Functions and Patterns by Lauren McCluskey

1. Functions and Patternsby 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.

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. Origin • The origin is where the x and y axes intersect. This is (0, 0). (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. 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: 0 -6 2 -3 4 0 6 3

17. Mapping Diagram Then draw lines between the coordinates. Domain: Range: 0 -6 2 -3 4 0 6 3 • 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 Function Notation: f(x) = 3x + 5 Rule for Function Function Rules by Christine Berg

20. Function Set up a table using the rule: f(x)= 3x+5 Function Rules by Christine Berg

21. Function Evaluate this rule for these x values: f(x)= 3x+5 So 3(2) + 5 = 11… 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. P = 6 P = 4 P = 3 P = 5 Number ofTrianglesOuter Perimeter (in units)1 3 2 4 3 5 4 6 ... …N p from the MCAS

26. Triangles P= 4 P= 6 * Write a rule for finding p, the outer perimeter for a design that uses n triangles. 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. P=4 P= 6 P=3 P=5 # ofTrianglesOuter Perimeter (in units) 1 3 (+1) 2 4 (+1+1) 3 5 (+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. 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.

33. 6 blocks 6 blocks 6 blocks 6 blocks 6 blocks 6 blocks Steps • 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. +4 blue +4 red +4 green +4 corners Square Tiles The constant difference is +4 so the rule is 4x + 4.

37. + 4 blue + 4 red + 4 green etc… + 4 corners Square Tiles • You can see this: +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? 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? 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 First, make a table… 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. 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?