Functions and Patterns by Lauren Mc Cluskey Exploring

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

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

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 1 2 out 4 8

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). ”

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

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

The Coordinate Plane l “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 l Always show time on the horizontal axis (x), because it is an independent variable.

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

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

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

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

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

Tests for Functions: l “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

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

Function Tests: l “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

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 l

Mapping Diagram Then draw lines between the coordinates. Domain: Range: 0 -6 2 -3 4 0 6 3 l 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. l So this is a function.

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

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

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

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

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

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

Patterns with Triangles l 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=3 P=6 P=5 from the MCAS

P=6 P=4 P=3 P=5 Number of Triangles 1 2 3 4. . . N Outer Perimeter (in units) 3 4 5 6 … p from the MCAS

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

P=4 P=3 P= 6 P=5 # of Triangles Outer 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

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!

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

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

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

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

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) 1 Output (y) 7 2 14 3 21 4 ? from the MCAS

l Hint: (Write a rule then evaluate. )

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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) 1 Output (y) 7 2 14 3 21 4 28 from the MCAS

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. l What is the rule? from the MCAS

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

Subway Cars f(x) = 30 x

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

Patterns in Real-life Situations l 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. l What’s the rule? from the MCAS

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

Library Fines lf(x) = $0. 15 x from the MCAS

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

Make a Table of the Coordinates (x) -2 -1 0 1 2 (y) from the MCAS

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

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

Make a Table of the Coordinates: (x) (y) -1 0 1 2 from the MCAS

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

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

How to Write a Rule: 1) 2) 3) 4) 5) 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?

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

Remember: to Write a Rule: 1) 2) 3) 4) 5) 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?