algebra icontent.njctl.org/courses/math/algebra-i/algfunctions/functions... · slide 1 / 113...

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Algebra I

Functions

2015-11-02

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Table of Contents

Relations and Functions

Domain and Range

Evaluating Functions

Explicit and Recursive Functions

Multiple Representations of Functions

click on the topic to go to that section

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Relations and Functions

Return to Table ofContents

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A function is a relation where each value in the domain hasexactly ONE value in the range.

The x-value does NOT repeat in a function.

A relation is any set of ordered pairs.

Vocabulary

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Determine if each of the relations below is a function and provide an explanation to support your answer:

Example

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234

7-3 8

x y

-1 2

589

x y

-2 3-5

4

x y


Example

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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11


Examplean

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On a graph, a function does not have a point in the same vertical location as another point.

The Vertical Line Test can determine if a graph represents a function. Place a ruler or imaginary vertical line on the graph and move it from left to right.

If the vertical line intersects only one point at a time on the ENTIRE graph, then it represents a function. If the vertical line intersects more than one point at ANY time on the graph, then it is NOT a function.

Graphs

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Function Not a Function

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

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Equations

An equation is a function only if when a number is substituted in for x, there is only 1 output y-value.

Function

y = 3x + 4

y = 5

Not a Function

x = 5

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Examplean

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Example

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1 Is the following relation a function?

Yes

No

{(3,1), (2,-1), (1,1)}

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Yes

No

-103

-2-1 0

x yan

swer

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Yes

No

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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Yes

No answ

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Yes

No

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Yes

No

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Yes

No

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Domain and Range


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The domain of a function/relation is the set of all possible input values (x-values).

Vocabulary

{1, 4, 6}

{2, 3, 5}

{1, 3, 1}

Relation Domain

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State the domain for each example below and tell whether the relation is a function.

234

7-3 8

x y

12

589

x y

-2 3-5

4

x y

Example

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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11

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Example

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Example

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State the domain for each example below:

Example

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9 What is the domain of the following:

(Choose all that apply)

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

-103

-2-1 0

x y answ

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x < 0

I x > 0

J All Real Numbers

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The range of a function/relation is the set of all possible output values (y-values).

Vocabulary

{1, 4, 6}

{2, 3, 5}

{1, 3}

Relation Domain

{-1, 3, 7}

{4}

{-2, 2, 6}

Range

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234

7-3 8

x y

12

589

x y

-2 3-5

4

x y

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State the range for each example below:

Example

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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11

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State the range for each example below:

Example

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State the range for each example below:Example

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ExampleState the range for each example below:

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15 What is the range of the following:


A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

{(3,1), (2,-1), (1,1)}

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

-103

-2-1 0

x y

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y < 0

I y > 0

J All Real Numbers

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Evaluating Functions


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Equations for relations have been in the form of y = 3x + 2

When a relation is a function, it can also be written in function notation:

f(x) = 3x + 2

f(x) = 3x + 2 is still a line with a slope of 3 and a y-intercept of 2.

When a relation is a function, y = can be substituted with the notation of f(x) =

Function Notation is read: "f of x"

Function Notation

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So why the new notation?

1) It lets the reader know the relation is a function.

2) A second function can be added, such as g(x) = 4x and the reader will be able to distinguish between the different functions.

3) The notation makes evaluating a value of x easier to read.

Function Notation

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Evaluating a Function

To Evaluate in y = Form:

Find the value of y = 2x + 1when x = 3

y = 2x + 1y = 2(3) + 1

y = 7 When x is 3 y = 7

To Evaluate in Function Notation

Given f(x) = 2x +1 find f(3)

f(3) = 2(3) + 1f(3) = 7

"f of 3 is 7"

Similar methods are used to solve but function notation makes asking and answering questions more concise.

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22 Given and Find the value of .

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Explicit and Recursive Functions


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Explicit and Recursive Arithmetic Sequences Video(Khan Academy Video)

Explicit and Recursive Arithmetic Sequences Video(Youtube Video)

Vocabulary

Arithmetic Sequence - is a sequence of numbers with a constant common difference.

a1 - is the first term of the sequence.

d - is the common difference

Examples:(5, 7, 9, 11, 13, ...) a1 = 5 and d = 2(23, 17, 11, 5, -1, ...) a1 = 23 and d = -6

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Explicit Function Form - a(n) = a1 + d(n-1)

Recursive Function Form - is written in two parts 1. The first part is the first term a1 2. The second part is a(n) = an-1 + d (previous term plus the common difference)

Note: In the recursive formula the previous term is used to produce the next term.

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Write the explicit form for the following sequences:1) 9, 16, 23, 30, ...

2) 1, 3, 6, 8, 11,...

3) 68, 57, 46, 35, ...

Example

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https://www.khanacademy.org/math/trigonometry/seq_induction/seq_and_series/v/explicit-and-recursive-definitions-of-sequences

http://njc.tl/t0

Write the recursive form for the following sequences:1) 9, 16, 23, 30, ...

2) -2, 1, 4, 7, 10,...

3) 68, 57, 46, 35, ...

Example

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Write the first five terms of the sequence given the explicit formula.

a(n) = 41 - 4(n - 1)

Examplean

swer

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Write the first five terms of the sequence given the recursive formula.

a1 = -10a(n) = an-1 + 8

Example

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28 What is the explicit formula of the sequence?

33, 42, 51, 60, ...

A a(n) = 9 + 33(n-1)

B a(n) 33 - 9(n-1)

C a(n) = 33 + 9(n-1)

D a(n) = 9 - 33(n-1)

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29 What is the recursive formula for the sequence?

11, 3, -5, -13, ...

A a1 = 11

a(n) = an-1 + 8

B a1 = 11

a(n) = an-1 + 7

C a1 = 11

a(n) = an-1 + 9

D a1 = 11

a(n) = an-1 - 8

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30 What is the sequence that corresponds to the formula?

f(n) = 42 - 3(n-1)

A 42, 45, 48, 51, ...

B 42, 39, 36, 34, ...

C 42, 46, 50, 54, ...

D 42, 39, 36, 33, ...

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f1 = 77

f(n) = fn-1 + 14

A 77, 90, 103, 116, 129, ...

B 77, 63, 49, 35, ...

C 77, 91, 105, 119, 133, ...

D 77, 62, 47, 32, ...

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Explicit and Recursive Geometric Sequences(Youtube Video)

Explicit and Recursive Geometric Sequences(Khan Academy Video)

VocabularyGeometric Sequence - is a sequence of numbers with a constant common ratio. a1 - is the first term of the sequence.r - is the common ratioExamples:(4, 12, 36, 108, ...) a1 = 4 and r = 3(28, 14, 7, 3.5, ...) a1 = 28 and r = 1/2

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Explicit Formula - a1(r)(n-1)

Recursive Formula - is written in two parts 1. The first part is the first term a1 2. The second part is an = r an-1 for n > 1 (common ratio times the previous term)

Note: In the recursive formula the previous term is used to produce the next term.

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http://njc.tl/t1

https://www.khanacademy.org/math/trigonometry/seq_induction/seq_and_series/v/geometric-sequences

Write an explicit formula for the following sequences:1) 7, 35, 175, 875, ...

2) 20, 5, 5/4, 5/16, ...

3) 2.5, 5, 10, 20, ...

Example

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Example

Write an recursive formula for the following sequences:1) 7, 35, 175, 875, ...

2) 20, 5, 5/4, 5/16, ...

3) 2.5, 5, 10, 20, ... answ

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Write the first five terms of the sequence given the explicit formula.a(n) = 27(1/3)n-1

Example

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Write the first five terms of the sequence given the recursive formula.a1 = -2

a(n) = (5)an-1 for n>1

Example

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32 What is the explicit formula of the sequence?

24, 12, 6, 3, ...

A f(n) = 24(2)n-1

B f(n) = 24(.5)n-1

C f(n) = 24 - 2(n-1)

D f(n) = 2(24)n-1

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33 What is the recursive formula for the sequence?

3, 18, 108, 648, ...

A B

C D

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f1 = -16

f(n) = (-1/4)fn-1 for n > 1

A -16, -4, -1, 1/4

B -16, 4, -1, 1/4

C -16, -4, 1, -1/4

D 16, -4, 1, -1/4

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f(n) = 2(4)n-1

A 2, 8, 32, 128, ...

B 4, 8, 16, 32

C 2, 1/2, 1/8, 1/16

D 2, 8, 32, 118

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Multiple Representations of

Functions


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Any function can be written as a table, graph, verbal model, or equation. We can find the rate of change and the y-intercept from any of these representations.

Remember, to find slope we can use the formula:

Slope =

To find the y-intercept (initial value) we look to where x = 0

y2 - y1

x2 - x1

Multiple Representations

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x -2 -1 0 1 2f(x) -5 -2 1 4 7

Look at the given table. We can use any two values to determine slope. We can find where x = 0 to determine the y-intercept.

Slope = y2 - y1

x2 - x1 2 - 17 - 4= 3

1= = 3

y-intercept = 1


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Sometimes a table will not show the x-coordinate of zero. In that case you need to figure it out. There are a few ways to do it.

x 1 2 3 4 5 6 7f(x) -1 2 5 8 11 14 17

f(x) = 3x - 4

There are two ways to find the y-intercept here.


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We can simply continue the table so that we can find y when x = 0. We see that the y's are moving at intervals of +3. In order to work backwards to where x = 0 we need to subtract 3 from y. -3 - (-1) = -4 so when x = 0, y = -4.

Note: This technique works best when the table is close to x = 0

x 1 2 3 4 5 6 7f(x) -4 -1 2 5 8 11 14 17

f(x) = 3x - 4

0

+3 +3 +3-3


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f(x) = 3x - 4

x 1 2 3 4 5 6 7f(x) -1 2 5 8 11 14 17

Another technique would be to substitute for x and solve for y using the equation.

f(x) = 3x - 4 f(0) = 3(0) - 4

f(0) = -4


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36 What is the slope of the following table?

x f(x)

-2 -1-1 10 31 52 73 9

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37 What is the y-intercept of the following table?

x f(x)-2 -1-1 10 31 52 73 9

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38 What is the slope of this table?

x f(x)10 411 512 613 714 8

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39 What is the y-intercept of this table?

x f(x)10 411 512 613 714 8

f(x) = x - 6

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Carla puts away a certain amount of money per week. She started out with a certain amount. How could we figure out what she started out with and what she puts in per week?

week 0 1 2 3 4 5 6

amount in

account74 82 90 98 106 114 122

Example

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# of weeks 4 5 6 7 8 9 10

account balance 74 82 90 98 106 114 122

If we look at the slope or rate of change we can figure out how much she puts in each week. So what is the slope of this table?

Example

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How much did Carla start out with?

# of weeks 4 5 6 7 8 9 10

account balance 74 82 90 98 106 114 122

Example

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Try One

Rob is training for a marathon. He is increasing his run each week. Based on the table:

How much more will he run each week?How many miles was he running before training?How many weeks until he runs a full marathon 26.2 miles?

# of weeks training

2 3 4 5 6 7

Miles 7 10 13 16 19 22

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40 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. Use the table to identify the cost of each topping.

# of toppings 1 2 3 4 5

Cost 15.25 16.50 17.75 19.00 20.25

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41 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. How much is a pie without any toppings.

# of toppings 1 2 3 4 5

Cost 15.25 16.50 17.75 19.00 20.25

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42 Sandra is going to a buffet. The meal is a fixed price but she has to pay for each soda she drinks. What is the initial value? Be prepared to explain how it relates to the scenario.

Number of drinks1 2 3 4 5 6

Cost

35

30

25

20

15

10

5

0

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43 Sandra is going to a buffet. The meal is a fixed price but she has to pay for each soda she drinks. What is the slope? Use the points (0, 15) and (6, 25). Be prepared to explain how it relates to the scenario.

Number of drinks1 2 3 4 5 6

Cost

35

30

25

20

15

10

5

0

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44 What does the circled coordinate mean?

A This tree grows 4 feet every year

B This tree was planted when it was 4 feet

C There are 4 trees planted

D The tree was planted when it was 4 years old.

Years since planting

Height

1 2 3 4 5 6 7 8

32

28

24

20

16

12

8

4

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Mica is having a pool party. The cost to rent the pool is $325 and $7.00 per person attending the party.

Notice that regardless of how many people come, Mica will have to pay $325. This is the initial value, the y-intercept, the "b", also known as the constant.

Also notice that it costs $7.00 per person. This amount will change as the number of guests changes. This will be the slope, the rate of change, or the "m".

So the equation of this problem becomes: y = 7x + 325

y = mx + b

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Try one!!

Raul is at the gas station. He is filling up his gas tank at $3.45 per gallon and is also buying $12 worth of food from the convenience store. Write an equation to show this scenario.

f(x) = 3.45x + 12

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45 Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping. Which equation would best fit this scenario?

A f(x) = 45x + 9B f(x) = 9x + 45C 45 = 9xD 9 = 45x an

swer

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46 How much does it cost to buy 5 necklaces?

ABCD

Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.

$9$45$225$234

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47 How many necklaces can a person buy with $377 ?

ABCD

Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.

46810

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We have learned how to represent a function several ways: Table/Ordered Pairs Graph Equation Verbal Description (Scenario)

Next we will compare two different models to each other. We will look at the relationship between the two models in terms of the rate of change and domain.


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In order to compare the rate of change of two different types of representations of functions we simply find the rate of change of each and compare them.

The higher the absolute value of the rate of change, the bigger it is.

For example, if a graph has a slope of -4 and an equation has a slope of 3, the slope of the graph is steeper because the absolute value of -4 = 4 and the absolute value of 3 = 3. 4 > 3 so The graph has a bigger slope, or rate of change.

Two Different Representations

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Let's try one!

(1, 1)

(2, 3)

(3, 5)

f(x) = -5x +6

Slope = -5

Slope = 3-1 2 2-1 1

= = 2

absolute value of -5 = 5 and absolute value of 2 = 2 5>2 so A had a greater rate of change than B.

A B

Which has a greater rate of change?

1

2

3

Greater Rate of Change

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Let's try to compare a table and a verbal model.

Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes. A

Bx 2 4 6 8 10 12 14

y 7 13 19 25 31 37 41

Which has the greater rate of change?


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Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes.

A

We can turn this into a function. 10 is a constant fee.2 changes depending on the amount of people at the party. So the equation is f(x) = 2x + 10.

The rate of change = 2


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Bx 2 4 6 8 10 12 14

y 7 13 19 25 31 37 41

To find the rate of change we can use the slope formula.

13 - 7 6 4 - 2 2= = 3 The rate of change is 3.


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Let's try to compare a table and a verbal model.

Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes. A

Bx 2 4 6 8 10 12 14

y 7 13 19 25 31 37 41

Which has the greater rate of change?

Rate of change of A = 2Rate of change of B = 3Therefore, B has the greater rate of change.


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48 Which has the greater rate of change?A {(1, 4), (2, 6), (3, 8), (4, 10), (5, 12)}

B answ

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49 Which has the greater rate of change?

A f(x) = 1/3x + 5

B The school store is selling book covers $1 for 2.

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50 Which has the greater rate of change?

A f(x) = x - 4

B x -9 -6 -3 0 3 6 9f(x) -4 -3 -2 -1 0 1 2

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51 Which has the greatest rate of change?

A {(1, 3), (2, 4), (3, 5), (4, 6), (5, 7)}

BRyan and Andrew jump down the stairs 3 steps at a time.

C f(x) = 1/8x - 2

D

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52 Which has the greatest rate of change?

A A cable company charges $12 for every 2 premium channel.

B f(x) = 5x + 6

C {(9, 3), (6, 2), (3, 1), (0, 0), (-3, -1)}

D

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