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FUNCTIONSWhen you see make sure you write down the notes, examples, graphs and all other information in your notes (pencil paper…no technology notes).
How do you know if a relationship is a linear function?
For example: Joe paid $6 for admission and $4 per ride at the fair!!
In this lesson you will learn how to recognize a linear function by
examining the four representations of a function.
Let’s ReviewLet’s Review
A function relation can be represented in 4 different ways.
Numeric GraphicAlgebraicy = 3x +
6 x y
-2 0
-1 3
0 6
1 9
2 12
Verbal
This function multiplies every input by 3 and
adds 6.
Let’s ReviewA Common Mistake
Four different functions !?!?
y = -2x – 1
x y
-2 3
-1 1
0 -1
1 -3
Multiply x by -2 and subtract 1
No, the SAME function!Four different ways to
represent it.
Let’s ReviewCore Lesson
$6 for admission and $4 per ride!!
Number of rides (x) Total cost (y)
012
43
61018
14
225 26
Constant rate of change = 4
Initial value=6
Let’s ReviewCore Lesson
Graph is a straight line.
Not Linear!Linear!
y = 4x + 6
y = x3 +
6
Let’s ReviewCore Lesson
Rate of change is constant between any two rows.
x y
0 6
1 10
2 14
3 18
4 22
x y
0 6
1 7
2 14
3 33
4 70
Linear!Not
Linear!
Let’s ReviewCore Lesson
Independent variable (input) exponent is 1.
Linear!Not
Linear!
y = 4x + 6
y = x3 +
6
Let’s ReviewCore Lesson
Words like “per,” “each,” and “every”
$6 for admission and $4 per ride!!
y = 4x + 6
The volume of a perfect cube plus 6 extra cubic units.
y = x3 + 6
In this lesson you have learned how to recognize a linear
function by examining the four representations of a function.
Let’s ReviewGuided Practice
Which representation is not linear?
You must pay $1 for a copy card and copies are $0.03 per sheet.
y = 0.03x + 1x y
0 1
1 1.3
2 2.6
3 5.2
4 10.4
Let’s ReviewQuick Quiz
A linear relation has a rate of change.
a) Changingb) Constantc) Zerod) Shifting
The graph at left is:a) linear, initial value = 1b) not linear, initial value = 1c) linear, rate of change = 1d) not linear, rate of change
= 1
In this lesson you will learn how to describe the rate of change in
a linear function by using the four representations.
Let’s ReviewLet’s Review
A linear function has a constant rate of change.
y = 3x + 4
y = mx + b
x y
0 4
1 7
2 10
3 13
4 16
Let’s ReviewCore Lesson
When can a function relation be classified as linear?
x y
-2 20
-1 15
0 10
1 5
2 0
y = -5x + 10
When it has a constant rate of change.
Let’s ReviewA Common Mistake
x y
0 4
1 7
4 16
6 22
7 25
3
3
96
Not LinearLinear !_ 3 = 3_ 1 = 3
_ 2 = 3_ 1 = 3
Let’s ReviewCore Lesson
Is y = 5 + 2x linear? Yes!
x y
-2 1
-1 3
0 5
1 7
2 9
y = 5 + 2x
2
2
22
_ 1 = 2_ 1 = 2
_ 1 = 2_ 1 = 2
Let’s ReviewCore Lesson Is y = 5 + 2x2 linear? No!
x y
-2 13
-1 7
0 5
1 7
2 13
y = 5 + 2x2
-6
6
-22
_ 1 = -2_ 1 = -6
_ 1 = 2_ 1 = 6
In this lesson you learned how to describe the rate of change in a linear function by using the four
representations.
Let’s ReviewGuided Practice
What is the constant rate of change for each of the function representations?
x y
-1 2
0 9
1 16
2 23
3 30
y = -10 – 3.5x
Let’s ReviewQuick Quiz
What is the constant rate of change for the relation at right?
What is the constant rate of change for the relation at left?
x y
-2 22
0 16
2 10
4 4
6 -2
How do you figure out if a parking garage represents a
function relation?
In this lesson you will learn how to identify function
properties by examining the input and output of real world
examples.
Let’s ReviewLet’s Review
A function can also be called:
- a relation- a rule- a relationship- a correspondence
Let’s ReviewLet’s Review
A function relation takes any input and yields exactly one output.
4321
Inputs
12963
Outputs
Function
Let’s ReviewA Common Mistake
Multiple inputs have the same output.
Not a function!1 2 3 4
4
Let’s ReviewCore Lesson
Let’s look at 3 real world examples.
Input (x) Output (y)
1
2
3
2
4
6
#1
Let’s ReviewCore Lesson
#2Input (x) Output (y)
. .across town
1
2
2
3
4
Let’s ReviewCore Lesson
#3 1 2 3 4 5 6
36 9 12 12 12
1 2 3 4 5 6
36 9 12
Input (x) Output (y)
Let’s ReviewCore Lesson
Input (x) Output (y)
123
246
Input (x) Output (y)
1
2
234
1 2 3 4 5 6
36 9 12
Input (x) Output (y)• Function• One-to-one• Constant
rate of change = 2
• Not a function• No other
properties apply
• Function• Not one-to-one• No constant
rate of change
Let’s ReviewCore Lesson
Input Output
123
246
1 2 3 4 5 6
36 9 12
Input Output
Not 1-1
1-1
In this lesson you have learned how to identify function
properties by examining the input and output of real world
examples
Let’s ReviewGuided Practice
Pick the relation that is one-to-one.
2 4 6 8
1 2 3 4
1 2 3 4
2 4 6 8
2 4 6 8
1 2 3 4
2 4 6 8
1 2 3 4
A) B) C) D)
Let’s ReviewExtension Activities
How could you make the parking garage from the lesson become a one-to-one function?
Let’s ReviewQuick Quiz
1) Pick all of the one-to-one function relations.
1 2 3 4
2 4 6 8
1 2 3 4
2 4 6 8
1 2 3 4
2 4 6 8
1 2 3 4
2 4 6 8
Let’s ReviewQuick Quiz
2) A one-to-one function relation is shown at right. If the input is 6, what is the output?
X Y
43
21 0.5
1
1.52
A) 2.5 B) 3 C) 3.5 D) 4
How do you know which equation is linear?
y = 3x + 4 ?
In this lesson you will learn how to identify a linear function relation by analyzing characteristics of a linear
function.
Let’s ReviewLet’s Review
Linear = “Straight” (graph is a straight line)
Let’s ReviewA Common Mistake
Only 1 slope and y-intercept for a linear function.
y = 5x + 3x – 4
Is the slope 5?Is the slope 3?Neither, it is 8.Combine all like terms.
y = 8x – 4
Let’s ReviewCore Lesson
Quadratic Functions
Linear
FunctionsCubic
Functions
Other types
Polynomials(many terms)
Let’s ReviewCore Lesson
The graph is a straight line
1
3
Let’s ReviewCore Lesson
The equation can give the slope and y-intercept
y = 3x + 4
y = mx + b
Let’s ReviewCore Lesson
7 – 4
1 – 0
The table shows a constant rate of change
x y
0 4
1 7
4 16
6 22
7 25
Constant rate of change = 33
1 – 0 7 – 4 4 – 1 16 –
7 16 – 7 4 – 1
6 – 4 22 – 16
22 – 16 6 – 4 3
In this lesson you have learned how to identify a linear
function relation by analyzing characteristics of a linear
function.
Let’s ReviewGuided Practice
Which equation is linear?
y = x
y = 7 – x2
y = 2x + 5
y = 3x – 1
Let’s ReviewExtension Activities
A local school is selling raffle tickets for $2 each and 3 tickets for $5.
Is this a linear relation?
Make a table, graph, and/or equation to help visualize and discuss your answer.
Let’s ReviewQuick Quiz
Which choice gives the constant rate of change for y = -7x + 3 ?
A) 3 B) 7 C) -3 D) -7
Let’s ReviewQuick Quiz
Which graph shows a linear relation?
a) Blackb) Greenc) Pinkd) Red
How do you determine which graph is not a function?
In this lesson you will learn identify a function by analyzing
its graph.
Let’s ReviewLet’s Review
Is a function1
2 Is one-to-one
3 Has a constant rate of change
Given a relation you can determine if it:
Let’s ReviewA Common Mistake
Input Output
123
246
No, input goes first!
Correct!
Let’s ReviewCore Lesson
Input Output
1
2
3
2
4
6
• Function• One-to-
one• Constant
rate of change = 2
Let’s ReviewCore Lesson
Input Output
1
1
3
2
4
6• Not a function
(input 1 has outputs 2 and 4)
Let’s ReviewCore Lesson
Input Output
1 2 3 4 5 6
3 6 9 12
• Function• Not one-to-one
(inputs 4, 5, 6 share same output)
Let’s ReviewCore Lesson
Function Not a function Function
One-to-one Not one-to-one
In this lesson you have learned how to identify a function by
analyzing a graph
Let’s ReviewGuided Practice
Which graph is not a function?
Let’s ReviewExtension Activities
Plot each relation on separate coordinate grids.
Decide which is the function and non-function, and discuss the reasons with a friend.
Input Output
1 234 56
3 3 3 3 3 3
Input Output
1 11111
0 1 2 3 4 5
Let’s ReviewQuick Quiz
Which graphs are functions?
Let’s ReviewQuick Quiz
Input Output
1 2 3 4
2 4 2 4
Which mapping matches the graph?
Input Output
1 2 3 4
8 6 4 2
Input Output
1 2 3 4
2 4 6 8
Input Output
1 2 3 4
2 4 4 4
