Transcript Function Rules - Math Makes Sense

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
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
 What
 It
does this mean?
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.
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:

“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
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)= 3x+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
 “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=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= 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 6x + 1.
Steps
6 blocks
6 blocks
 You
can see the
constant difference.
6 blocks
6 blocks
6 blocks
6 blocks
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:
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
4x + 4.
2 12
3 16
+4 blue
+4 red
+4 corners
+4 green
Square Tiles
 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

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 the table on the next
slide to show the total number of seats for
different numbers of subway cars.

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) = 30x
Try it!

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
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:
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) = $15x
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 + $2x
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.
4
6
…
Fines for Overdue
Library Books
2
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
f(x)
= $0.15x
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
f(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
f(x)
= .5x -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
f(x)
= 4x + 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?