1. Functions and Patterns by Lauren McCluskey
Exploring the connection between input / output tables, patterns, and functions…

2. Credits Function Rules by Christine Berg
Algebra I from Prentice Hall, Pearson Education
The Coordinate Plane by Christine Berg

3. Relation
According to Prentice Hall: “A relation is a set of ordered pairs.” Or
A relation is a set of input (x) and output (y) numbers. in
out 1 4 2 8

4. Function
According to Prentice Hall: “A function is a relation that assigns exactly one value in the range (y) to each value in the domain (x).”

5. Functions What does this mean?
It means that for every input value there is only one output value.

6. More on that later, but first let’s review coordinate planes…

7. The Coordinate Plane
“You can use a graph to show the relationship between two variables…. When one variable depends on another, show the dependent quantity on the vertical axis (y).” Prentice Hall
Always show time on the horizontal axis (x), because it is an independent variable.

8. Remember: - + The x-axis is a horizontal number line.
It is positive to the right and negative to the left. - +
The Coordinate Plane by Christine Berg

9. Y-axis The y-axis is a vertical number line. + -
It is positive upward and negative downward. -
The Coordinate Plane by Christine Berg

10. The origin is where the x and y axes intersect. This is (0, 0).
The Coordinate Plane by Christine Berg

11. Quadrants
The x and y axes divide the coordinate plane into 4 parts called quadrants. I II
III IV
The Coordinate Plane by Christine Berg

12. A pair of numbers (x , y) assigned to a point on the coordinate plane.
Ordered Pair
A pair of numbers (x , y) assigned to a point on the coordinate plane.
The Coordinate Plane by Christine Berg

13. Tests for Functions:
“One way you can tell whether a relation is a function is to analyze the graph of the relation using the vertical-line test. If any vertical line passes through more than one point of the graph, the relation is not a function.” Prentice Hall

14. Vertical-Line Test
This is a function because a vertical line hits it only once.

15. Function Tests:
“Another way you can tell whether a relation is a function is by making a mapping diagram. List the domain values and the range values in order. Draw arrows from the domain values to their range values.” Prentice Hall

16. Mapping Diagram
(0, -6), (4, 0), (2, -3), (6, 3) are all points on the previous graph. List all of the domain to the left; all of the range to the right (in order):
Domain: Range:

17. Mapping Diagram Then draw lines between the coordinates.
Domain: Range:
If there are no values in the domain that have more than one arrow linking them to values in the range, then it is a function.
So this is a function.

18. Function Notation f(x) = 3x + 5 Output Input
Function Rules by Christine Berg

19. Function f(x) = 3x + 5 Function Notation: Rule for Function
Function Rules by Christine Berg

20. Set up a table using the rule: f(x)= 3x+5
Function
Set up a table using the rule: f(x)= 3x+5 x
(Input) 1 2 3 4 5 y
(Output) 8
Function Rules by Christine Berg

21. Evaluate this rule for these x values: f(x)= 3x+5
Function
Evaluate this rule for these x values: f(x)= 3x+5
So 3(2) + 5 = 11… x
(Input) 1 2 3 4 5 y
(Output) 8 11 14 17 20
Function Rules by Christine Berg

22. Functions
“You can model functions using rules, tables, and graphs.” Prentice Hall
Each one shows the relationship from a different perspective. A table shows the input / output numbers, a graph is a visual representation, a function rule is concise and easy to use.

23. Patterns Patterns are functions. They’re predictable.
Patterns may be seen in:
Geometric Figures
Numbers in Tables
Numbers in Real-life Situations
Linear Graphs
Sequences of Numbers

24. Patterns with Triangles
Jian made some designs using equilateral triangles, as shown below. He noticed that as he added new triangles, there was a relationship between n, the number of triangles, and p, the outer perimeter of the design.
P= 4
P=6
P=3
P=5
from the MCAS

25. Number of Triangles Outer Perimeter
(in units) … N p
from the MCAS

26. Triangles
* Write a rule for finding p, the outer perimeter for a design that uses n triangles.
P= 4
P= 6
P = 3
P = 5
P= 3
P= 5
from the MCAS

27. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

28. to get the output number.
# of Triangles Outer Perimeter
(in units) (+1)
(+1+1)
(+1+1+1)
**The constant difference is +1.
So multiply x by 1
then add 2
to get the output number.
from the MCAS

29. f(x)= X + 2 It works! So evaluate and you get: 2+1= 3; 2+2 = 4;
P = 6
P= 4
P = 3
P = 5
f(x)= X + 2
So evaluate and you get:
2+1= 3;
2+2 = 4;
and 3+2 = 5.
It works!

30. Brick Walls
Now you try one:
What’s my rule?
from the MCAS

31. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

32. Steps The constant difference is +6, so the rule is 6x + 1. x
f(x) or y 1 7 2 13 3 19
The constant difference is +6,
so the rule is 6x + 1.

33. Steps You’re adding 6 blocks each time. You can see the
constant difference.
You’re adding 6 blocks each time.

34. Square Tiles The first four figures in a pattern are shown below.
* What’s my rule?
from the MCAS

35. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

36. Square Tiles +4 corners 4x + 4. The constant difference
is +4 so the rule is
4x + 4. x
f(x) or y 1 8 2 12 3 16
+4 blue
+4 red
+4 green
+4 corners

37. Square Tiles You can see this: + 4 corners + 4 blue + 4 red + 4 green
etc…
+ 4 corners
+4 blue
+4 red
+4 green

38. Extending Patterns in Tables
Based on the pattern in the input-output table below, what is the value of y when x = 4?
Input (x)
Output (y) 1 7 2 14 3 21 4 ?
from the MCAS

39. Hint: (Write a rule then evaluate.)

40. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

41. Extending Patterns in Tables
Based on the pattern in the input-output table below, what is the value of y when x = 4?
Input (x)
Output (y) 1 7 2 14 3 21 4 28
from the MCAS

42. Patterns in Tables
A city planner created a table to show the total number of seats for different numbers of subway cars. Copy the table.
What is the rule?
from the MCAS

43. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

44. Subway Cars 6 180 8 240 10 300 … n s First, make a table…
Number of Subway Cars
Total Number of Seats 6
180 8
240 10
300 … n s
from the MCAS

45. Subway Cars
f(x) = 30x

46. Try it!
Write a rule that describes the relationship between the input (x) and the output (y) in the table below.
Input (x) 2 5 10 11
Output (y) 21 23
from the MCAS

47. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

48. Input / Output Table
f(x)=2x + 1

49. Patterns in Real-life Situations
Lucinda earns $20 each week. She spends $5 each week and saves the rest. The table below shows the total amount that she saved at the end of each week for 4 weeks.
What’s the rule?
from the MCAS

50. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

51. Lucinda’s Savings
f(x) = $15x
from the MCAS

52. Write a rule for the cost of n rides:
from the MCAS

53. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

54. Fall Carnival
f(x) = $10 + $2x

55. Patterns in Real-Life Situations:
The local library charges the same fine per day for each day a library book is overdue. The table below shows the amount of the fine for a book that is overdue for different numbers of days.
Fines for Overdue Library Books 2 4 6 …
Amount of Fine
$0.30
$0.60
$0.90
What’s the rule? What do they charge for 1 day?
from the MCAS

56. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

57. Library Fines
f(x) = $0.15x
from the MCAS

58. Patterns in Graphs #1
What’s the rule?
from the MCAS

59. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

60. Make a Table of the Coordinates
(x)
(y) -2 -1 1 2
from the MCAS

61. Patterns in Graphs #1
f(x) = x - 4

62. Patterns in Graphs #2
What’s my rule?
from the MCAS

63. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

64. Make a Table of the Coordinates:
(x)
(y) -1 1 2
from the MCAS

65. Patterns in Graphs #2
f(x) = 2x -1

66. Patterns in Sequences of Numbers:
12, 16, 20, 24…
What’s my rule?

67. How to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Does it work?

68. Patterns in Sequences of Numbers
f(x) = 4x + 8

69. Remember: to Write a Rule:
Make a table.
Find the constant difference.
Multiply the constant difference by the term number (x).
Add or subtract some number in order to get y.
Check your rule for at least 3 values of x.
*Then ask: Does it work?