x) = x2 is even:

106. Odd Functions
A function f(x) is “odd” if f(-x) = - f(x) for every x in the domain
Test to determine if a function is odd:
If substituting “–x” for “x” makes f(-x) = - f(x), the function is “odd”
Odd functions have graphs that are symmetric with respect to the origin
every line through the origin that intersects the graph at a distance of “d” on one side of the origin will also intersect the graph at a distance of “d” on the other side of the origin

107. Determine if f(x) = x3 is odd:

108. Homework Problems Section: 2.6 Page: 265 Problems: All: 19 – 30
MyMathLab Assignment 51 for practice
MyMathLab Quiz 2.6 is due for a grade on the date of our next class meeting

109. Resultant Functions from Operations on Functions
Given functions f and g we can define sum, difference, product and quotient functions as follows:
Sum function:
Difference function:
Product function:
Quotient function:

110. Domains of Sum, Difference, Product and Quotient Functions
The domain of each of these functions is the intersection of their individual domains, with the exception that for the quotient function, those values of x are excluded for which g(x) = 0
Note: Domains must be determined from individual functions not from looking only at the resultant function

111. Examples of Operations on Functions
Given:
Find definition and domain of:

112. Examples of Operations on Functions
Given information from previous slide:

113. Example: Determining Domain
As previously noted: Domains must be determined from individual functions not from looking only at the resultant function
Example: .

114. Homework Problems Section: 2.7 Page: 276 Problems: All: 9 – 14
MyMathLab Assignment 52 for practice

115. Evaluating Resultant Functions from Graphs

116. Homework Problems Section: 2.7 Page: 277 Problems: All: 23 – 26
MyMathLab Assignment 53 for practice

117. Difference Quotient
If (x, f(x)) represents a point on the graph of y = f(x) and “h” is a positive number, then (x+h, f(x+h)) is a second point on the graph and, from the slope formula, the slope of the line joining these points is:

118. Finding Difference Quotient
Given find difference quotient

119. Homework Problems Section: 2.7 Page: 278 Problems: Odd: 33 – 39
MyMathLab Assignment 54 for practice

120. Composition of Functions
Given two functions f(x) and g(x), a third function, called the composition function:
may be defined as:
The domain of the composition function is the set of all “x” in the domain of g , such that g(x) is in the domain of f
Note: In general:

121. Example of Forming Composite Functions
Given:
Find:

122. Example to Show that Composition is Not Commutative
Given:
Find:
Not the same!

123. Homework Problems Section: 2.7 Page: 278
Problems: Odd: 41 – 53, 57 – 63
MyMathLab Assignment 55 for practice
MyMathLab Quiz 2.7 is due for a grade on the date of our next class meeting