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

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

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Relation According to Prentice Hall: A relation is a set of ordered pairs. Or Or A relation is a set of input (x) and output (y) numbers. inout 14 28

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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).

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

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More on that later, but first lets review coordinate planes… More on that later, but first lets review coordinate planes…

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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 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. Always show time on the horizontal axis (x), because it is an independent variable.

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

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

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

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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

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

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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 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

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

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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 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

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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): (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:

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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. 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. So this is a function.

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Function Notation f(x) = 3x + 5 OutputInput Function Rules by Christine Berg

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

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Function Set up a table using the rule: f(x)= 3x+5 x(Input)12345 y(Output)8 Function Rules by Christine Berg

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Function Evaluate this rule for these x values: f(x)= 3x+5 So 3(2) + 5 = 11… x(Input)12345 y(Output) Function Rules by Christine Berg

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Functions You can model functions using rules, tables, and graphs. Prentice Hall 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. 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.

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

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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. 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. from the MCAS P=3 P= 4 P=5 P=6

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Number of Triangles Outer Perimeter (in units) … N p (in units) … N p from the MCAS P = 3 P = 4 P = 5 P = 6

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

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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# of Triangles Outer Perimeter # of Triangles Outer Perimeter (in units) 1 3 (+1) (in units) 1 3 (+1) 2 4 (+1+1) 2 4 (+1+1) 3 5 (+1+1+1) 3 5 (+1+1+1) ** The constant difference is +1. So multiply x by 1 So multiply x by 1 then add 2 then add 2 to get the output number. to get the output number. from the MCAS P=3 P=4 P=5 P= 6

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

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Brick Walls Brick Walls Whats my rule? from the MCAS Now you try one:

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Steps x f(x) or y The constant difference is +6, so the rule is 6x + 1.

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Steps You can see the You can see the constant difference. Youre adding 6 blocks each time. 6 blocks

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

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Square Tiles Square Tiles x f(x) or y The constant difference is +4 so the rule is 4x blue+4 red+4 green +4 corners

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You can see this: You can see this: Square Tiles Square Tiles +4 blue+4 red+4 green + 4 blue+ 4 red+ 4 greenetc… + 4 corners

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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) ? from the MCAS

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

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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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) from the MCAS

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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? What is the rule? from the MCAS

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Subway Cars Subway Cars Number of Subway Cars Total Number of Seats …… ns from the MCAS First, make a table…

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Subway Cars f(x) = 30x

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Try it! Write a rule that describes the relationship between the input (x) and the output (y) in the table below. Write a rule that describes the relationship between the input (x) and the output (y) in the table below. Input (x) Output (y) from the MCAS

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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

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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. 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. Whats the rule? Whats the rule? from the MCAS

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Lucindas Savings f(x) = $15x from the MCAS

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

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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

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Patterns in Real-Life Situations: 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 246… Amount of Fine $0.30$0.60$0.90… from the MCAS Whats the rule? What do they charge for 1 day?

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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

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Patterns in Graphs #1 from the MCAS Whats the rule?

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Make a Table of the Coordinates (x)(y) from the MCAS

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

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Patterns in Graphs #2 from the MCAS Whats my rule?

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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Make a Table of the Coordinates: (x)(y) from the MCAS

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

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Patterns in Sequences of Numbers: 12, 16, 20, 24… Whats my rule?

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How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Does it work?

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

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Remember: to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y. 5) Check your rule for at least 3 values of x. *Then ask: Does it work?

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