1. FUNCTIONS AND GRAPHS

2. Aim #1.2: What are the basics of functions and their graphs?
Let’s Review:
What is the Cartesian Plane or Rectangular Coordinate Plans?
How do we find the x and y-intercepts of any function?
How do we interpret the viewing rectangle
[-10,10, 1] by [-10, 10,1]?

3. IN THIS SECTION WE WILL LEARN:
How to find the domain and range?
Determine whether a relation is a function
Determine whether an equation represents a function
Evaluate a function
Graph functions by plotting points
Use the vertical line to identify functions

4. WHAT IS A RELATION? A relation is a set of ordered pairs.
Example: (4,-2), (1, 2), (0, 1), (-2, 2)
Domain is the first number in the ordered pair. Example:(4,-2)
Range is the second number in the ordered pair.
Example: : (4,30)

5. EXAMPLE 1: Find the domain and range of the relation.
{(Smith, %), (Johnson, 0.810%), (Williams, 0.699%), (Brown, 0.621%)}

6. PRACTICE:
Find the domain and range of the following relation.

7. HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?
A relation is a function if each domain only has ONE range value.
There are two ways to visually demonstrate if a relation is a function.
Mapping
Vertical Line Test

8. DETERMINE WHETHER THE RELATION IS A FUNCTION
{(1, 6), (2, 6), (3, 8), (4, 9)}
{(6, 1), (6, 2), (8, 3), (9, 4)}

9. FUNCTIONS AS EQUATIONS
Functions are usually given in terms of equations instead of ordered pairs.
Example: y =0.13x x + 8.7
The variable x is known the independent variable and y is the dependent variable.

10. HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION?
x2 + y = 4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.

11. HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION
x2 + y2 =4
Steps:
Solve the equation for y in terms of x.
Note:
If two or more y values are found then the equation is not a function.

12. PRACTICE:
Solve each equation for y and then determine whether the equation defines y as a function of x.
2x + y = 6
x2 + y2 = 1

13. WHAT IS FUNCTION NOTATION?
We use the special notation f(x) which reads as f of x and represents the function at the number x.
Example: f (x) = 0.13x x +8.7
If we are interested in finding f (30), we substitute in 30 for x to find the function at 30.
f (30)= 0.13(30) (30) + 8.7
Now let’s try to evaluate using our calculators.

14. HOW DO WE EVALUATE A FUNCTION?
F (x) = x2 + 3x + 5
Evaluate each of the following:
f (2)
f (x + 3)
f (-x)
Substitute the 2 for x and evaluate.
Then repeat.

15. GRAPHS OF FUNCTIONS
The graph of a function is the graph of the ordered pairs.
Let’s graph:
f (x) = 2x
g (x) = 2x + 4

16. USING THE VERTICAL LINE TEST
The Vertical Line Test for Functions
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

17. Practice:
Use the vertical line test to identify graphs in which y is a function of x.

18. SUMMARY: ANSWER IN COMPLETE SENTENCES.
What is a relation?
What is a function?
How can you determine if a relation is a function?
How can you determine if an equation in x and y defines y as a function of x? Give an example.

19. AIM #1.2B: WHAT KIND OF INFORMATION CAN WE OBTAIN FROM GRAPHS OF FUNCTIONS?
Note the closed dot indicates the graph does not extend from this point and its part of the graph.
Open dot indicates that the point does not extend and the point is not part of the graph.
Ask the questions at the bottom of the graph to the whole class. ; In addition use the graph for check point 7 on page 157

20. HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A FUNCTION’S GRAPH?
Domain: set of inputs
Found on x –axis
Range: set of outputs
Found on y -axis

21. Using set builder notation it would look like this for the domain:
Using Interval Notation: [-4, 2]
What would it look like for the range using both set builder and interval notation?

22. IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH
Use Set Builder Notation.
Domain:
Range:

23. IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH

24. IDENTIFYING INTERCEPTS FROM A FUNCTION’S GRAPH
We can say that -2, 3, and 5 are the zeros of the function. The zeros of the function are the
x- values that make
f (x) = 0.
Therefore, the real zeros are the x-intercepts.
A function can have more than one x-intercept, but at most one y-intercept.

25. SUMMARY: ANSWER IN COMPLETE SENTENCES.
Explain how the vertical line test is used to determine whether a graph is a function.
Explain how to determine the domain and range of a function from its graph.
Does it make sense? Explain your reasoning.
I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.

26. AIM #1.3: HOW DO WE IDENTIFY INTERVALS ON WHICH A FUNCTION IS INCREASING OR DECREASING?
Increasing, Decreasing and Constant Functions
A function is increasing on a open interval, I if
f (x1) < f(x2) whenever x1<x2 for any x1 and x2 in the interval.

27. A function is decreasing on an open interval, I, if f(x1) > f (x2) whenever x1 > x2
for any x1 and x2 in the interval.

28. A function is constant on an open interval, I, f(x1) = f (x2) for any x1 and x2 in the interval.

29. Note:
The open intervals describing where function increase, decrease or are constant use
x-coordinates and not y-coordinates.

30. Example 1: Increases, Decreases or Constant
State the interval where the function is increasing, decreasing or constant.

31. Practice:
State the interval where the function is increasing, decreasing or constant.

32. WHAT IS A RELATIVE MAXIMA?
Definition of a Relative Maximum
A function value f (a) is a relative maximum of f if there exists an open interval containing a such that f (a) > f (x) for all x ≠ a in the open interval.

33. WHAT IS A RELATIVE MINIMA?
Definition of a Relative Minimum
A function value f (b) is a relative minimum of f if there exists an open interval containing b such that f (b) < f (x) for all x ≠ b in the open interval.

34. HOW DO WE IDENTIFY EVEN AND ODD FUNCTIONS AND SYMMETRY?
Definition of Even and Odd Functions
The function f is an even function if
f (-x)= f (x) all x in the domain of f.
The right side of the equation of an even function does not change if x is replaced with –x.
The function f is an odd function if f (-x) = -f (x)
for all x in the domain of f.
Every term on the right side of the equation of an odd function changes its sign if x is replaced
with –x.

35. DETERMINE IF FUNCTION IS EVEN,ODD OR NEITHER
f (x) = x3 - 6x
Steps:
Replace x with –x and simplify.
If the right side of the equation stays the same it is an even function.
If every term on the right side changes sign, then the function is odd.

36. DETERMINE IF FUNCTION IS EVEN, ODD OR NEITHER
g (x) = x4 - 2x2
h(x) = x2 + 2 x + 1
Steps:
Replace x with –x and simplify.
If the right side of the equation stays the same it is an even function.
If every term on the right side changes sign, then the function is odd.

37. PRACTICE: Determine if function is Even, Odd or Neither f (x) = x2 + 6
g(x) = 7x3 – x
h (x) = x5 + 1

38. The function on the left is even.
What does that mean in terms of the graph of the function?
The graph is symmetric with respect to the y-axis. For every point (x, y) on the graph, the point (-x, y) is also on the graph.
All even functions have graphs with this kind of symmetry.

39. The graph of function f (x) = x3 is odd.
It may not be symmetrical with respect to the y-axis. It does have symmetry in another way.
Can you identify how?

40. For each point (x, y) there is a point (-x, -y) is also on the graph.
Ex. (2, 8) and (-2, -8) are on the graph.
The graph is symmetrical with respect to the origin.
All ODD functions have graphs with origin symmetry.

41. SUMMARY: ANSWER IN COMPLETE SENTENCES.
What does it mean if a function f is increasing on an interval?
If you are given a function’s equation, how do you determine if the function is even, odd or neither?
Determine whether each function is even, odd or neither.
a. f (x) = x2- x4 b. f (x) = x(1- x2)1/2

42. AIM #1.3B: HOW DO WE UNDERSTAND AND USE PIECEWISE FUNCTIONS?
A piecewise function is a function that is defined by two (or more) equations over a specified domain.

43. Example: ( DO NOT COPY) READ
A cellular phone company offers the following plan:
$20 per month buys 60 minutes
Additional time costs $0.40 per minute

44. HOW DO WE EVALUATE A PIECEWISE FUNCTION?

45. PRACTICE: Find and interpret each of the following:
a. C (40) b. C (80)

46. HOW DO WE GRAPH A PIECEWISE FUNCTION?

47. We can use the graph of a piecewise function to find the range of f.
What would the range be for the piecewise function? ( For previous piecewise function)

48. Some piecewise functions are called step functions because the graphs form discontinuous steps.
One such function is called the greatest integer function, symbolized by int (x) or
int (x) = greatest integer that is less than or equal to x.
For example:
a. int (1) = 1, int (1.3) = 1, int (1.5) = 1, int (1.9)= 1
b. int (2) = 2, , int (2.3) =2 , int (2.5) = 2, int (2.9)= 2

49. Graph of a Step Function

50. FUNCTION AND DIFFERENCE QUOTIENTS
Definition of the Difference Quotient of a Function:
The expression for h≠0 is called
the difference quotient of the function f.

51. HOW DO WE EVALUATE AND SIMPLIFY A DIFFERENCE QUOTIENT?
If f (x) = 2x2 – x + 3, find and simplify each expression:
f ( x + h)
Try:
Steps:
Replace x with (x + h) each time x appears in the equation.

52. PRACTICE: If f (x)= -2x2 + x + 5, find and simplify each expression:
f (x + h)

53. SUMMARY: ANSWER IN COMPLETE SENTENCES.
What is a piecewise function?
Explain how to find the difference quotient of a function f,
If an equation for f is given.

54. AIM #1.4 HOW DO WE IDENTIFY A LINEAR FUNCTION AND ITS SLOPE?
Data presented in a visual form as a set of points is called a scatter plot. A scatter plot shows the relationship between two types of data.

55. Regression line is the line that passes through or near the points
Regression line is the line that passes through or near the points. This is the line that best fits the data points.
We can write an equation that models the data and allows us to make predictions.

56. WHAT IS THE DEFINITION OF A SLOPE OF A LINE?

57. PRACTICE: Find the slope of a line. (-3, 4) and (-4, -2)

58. POINT-SLOPE FORM EQUATION
The point-slope of the equation of a nonvertical line with slope m that passes through the point (x1,y1) is:
y –y1=m (x – x1)

59. HOW DO WE WRITE AN EQUATION IN POINT-SLOPE FORM?
Write an equation in point-slope form for the line with slope 4 that passes through the point (-1, 3).
Then solve the equation for y.
Steps:
Write the point-slope form equation.
y –y1=m (x – x1)
Substitute the given values.
Then solve for y.

60. PRACTICE:
Write an equation in point-slope form for the line with slope 6 that passes through the point (2, -5).
Then solve the equation for y.

61. HOW DO WE WRITE AN EQUATION FOR A LINE WHEN WE ONLY HAVE TWO POINTS?
Write an equation in point-slope form for the line that passes through the points (4, -3) and (-2, 6).
Then solve the equation for y.
What would you need to do first?
Steps:
Find the slope.
Then choose any pair of points and substitute into the point-slope form equation.

62. PRACTICE:
Write an equation in point-slope form for the line that passes through the points (-2,-1) and (-1, -6).
Then solve the equation for y.

63. SUMMARY: ANSWER IN COMPLETE SENTENCES.
What is the slope of a line and how is it found?
Describe how to write an equation of a line if you know at least two points on that line.
How do you derive the slope-intercept equation from y –y1=m (x – x1)?

64. AIM # 1.4B: HOW DO WE WRITE AND GRAPH LINEAR EQUATIONS IN THE FORM OF Y = MX+B?
Slope intercept equation is y = mx + b where m is the slope and b is the y-intercept of the equation.
Graphing y = mx + b using the slope and y- intercept.
Graph the y-intercept first. (0, b)
Then use the slope to get to the other points on the line.
Let’s try:

65. EQUATIONS OF HORIZONTAL LINES
A horizontal line has a m=0. Therefore the equation is y=0x + b which can be simplified to
y = b.
Graph y = 3 or f (x) = 3
Note: This is a constant function

66. EQUATIONS OF VERTICAL LINES
The slope of a vertical line is undefined. The equation of a vertical line is x = a, where a is the x-intercept of the line.
Note:
No vertical line represents a function.

67. WHAT IS THE GENERAL FORM OF THE EQUATION OF A LINE?
Every line has an equation that can be written in the general form:
Ax + By = C or Ax + By- C = 0
Where A, B, C are real numbers and A and B are not both zeros.

68. FINDING THE SLOPE AND THE Y-INTERCEPT
Find the slope and the y-intercept of the line whose equation is 3x + 2y – 4 =0.
Steps:
The equation is given in general form. Express in y = mx + b by solving for y.
Then the slope and y-intercept can be identified.

69. PRACTICE:
Find the slope and the y-intercept of the line whose equation is 3x + 6y – 12 =0.
Then use slope and y-intercept to graph the line.

70. HOW DO WE FIND THE INTERCEPTS FROM THE GENERAL FORM OF THE EQUATION OF A LINE?
Graph using the intercepts: 4x – 3y – 6=0
Steps:
To find the x-intercept. Set y = 0 and solve for x.
To find the y-intercept. Set x = 0 and solve for y.
Graph the points and draw a line connecting these points.

71. PRACTICE:
Graph using the intercepts: 3x – 2y – 6=0

72. REVIEW OF THE VARIOUS EQUATIONS OF LINES

73. SUMMARY: ANSWER IN COMPLETE SENTENCES.
How would you graph the equation x = 2. Can this equation be expressed in slope-intercept form? Explain.
Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
How do you use the intercepts to graph the general form of a line’s equation?

74. Aim # 1.5: How do we find the average rate of change?
Slope as a rate of change:
Example 1:
The line on the graph for the number of women and men living alone are shown in the graph.
Describe what the slope represents.

75. Solution: Note x – represents the year and y- number of women.
Choose two points from the women’s graph.
Find the slope and then describe the slope.
Remember to include the units.

76. Practice:
Use the ordered pairs and find the rate of change for the green line or men graph. Express slope two decimal places and describe what it represents.

77. Average Rate of Change
If the graph of the function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points.
This line is called the secant line.

78. Problem:
Looking at the graph, what is the man’s average growth rate between the ages 13 and 18.

79. The Average Rate of Change of a Function

80. Example 1: Find the average rate of change of f (x) = x2 from Steps:
Use

81. Find the average rate of change of f (x) = x2 from

82. Average Velocity of an Object
Suppose that a function expresses an object’s position, s (t), in terms of time, t. The average velocity of the object from t1 to t2 is:

83. Example 2:
The distance, s (t), in feet, traveled by a ball rolling down a ramp is given by the function:
s (t)= 5t2,
where t is the time, in seconds after the ball is released. Find the ball’s average velocity from
t1 = 2 seconds to t2 = 3 seconds
t1 = 2 seconds to t2 = 2.5 seconds
t1 = 2 seconds to t2 = 2.01 seconds

84. Summary: Answer in complete sentences.
If two lines are parallel, describe the relationship between their slopes and y-intercept.
If two line are perpendicular, describe the relationship between their slopes.
What is the secant line?
What is the average rate of change of a function?

85. Aim #1.6: How do we recognize transformations?
Review Algebra’s Common Graphs (distribute)
Vertical Shift

86. Vertical Shifts
In general if c is positive, y = f (x) +c shifts upward c units. If c is negative it shifts downward c units.

87. Example 1:

88. Practice:
Use the graph of to obtain the graph of

89. Horizontal Shifts
In general, if c is positive, y = f (x + c) shifts the graph of f to the left c units and y = f (x – c) shifts the graphs of f to the right c units.
These are called horizontal shifts of the graph of f.

90. Example 2:

91. Note:

92. Example 3:
Use the graph of f (x ) = x2 to obtain the graph of h (x) = (x + 1)2 – 3.
Steps to combining a shift:
Graph the original function.
Then shift horizontally.
Then shift vertically.

93. Reflection about the x-axis:
Reflections of Graphs
Reflection about the x-axis:
The graph of y = - f (x) is the graph of y = f (x) reflected about the x- axis.

94. Example 4:
Use the graph of to obtain the graph of

95. Reflections of Graphs Reflection about the y-axis:
The graph of y = f (-x) is the graph of y = f (x) reflected over the y –axis.
Example: The point (2, 3) reflected over the y-axis is (-2, 3).

96. Vertical Stretching By multiplying the y-coordinates by c.
Let f be a function and c be a positive real number.
If c > 1 the graph of y = c f (x) is the graph y = f (x) vertically stretched.
How?
By multiplying the y-coordinates by c.

97. Vertical Shrinking By multiplying the y-coordinates by c.
Let f be a function and c be a positive real number.
If 0 < c < 1 the graph of y = c f (x) is the graph y = f (x) vertically shrunk.
How?
By multiplying the y-coordinates by c.

98. Example 6:
Use the graph of f (x) = x3 to obtain the graph of
h (x) =

99. By dividing x- coordinates by c.
Horizontal Shrinking
Let f be a function and c be a positive real number.
If c > 1 the graph of y = f (cx) is the graph
y = f (x) horizontal shrink.
How?
By dividing x- coordinates by c.

100. Horizontal Stretching
Let f be a function and c be a positive real number.
If 0 < c < 1 the graph of y = f (cx) is the graph
y = f (x) horizontal stretch.
How?
By dividing x- coordinates by c.

101. Example 7:
Use the graph y = f (x) to obtain each of the following graphs.
a. g (x) = f (2x) b. h( x) = f( 1/2x)

102. Sequences of Transformations
Transformations involving more than one transformation can be graphed performing the transformations in the following order:
Horizontal shifting
Stretching or shrinking
Reflecting
Vertical shifting

103. Summary: Answer in complete sentences.
What must be done to a function’s equation so that its graph is shifted vertically upward?
What must be done to a function’s equation so that its graph is shifted horizontally to the right?
What must be done to a function’s equation so that its graph is reflected about the x-axis?
What must be done to a function’s equation so that its graph is stretched vertically?

104. Aim# 1.7: What are composite functions?
Finding the domain of a function (w/ no graph).
Example 1: Find the domain of each function.

105. The Algebra of Functions
We can combine functions by addition, subtraction, multiplication and division by performing operations with the algebraic expressions that appear on the right side of the equations.
Example 2: Let f (x)=2x – 1 and g (x) = x2 + x – 2
Find the following function.
Sum : (f + g ) (x)= f (x ) + g (x)
( 2x – 1) + (x2 + x – 2)= Substitute given functions.
2. Simplify.

106. The Algebra of Functions

107. Example 3: Adding Functions and Determining the Domain
Find each of the following:
(f + g) (x)
The domain of (f + g)

108. Composite Functions This is another way to combine functions.
Example: f (g (x)) can be read as f of g of x and it is written as

109. Example 4: Forming Composite Functions
Given f (x)= 3x – 4 and g (x) = x2 – 2x + 6, find each of the following:

110. Example 5: Forming a Composite Function and Finding Its Domain

111. Practice: Given and g(x)= Find each of the following: 1.
2. the domain of

112. Summary: Answer in complete sentences.
Explain how to find the domain of a function of a radical and a rational equation.
If equations for f and g are given, explain how find f- g.
If equations for two functions are given, explain how to obtain the quotient of the two functions and its domain. ( ex. f/g(x))
Describe a procedure for finding .
What is the name of this function?

113. Aim# 1. 8 What is an inverse function?

114. Example 1: Verifying Inverse Functions
Show that each function is the inverse of the other:
Steps:
To show that f and g are inverses of each other, we must show that f (g (x)) = x and f (g (x)) = x.
Begin with f (g (x))
Then with g (f (x))
Do you get x?

115. Practice:
Show that each function is the inverse of the other:

116. How do we find the inverse?
Find the inverse of
f (x) = 7x – 5
Steps:
Replace f (x) with y.
Switch x and y.
Solve for y.
Then replace y with f-1 (x)

117. How do we find the inverse?
Find the inverse of:
f (x ) = x3 + 1
Steps:
Replace f (x) with y.
Switch x and y.
Solve for y.
Then replace y with f-1 (x)

118. How do we find the inverse?
Find the inverse of:
Steps:
Replace f (x) with y.
Switch x and y.
Solve for y.
Then replace y with f-1 (x)

119. Practice: Find the inverse for the following: Steps:
Replace f (x) with y.
Switch x and y.
Solve for y.
Then replace y with f-1 (x)

120. Properties of Inverse Functions
1. A function f has an inverse that is a function, f-1, if there is no horizontal line that intersects the graph of the function f at more than one point.

121. Properties of Inverse Functions
2. The graph of a function’s inverse is a reflection of the graph of f about the line y = x.

122. Summary: Answer in complete sentences.
Explain how to determine if two functions are inverse of each other.
Describe how to find the inverse of a function.
What are some properties of a function and its inverse? Draw an illustration to support your written statement.

123. Aim #1.9: How do we find the distance and midpoint of a segment?
Formulas: