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Functions and Patterns by Lauren McCluskey. Exploring the connection between input / output tables, patterns, and functions…. Credits. Function Rules by Christine Berg Algebra I from Prentice Hall, Pearson Education The Coordinate Plane by Christine Berg. Relation.
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Functions and Patternsby Lauren McCluskey Exploring the connection between input / output tables, patterns, and functions…
Credits • Function Rules by Christine Berg • Algebra I from Prentice Hall, Pearson Education • The Coordinate Plane by Christine Berg
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.
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 • What does this mean? • It means that for every input value there is only one output value.
More on that later, but first let’s review coordinate planes…
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.
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). (0, 0) The Coordinate Plane by Christine Berg
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
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: • “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: • “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
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.
Function Notation f(x) = 3x + 5 Output Input Function Rules by Christine Berg
Function Function Notation: f(x) = 3x + 5 Rule for Function Function Rules by Christine Berg
Function Set up a table using the rule: f(x)= 3x+5 Function Rules by Christine Berg
Function Evaluate this rule for these x values: f(x)= 3x+5 So 3(2) + 5 = 11… Function Rules by Christine Berg
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.
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
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
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
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
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?
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
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: • 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 The constant difference is +6, so the rule is 6x + 1.
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.
Square Tiles • The first four figures in a pattern are shown below. * What’s my rule? from the MCAS
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?
+4 blue +4 red +4 green +4 corners Square Tiles The constant difference is +4 so the rule is 4x + 4.
+ 4 blue + 4 red + 4 green etc… + 4 corners Square Tiles • You can see this: +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? from the MCAS
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?
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
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
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?
Subway Cars First, make a table… from the MCAS
Subway Cars f(x) = 30x
Try it! • Write a rule that describes the relationship between the input (x) and the output (y) in the table below. from the MCAS
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?
Input / Output Table • f(x)=2x + 1
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
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?