1. Unit 1B -- Functions

2. What is a Function? What is a Relation?
A "relation" is just a relationship between sets of information.
A function is a relationship between two variables. The first variable determines the value of the second variable. It is a “well-behaved” relation.
Two quantities are related to each other by some rule of correspondence.

3. What Will You Learn?
Basic Function Characteristics- (parent functions- linear, quadratic, exponential, square root, cube root, absolute value) determine if something is a function, graphs, tables and ordered pairs
What are Domain and Range?

5. Real-World Examples Output/Range Dependent variable x
Input/Domain/
Independent variable

6. The “Pairing” of names and heights is a relation.
Pairs of names and heights are “ordered” in functions.
(Person, Height)
One-to-One

7. So is this a function?
(Height, Person)???
Why or Why Not???

8. Remember!!!
Definition of a Function: A special relation such that for all values of x there exists one and only one y-value.

9. So which of these below are Functions?
Function? Yes or No
1. {(2, 3), (4, 5), (5, 6)}
2. {(2, 3), (4, 3), (5, 6)}
3. {(2, 3), (2, 5), (5, 6)}

10. https://video. search. yahoo. com/yhs/search
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11. Graph – Vertical Line Test

12. Graph – Vertical Line Test

13. Is it a function?

14. Function?

15. Function Notation
Linear Function
Y = 2x – 1
F(x) = 2x-1

16. Types of Functions-Foldable

21. Warm-Up---Find the Missing Values and Sketch the Graph

22. What is Meant by Domain and Range
The Domain of a Function is the set of all possible x-values of a function 
(independent variable) (input)
The Range of a Function is the set of y-values of a function 
(dependent variable) (output)
But First….you need to know about Interval Notation

28. 4) 5) 3)

29. What is a Parent Function?
A parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form:
y = a 𝒙 𝟐 + bx + c
….the simplest quadratic function is
y = 𝒙 𝟐
…. This is therefore the parent function of the family of quadratic equations.

30. Parent Functions -2 -1 1 2

31. 2 1 1 2

32. 4 1 1 4

33. -8 -1 1 8

34. 1 4
1 2 1 2 4

35. 1 2 3 4

36. What are the Characteristics of Functions?
Maximum (max) and Minimum (min)
Increasing vs Decreasing Intervals
Relative Max and Relative Min,
Even vs Odd
x- and y- Intercepts
Zeros, Roots and Solutions

37. Definitions of Critical Vocabulary
The RELATIVE MAXIMUM of a function is the y-value of the highest point on a particular region of a graph
The RELATIVE MINIMUM of a function is the y-value of the lowest point on a particular region of a graph
ZEROs, ROOTs, x-INTERCEPTs, SOLUTIONs, are all terms that refer to the x-value of a graph when the y-value is zero
x-Intercepts and y-Intercepts are the points at which a graph crosses the x- and y-axis correspondingly
A function is INCREASING on an interval if the y–value increases as the x-value increases
A function is DECREASING on an interval if the y-value decreases as the x-value increases

43. Even or Odd?
Odd

44. Even or Odd?
Even

45. h(x) = 2x - 1

46. g (x) = - 𝑥

47. Review of Properties of Functions
The RELATIVE MAXIMUM of a function is the y-value of the highest point on a particular region of a graph
The RELATIVE MINIMUM of a function is the y-value of the lowest point on a particular region of a graph
ZEROs, ROOTs, x-INTERCEPTs, SOLUTIONs, are all terms that refer to the x-value of a graph when the y-value is zero
x-Intercepts and y-Intercepts are the points at which a graph crosses the x- and y-axis correspondingly
A function is INCREASING on an interval if the y–value increases as the x-value increases
A function is DECREASING on an interval if the y-value decreases as the x-value increases

48. Review of Properties of Functions
Odd—a function is “ODD” if f(-x) = -f(x). All odd functions are symmetrical around the origin
Even—a function is “EVEN” if f(-x) = f(x). All even functions are symmetrical around the y-axis
** NEW!! Symmetry—symmetry is defined as the orientation of a graph relative to a specific point or line or axis. Symmetry around the y-axis is like a “reflection”. Symmetry around a point (like the origin) is when every point has a matching point on the “opposite side” that is the same distance from the origin but is in the opposite direction….
** NEW!! End Behavior—what is happening to the value of “y” or “f(x)” or the function as the x-value increases or decreases…..

52. Increasing and Decreasing Functions
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Using Interval Notation

53. intervals increasing and decreasing

60. Increasing/Decreasing
A function is increasing if its y-values increase 
from left to right.
(from x, to x)
Determine the intervals on which this 
function is increasing.
Determine the intervals on which this 
function is decreasing.

61. Maximum: highest Y value a function has
Minimum: lowest Y value a function has
absolute max: highest points
relative/local max: highest point within the area shown
absolute min: highest points
relative/local min: highest point within the area shown

63. Let’s look at two graphs
What is Happening to the y-value
As the x-value is changing?

64. Let’s look at two graphs
What is Happening to the y-value
As the x-value is changing?

65. variation
Variation is that way we describe the relationship between the x-and y-values as one of them changes…..
In other words as the x-value changes, how does that affect the y-value?

66. Direct variation First, let’s look at Direct Variation:
In Direct Variation, as the x-value increases, the y-value increases by some factor
In Direct Variation, as the x-value decreases, the y-value decreases by the exactly same factor
We call this factor the “constant of variation”
The formula for Direct Variation is: y = kx

67. Direct variation
An equation is a Direct Variation if it can be written in the form y = kx OR it’s graph is a line that passes through zero
In a direct variation, the values of the two variables change in the same manner –
As one value increases, the other increases
Or as one value decreases, the other decreases

68. Let’s look at two graphs
What is Happening to the y-value
As the x-value is changing?

69. So let’s take a few minutes to graph a few of these Direct variations
Enter y=kx……how does the graph change as you enter a different constant, k?
Step 1: press the y= button
Step 2: enter 3x and then graph
Try a few more….

70. Direct variation So what do you notice?????
What is happening to the function?
What is happening to “y” as you change the “k” value?
What is happening to the graph?

71. Direct variation So is this a Direct Variation? 5x – 3y = 6?
Why or why not?
Can you find the constant of variation, k? What is it if you can?
How about this one? 7y = 2x

72. Direct variation
Can you write an equation for the direct variation given a point on the graph?
Let’s try….. (4,-3) but remember: a direct variation is always given in the form y=kx

73. Direct variation
Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see the lightning, you are about 2 miles from the lightning.
Write an equation for the relationship between time and distance.

74. Direct variation
Relate: The distance varies directly with the time. When x = 10, y = 2. Define: Let x = number of seconds between seeing lightning and hearing thunder. Let y = distance in miles from lightning.

75. INVERSE VARIATION Y = 𝑲 𝑿 or xy=k
WHERE “X” AND “Y” ARE VARIABLE THAT HAVE CHANGING VALUES
AND, “K” IS A CONSTANT (often called the constant of variation)
SO WHAT HAPPENS AS X CHANGES?

76. Let’s look at two graphs
What is Happening to the y-value
As the x-value is changing?

77. INVERSE VARIATION
Suppose that y varies inversely with x…..and x goes through the point (3, -5). Find the function that models this variation…
if x = 3 and y = -5 , then Y = 𝐾 𝑋 means = 𝐾 3
so, k = -15
And if k = -15, then the equation for the inverse variation is: Y = −15 𝑋

78. INVERSE VARIATION
So let’s look at some graphs of these Inverse Variations…..
First, enter y = 𝑘 𝑥 into your calculator using the “y=“ button
Then, press graph……
Try y = 3 𝑥 and y = −7 𝑥 then y = 1 2𝑥
what is k =????

79. inverse variation So what do you notice?????
What is happening to the function?
What is happening to “y” as you change the “k” value?
What is happening to the graph?

80. inverse variation Warm-Up Problem
A biker traveling at 8 mph can cover 8 miles in 1 hour. If the biker's speed decreases to 4 mph, it will take the biker 2 hours to cover the same distance. As his speed decreases, the time increases

81. inverse variation
In an inverse variation, the values of the two variables change in an opposite manner as one value increases, the other decreases

82. inverse variation
So can you write an equation for an Inverse Variation given one point on its graph?
Try….. Suppose y varies inversely with x and y = 7 when x = 5.
Remember inverse variations are given by the formula xy = k
You try using the point (3,-2)

83. Applications of Variations
Heart rates and life span of most mammals are inversely related. A cat lives about 15.2 years on average and has a heart rate of 126 beats per minute.
What is the constant of variation?
A hamster has a heart rate of about 634 beats per minute. About how long will the hamster live?
An elephant lives for about 70 years. Find the elephant’s heart rate.

84. Applications of Variations
The grade you earn in math varies inversely with the number of minutes you watch television per night. If you watch 90 minutes of TV per night, you get a 60% in math.
How much TV can you watch if you want to earn a 70%
If you cut back to only 75 minutes a night watching TV, what grade would you expect to earn?

85. END BEHAVIOR of a FUNCTION
The appearance of a graph as it is followed farther and farther in either direction.
For polynomials, the end behavior is indicated by drawing the positions of the arms of the graph, which may be pointed up or down. Other graphs may also have end behavior indicated in terms of the arms, or in terms of asymptotes or limits.
Polynomial End Behavior:
1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both down. 2. If the degree n is odd, then one arm of the graph is up and one is down. 3. If the leading coefficient an is positive, the right arm of the graph is up. 4. If the leading coefficient an is negative, the right arm of the graph is down.

86. Symmetry Even - Graph is symmetric about the y-axis
Odd- Graph is symmetric about the origin

87. Even, odd, or neither

88. https://www. khanacademy
functions/polynomial-end-behavior/v/polynomial-end-behavior