math 1201 unit 5: relations & functions

1

Math 1201

Unit 5: Relations & Functions

Read Building On, Big Ideas, and New Vocabulary, p. 254 text.

Ch. 5 Notes

§5.1 Representing Relations (0.5 class)

Read Lesson Focus p. 256 text.

Outcomes

1. Define and give an example of a set. p. 257

2. Define and give an example of an element of a set. p. 257

3. Define and give an example of a relation. p. 257

4. Represent a relation in various ways. pp. 257-261

nDef : A set is a collection of distinct objects. For example, the types of animals found in

Newfoundland could be a set.

nDef : An element of a set is one of the objects in the set. For example, in the set of the types of animals

found in Newfoundland, beaver would be an element. Some other elements of this set would be moose,

rabbits, and bears.

Note: The elements of a set are often listed inside a set of curly brackets called braces. For example, the

set of the types animals found in Newfoundland could be written as

beaver, moose, rabbit, bear, duck, geese, ...

nDef : A relation shows how the elements in 2 or more sets are connected to each other. For example, if

we have 2 sets moose, duck and walk, fly then one relation could be written as a set of ordered

pairs moose, walk , duck, fly which shows how some animals in Newfoundland can move. Note

that the previous relation was written as a set of ordered pairs. However, a relation can be written in

other ways.

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Ways to Represent a Relation

Ordered Pairs

Arrow (Mapping) Diagram

Table or Table of Values (TOV)

Graph (E.g.: Line Graph, Bar Graph, Histogram)

Equation

OR

Words For the sets 2010, 2011, 2012, 2013 and

22, 24,18, 20 , the relation

2010,22 , 2011,24 , 2012,18 , 2013,20 could

represent the number of students in Math 1201 at

Heritage Collegiate for the given years.

Do #’s 3 a, 4, 6 a, b, 10, pp. 262-263 text in your homework booklet.

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§5.2 Properties of Functions (2 classes)


Outcomes

1. Define and give an example of a function. p. 265

2. Determine if a relation is a function. pp. 265-266

3. Represent a function in a variety of ways. pp. 264-270

4. Determine the domain of a function. p. 265

5. Explain how the domain of a function is related to the independent variable. p. 267

6. Determine the range of a function. p. 265

7. Explain how the range of a function is related to the dependent variable. p. 267

Recall that a relation shows how the elements in two sets are related or connected.

nDef : The elements in the first set is called the domain. nDef : The elements in the second set is called the range.

E.g.: In the relation described by the arrow (mapping) diagram below, the domain is 4, 2, 0, 3 and

the range is 3, 3, 5,6,7 .

E.g.: In the relation described by the arrow (mapping) diagram below, the domain is 8, 9,10,13 and

the range is 3, 1, 5 .

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E.g.: In the relation described by the bar graph below, the domain is

comedy, action, romance, drama, sci-fi and the range is 1, 4, 5, 6 .

Note that the order of the elements does not matter

and that if an element is repeated, it is only written

once in the domain and/or range.

In this example, you may have wondered why the domain was the movie type and why the range was

the number of people choosing that movie type as their favorite instead of the reverse. The answer lies

what the independent and dependent variables are in this relation. The domain is the elements that

make up the independent variable and the range is the elements that make up the dependent variable.

In this relation, the independent variable is movie type so the domain is

comedy, action, romance, drama, sci-fi and the dependent variable is the number selecting that movie

type as their favorite so the range is 1, 4, 5, 6 .

We can now expand our definitions of domain and range.

nDef : The elements/values associated with the independent variable is called the domain. nDef : The elements/values associated with the dependent variable is called the range.

Function – A Special Type of Relation

nDef : A function is a special type of relation. It is a relation in which each element in the domain is

associated with exactly one element in the range.

E.g.: The relation described by the arrow (mapping) diagram below is a function because each element

in set A is associated with exactly one element in set B.

a is associated with x

b is associated with y

c is associated with y

d is associated with z

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E.g.: The relation described by the arrow (mapping) diagram below is NOT a function because the

element 3 is associated with 5 and with 7.

E.g.: Is the relation below a function? Why or why not?

E.g.: Explain why the left relation is a function while the right relation is not as function.

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Complete the following table. The first row is done for you.

Relation Function?

(Y or N)

Independent

Variable

Domain Dependent

Variable

Range

N x 0,1, 2, 3 y 2,1, 2, 4

x 2,1, 0,1, 2

y 0,1, 4

N/A N/A

Student

Name

9,10

Number of

Students

Do #’s 4, 5 c, d, 9, pp. 270-271 text in your homework booklet.

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A Function as an Input-Output Machine

A function can be thought of as a machine that turns an element in the domain (input) into exactly one

element in the range (output) (see below).

A specific example of changing in an input value into an output value can be seen below. The input

value (3) goes into the function machine which multiplies it by 3, subtracts 4, and outputs the value 5.

Note that the 3 that is input must come from the domain of the relation and the 5 that is output becomes

part of the range of the relation.

E.g.: Complete the table of values (TOV) below for the function machine directly above.

Input Value (# that goes into the function machine)

OR

(# that replaces the x in 3x – 4)

Output Value (# that comes out of the machine)

-5 -19

-2

1

4

E.g.: Complete the table of values (TOV) below for the function machine above.

Input Value (# that goes into the function machine)

OR

(# that replaces the x in 3x – 4)

Output Value (# that comes out of the machine)

-16

-4

2

26

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Function Notation

Functions typically take the form of a rule of some sort (see below).

A value from the domain (the input value) is changed according to a rule into an output value that

becomes part of the range. For example, if the input value was 10 and the function rule was “multiply

by 2” then the output would be 20.

E.g.: Complete the table below.

Input

Function Rule Output

10 “Cut in half”

-5 “Triple”

9 “Square”

49 “Take the square root”

We give the rule a name, often “f” because it is the first letter in the word “function”, but it could be any

name. The function rule in the function machine below is called “Fred”.

Now the function rule is often written as an algebraic expression of some sort, like 3x – 4, so when we

give this rule a name we might write something like

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This is called function notation and it is pronounced “f of x” equals …

Function notation comes in handy when we want to quickly paraphrase an English phrase

mathematically. Compare the statements in the table below.

English Phrase Mathematical Equivalent

If the function rule is “multiply by 3 then subtract

4” find the output value when the input value is 5. If 3 4f x x , find 5f .

If the function rule is “double the number” find the

output value when the input value is -8. If 2g x x , find 8g .

If the function rule is “square the number” find the

output value when the input value is 11. If 2h x x , find 11h .

If the function rule is “find the square root of the

number” find the output value when the input

value is 49.

If r x x , find 49r .

E.g.: Complete the table for the given functions. The first two are done for you.

Function Function

Name

Function Rule Variable in

the Rule

Input

Value

Output Value

3 4f x x f 3x – 4 x 6 6 OR 14f

20.5 6d t t d 20.5 6t t -4 4 OR 2d

24.9 25 100h t t t

0

r x x 1

4

We have used notation similar to function notation quite often in this course. For example, when we

wrote an area or volume formula, we often wrote it using function notation without indicating the

variable(s) used in the function rule. Some examples are given below.

2D or 3D Figure Common Form Function Notation Form

Circumference of a Circle 2C r 2C r r

Area of a Square 2A s 2A s s

Area of a Circle 2A r 2A r r

Area of a Triangle

2

bhA ,

2

bhA b h

Variable in the function rule

Name of function Function Rule

( ) 3 4f x x

24.9 25 100t t

10

Area of a Rectangle A lw ,A l w lw

Surface Area of a Cube 26SA s 26SA s s

Surface Area of a Cylinder 22 2SA r rh 2, 2 2SA r h r rh

Surface Area of a Sphere 24SA r 24SA r r

Volume of a Cube 3V s 3V s s

Volume of a Sphere 34

3V r 34

3V r r

Do #’s 6, 20, p. 271-273 text in your homework booklet.

Finding the Output Value given the Input Value

E.g.: If 3 4f x x , find 5f .

The input value is 5, so 5 is substituted for x.

5 3 5 4 15 4 11f

So,

If the input value is 5 and the function rule is 3 4x then the output value is 11.

E.g.: If 36SA s s , find 2SA .

The input value is 2, so 2 is substituted for s.

3

2 6 2 6 8 48SA

So,

2 48SA

If the input value is 2 and the function rule is 36s then the output value is 48.

Do #’s 14, 19 a, p. 272 text in your homework booklet.

Finding the Input Value given the Output Value

E.g.: If 3 4f x x , find the value of x for which 26f x .

Since 26f x , we substitute 26 for f x in 3 4f x x to get 26 3 4x , and solve for x.

5 11f

11

26 3 4

26 4 3 4 4

30 3

30 3

3 3

10

x

x

x

x

x

So if the output value is 26 and the function rule is 3 4f x x then the input value was 10.

E.g.: If 6 8g n n , find the value of n for which 1g n .

Since 1g n , we substitute -1 for g n in 6 8g n n to get 1 6 8n , and solve for n.

1 6 8

1 8 6 8 8

9 6

9 6

6 6

1.5

n

n

n

n

n

So if the output value is -1 and the function rule is 6 8g n n then the input value was 1.5.

Do #’s 15, 19 b, p. 272 text in your homework booklet.

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§5.3 Interpreting and Sketching Graphs (1 class)


Outcomes

1. Interpret a given graph. p. 278

2. Describe a situation for a given graph. p. 279

3. Sketch a graph for a given situation. p. 280

Interpreting a Given Graph

E.g.: Use the graph below to answer the following questions.

a) Over how many days was the water temperature taken? _____

b) On what two consecutive days was the water temperature the same? __________

c) What was the highest temperature recorded? _____

d) What was the lowest temperature recorded? _____

e) On how many days was the temperature lower than the day before? _____

f) On how many days was the temperature higher than the day before? _____

g) The greatest temperature increase occurred between what two consecutive days?_____

Before you do the next example, you may want to look at the graph on the bottom of page 277 of the

text and note which line segments indicate a steep or moderate increase, which segments indicate a steep

or moderate decrease, and which segment indicates no change.

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E.g.: Use the graph below to answer the following questions.

a) When did the journey start? ______

b) How far was Picnic Park from home? ______

c) How long did it take to get to Picnic Park? ______

d) How long was the stop at Picnic Park? ______

e) How far was the Campground from Picnic Park? ______

f) How long did it take to drive from home to the Campground?

g) How long was the stop at the Campground? ______

h) How long was the entire trip? ___________

i) Was the speed greater from home to Picnic Park or from Picnic Park to the Campground?

______________________________________________

j) When was the speed greatest? ____________________________

Do #’s 3, 4, 8, pp. 281-282 text in your homework booklet.

Describing a Given Graph

E.g.: Describe what is happening for each line segment in the graph below.

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i. From 10am to 11am, the number of people in the store increased by 3 from 2 to 5 people.

ii. From 11am to 12pm, the number of people in the store increased by 5.

iii. From 12pm to 1pm, the number of people in the store increased by 12.

iv. From 1pm to 2pm, the number of people in the store decreased by 7.

v. From 2pm to 3pm, the number of people in the store decreased by 10.

vi. From 3pm to 4pm, the number of people in the store decreased by 1.

vii. From 4pm to 5pm, the number of people in the store stayed the same.

viii. From 5pm to 6pm, the number of people in the store decreased by 1 to 3 people.

E.g.: Describe what the graph below indicates with respect to women’s smoking rates.

The rate of smoking for women doubled between 1960 and 1965 from about 90 smokers per 1000 to

about 180 smokers per 1000. It increased only slightly between 1965 and 1970. The smoking rate for

women increased sharply again between 1970 and 1975 to 320 smokers per 1000 and then remained

stable until 1980. It then decreased at a nearly constant rate to 280 smokers per 1000 over the next 10

years, declined more sharply to 220 smokers per 1000 between 1990 and 1995, and declined further to

200 smokers per 1000 by the year 2000.

E.g.: Describe what is happening in the distance-time graph below.

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Dan walks with a speed of 2km/h for the first hour and then stops to rest for 30 minutes. He then walks

with a speed of 2km/h for another hour and stops for an hour. Dan then walks all the way home in two

hours with a constant speed of 2km/h.


Sketching a Graph for a Given Situation

E.g.: Sketch a distance-time graph that represents the following situation.

A cyclist leaves home at 9am and travels at a constant speed of 20km/h for one hour. The cyclist then

travels at a slower constant speed of 6.6km/h for the next 45 minutes. She then stops at 10:45am to rest

for 1 hour. The cyclist then starts to travel home at a constant speed of 30km/h for 30 minutes, stops to

rest for 15 minutes, and then continues with a constant speed of 40km/h, arriving home at 12:45pm.


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§5.4 Graphing Data (1 class)


Outcomes

1. Graph a given set of data. pp. 284-286

2. Determine if the data is continuous or discrete. pp. 284-286

3. Determine if the data on the graph should be joined. pp. 284-286

4. Determine the domain of the relation from the graph of the data and write it using set

notation and interval notation. pp. 284-286

5. Determine the range of the relation from the graph of the data and write it using set notation

and interval notation. pp. 284-286

E.g.: For the data given in the table below:

Altitude (A) (1000 ft.) 1 5 10 20 25 30

Air Temperature (T) F 55 43 24 -13 -30 -47

i. Determine if the data is discrete or continuous.

ii. Decide if the points can be joined.

iii. Graph the data.

iv. Give the domain of the relation.

v. Give the range of the relation.

i. The data is continuous because the air temperature can be measured at any height.

ii. Since the data is continuous, the points can be joined.

iii.

iv. The domain is all the altitude values from 1000ft. to 30000ft. This can be written as

A|1000 A 30000 using set notation or as 1000, 30000 using interval notation.

v. The range is all the temperature values from 47 F to 55 F . This can be written as

T| -47 T 55 using set notation or as 47, 55 using interval notation.

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E.g.: For the data given in the table below:

Cost per Donut (C) (cents) 55 50 45 40 35 30

# Donuts Sold (n) 120 275 450 630 825 1020

i. Determine if the data is discrete or continuous.

ii. Decide if the points can be joined.

iii. Graph the data.

iv. Give the domain of the relation.

v. Give the range of the relation.

i. The data is discrete because part of a donut would not be sold.

ii. Since the data is discrete, the points cannot be joined.

iii.

iv. The domain is all the values for the cost of a donut 30,35,40,45,50,55 .

v. The range is all the values for the number of donuts sold 120,275,450,630,825,1020 .

Do # 2, p. 286 text in your homework booklet.

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§5.5 Graphs of Relations and Functions (2 classes)


Outcomes

1. Indentify the independent and dependent variables in a given context. p. 288

2. Explain how the domain of a relation is related to the independent variable. p. 289

3. Explain how the range of a relation is related to the dependent variable. p. 289

4. Given the graph of a relation, determine if a relation is a function. p. 289

5. Determine the domain of the relation from the graph of the data and write it using set

notation and interval notation. p. 289

6. Determine the range of the relation from the graph of the data and write it using set notation

and interval notation. p. 289

Independent and Dependent Variables

Recall that in an experiment, the independent variable is the variable that is varied or manipulated by

the researcher, and the dependent variable is the response that is measured.

For example, in a study to determine how the amount of time studying affects test marks, the amount of

study time would be the independent variable and the test marks would be the dependent variable.

E.g.: Complete the table below. The first one is done for you.

Situation Independent Variable Dependent Variable

The amount of Vitamin C one consumes can

influence life expectancy.

Amount of Vitamin C Life Expectancy

A farmer wants to determine the influence of

different quantities of fertilizer on plant growth.

The time spent studying will influence test scores.

The weight of a package will determine the

amount of postage paid.

Shots on net will influence the number of goals

scored.

Determining if a Relation is a Function

Recall that if a relation is written as a set of ordered pairs, as a TOV, or as a mapping diagram, you can

tell that the relation is a NOT function if the first coordinate appears more than once.

Note that the relation on the right is NOT a function because -2 occurs as the first coordinate more than

once in the TOV. The relation on the left is a function because -3, 0, 3, 8, and -10 only occur once.

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Determining from a Graph if a Relation is a Function

If you are given a graph instead of a set of ordered pairs, a TOV, or an arrow (mapping) diagram, you

can still easily determine if the relation is a function. Let’s look at the graph below and list the

coordinates of some of the points on the graph.

Average Temp F # Beach Visitors

84 225

86 300

86 350

92 450

94 500

94 600

If you look at the TOV closely, you should see that 86 (and 94) appears more than once as a first

coordinate. This means that the relationship between the average daily temperature and the number of

beach visitors is NOT a function. If you look at those points on the graph, you should see that the two

points with a first coordinate of 86 can be joined by a vertical line. This is also true for the two points

with a first coordinate of 94. This property leads us to a simple way of determining if a relation is a

function given the graph of the relation.

Vertical Line Test (VLT) for a Function

A graph represents a function if no points on the graph can be joined with a vertical line. So the graph to

the left is a function because it passes the vertical line test whereas the graph in the centre is NOT a

function because it does NOT pass the vertical line test. Likewise, the graph of a circle is NOT a

function because it does NOT pass the vertical line test.

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Complete the following table.

Graph Function ? (Y or N)

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Determining the Domain and Range of a Relation from a Graph

E.g.: For the graph given to the right:

i. Determine if the graph represents a function.

ii. Give the domain and range of the relation.

i. This graph represents a function because the graph

passes the VLT.

ii. Domain = 1,3,4,6 ; Range = 2,2,5


i. Determine if the graph represents a function.

ii. Give the domain and range of the relation.

i. This graph does NOT represent a function because

the graph does NOT pass the VLT.

ii.

Domain = 4 4 OR 4,4

Range = 2.7 3 OR 2.7,3

x x

y y

Do #’s 7, 6, 4, 8, 9, 12, pp. 294-295 text in your homework booklet.


i. Identify the independent variable.

ii. Identify the dependent variable.

iii. Determine the domain and range of the relation.

iv. Determine if the relation is a function.

i. Time of Day

ii. Average Water Level

iii.

Domain = Time 12 midnight Time 12 midnight OR 12am, 12am

Range = Height 4.2 Height 5.1 OR -4.2, 5.1

iv. The relation is a function because the graph passes the VLT.

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i. Identify the independent variable.

ii. Identify the dependent variable.

iii. Determine the domain and range of the relation.

iv. Determine if the relation is a function.

i. Weight

ii. Cost

iii. Domain = Weight 0 Weight 6 OR (0,6]; Range = 39, 41, 43, 45, 47, 48

iv. The relation is a function because the graph passes the VLT. (See below)


Determining the Domain Values and the Range Values from the Graph of a Function

E.g. Use the graph below to find:

i. the range (output) value when the domain (input) value is

-1.

ii. the range (output) value when the domain (input) value is

-2.

iii. the domain (input) value when the range (output) value is

-2.

i. Since 1,1 is a point on the graph, if the domain value

is -1 then the range value is 1.

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ii. Since 2,2 is a point on the graph, if the domain value is -2 then the range value is 2.

iii. Since 4, 2 is a point on the graph, if the range value is -2 then the domain value is -4.

E.g. Use the graph below to find:

i. the range (output) value when the domain (input) value is -3.

ii. the range (output) value when the domain (input) value is 0.

iii. the domain (input) value when the range (output) value is 4.

i. Since 3,0.25 is a point on the graph, if the domain value

is -3 then the range value is 0.25.

ii. Since 0,1 is a point on the graph, if the domain value is -0

then the range value is 1.

iii. Since 3,4 is a point on the graph, if the range value is 4

then the domain value is 3.

E.g. Use the graph to the right to find:

i. 2f

ii. 0f

iii. 1f

iv. 2f

v. A range of values for x for which

10f x .

vi. A range of values for x for which 0f x .

vii. A range of values for x for which 0f x .

i. Since 2,10 is a point on the graph, then 2 10f .

ii. Since 0,10 is a point on the graph, then 0 10f .

iii. Since 1,0 is a point on the graph, then 1 0f .

iv. Since 2, 10 is a point on the graph, then 2 10f .

v. 10f x for 2 3x

vi. 0f x for 2 1x

vii. 0f x for 1 3x

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E.g. Use the graph to the right to find:

i. 1f

ii. 1f

iii. 3f

iv. x such that 4f x .

v. x such that 1f x .

i. Since 1

1,2

is a point on the graph, then 1

12

f .

ii. Since 1,2 is a point on the graph, then 1 2f .

iii. Since 3,8 is a point on the graph, then 3 8f .

iv. Since 2,4 is a point on the graph, then 4f x when 2x .

v. Since 0,1 is a point on the graph, then 1f x when 0x .

Do #’s 16, 17, p. 296 text in your homework booklet.

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§5.6 Properties of Linear Relations (2 classes)


Outcomes

1. Determine if a TOV represents a linear relation. p. 301

2. Determine if a graph represents a linear relation. p. 302

3. Determine the rate of change of a linear relation from its graph. p. 306

4. Determine the domain and range of a linear relation and express them in a variety of ways.

pp. 300-307

Determining if a Table of Values (TOV) Represents a Linear Relation

Examine the two TOV’s below.

Both scatter plots above seem to indicate that each relation is linear, but one of them is not.

In the left TOV’s, the change in the independent (x) variable is constant (it is -6). Each time the value of

x decreases by 6. The change in the dependent (y) variable is also constant (it is -4). Each time the value

of y decreases by 4. This TOV represents a linear relation.

In the right TOV, the change in the independent (x) variable is constant (it is 1). Each time the value of x

increases by 1. However, the change in the dependent (y) variable is NOT constant. This TOV does

NOT represent a linear relation.

*** If a constant change in the independent variable produces a constant change in the dependent

variable, then the relation is linear. ***

Watch https://www.youtube.com/watch?v=1xfbBHE3UCA (0-7:25)

https://www.youtube.com/watch?v=1xfbBHE3UCA

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E.g.: Complete the table below. The first one is done for you.

TOV Linear Relation? (Y or N) Linear Function? (Y or N)

Yes. As the value of x

increases by 1 the value of

y increases by 0.5.

Yes, no x-coordinate

appears more than once.

x -2 0 2 4

y 5 5 5 5

Do #’s 3, 4, 11, pp. 308-309 text in your home work booklet.

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Determining if an Equation Represents a Linear Relation

You can also tell if a relation is linear by looking at the equation.

E.g.: Draw the graph for each equation below and determine if the equation represents a linear function.

i. 2 18 or 2 18f x x y x

ii. 2 27 18 or 7 18f x x x y x x

iii. 3 2 3 29 9 or 9 9f x x x x y x x x

iv. 3 or 3f x y

v. 4x

Equation Graph Linear

Function?

(Y or N)

2 18 or 2 18f x x y x

2 27 18 or 7 18f x x x y x x

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3 2 3 29 9 or 9 9f x x x x y x x x

3 or 3f x y

4x

The graphs of 2 18 or 2 18f x x y x , 3 or 3f x y , and 4x are linear relations

because their graphs are lines. The first equation represents an oblique line, the second a horizontal

line, and the third a vertical line. In general, equations of the form:

or , , are constantf x ax b y ax b a b (oblique line)

or , is constantf x k y k k (horizontal line)

, is constantx k k (vertical line)

represent linear relations

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E.g.: Complete the table below.

Equation Linear Relation?

(Y or N)

Linear Function?

(Y or N)

3 7 or 3 7f x x y x y y

2 22 6 5 or 2 6 5f x x x y x x n n

3 2 3 23 2 6 4 or 3 2 6 4 f x x x x y x x x n n

11 or 11f x y y y

6x y n

2 7 11y y y

2 8 6x y n

Watch https://www.youtube.com/watch?v=1xfbBHE3UCA (7:26-end)

Watch https://www.youtube.com/watch?v=oApNazdkqhE (0-3:00)

Do #’s 6 b, 13, 16, 18, pp. 308-310 text in your home work booklet.

Determining the Rate of Change of a Linear Relation from Its Graph

To determine the rate of change from the linear graph we need to find the coordinates of 2 points on the

line and determine the rise and the run between the points. If the graph rises to the right, the rate of

change is positive and if the graph falls to the right, the rate of change is negative.

**Rises to right = positive rate of change

**Falls to right = negative rate of change

E.g.: For the graph to the right:

Rise = 7 – 2 = 5

Run = 4 – 1 = 3

Rate of change 5

3

https://www.youtube.com/watch?v=1xfbBHE3UCA

https://www.youtube.com/watch?v=oApNazdkqhE

30

E.g.: For the graph to the right:

Rise = -2 – 4 = -6

Run = 8 – (-4) = 12

Rate of change 6 1

12 2

In general, the rate of change is the:

*****change in the -coordinate

change in the -coordinate

y

x*****

E.g.: For the graph to the right find

the:

a) dependent variable.

b) independent variable.

c) rate of change.

a) The dependent variable is

engine repair cost.

b) The independent variable is

the number of oil changes

per year.

c) The coordinates of 2 points on

the line would be 4,400 and

5,320

The rise is 400 – 320 = $80

The run is 4 – 5 = -1 oil changes.

The rate of change is $80

$80/ oil change1 oil changes

This tells us that, on average, every oil change decreases the cost of engine repair by $80.

31

E.g.: For the graph to the right find the:



c) rate of change.

a) The dependent variable is price.

b) The independent variable is the

distance.

c) The coordinates of 2 points on the line

would be 10,1.5 and 20,3.1

The rise is 3.1 – 1.5 = $1.6

The run is 20 – 10 = 10 miles.

The rate of change is $1.6

$0.16/mile10miles

This tells us that, on average, each mile driven increases the cost by $0.16 or 16 cents.

E.g.: For the graph to the right find the:



c) rate of change.

a) The dependent variable is jump height.

b) The independent variable is the bike

weight.

c) The coordinates of 2 points on the line

would be 19.5,10.3 and 22.5,9.8

The rise is 10.3 – 9.8 = 0.5ft

The run is 19.5 – 22.5 = -3.

The rate of change is $0.5ft

0.17ft/lbs3lbs

This tells us that, on average, each pound of weight of the bike decreases the jump height by

0.17ft.

Watch https://www.youtube.com/watch?v=oApNazdkqhE (3:00-end)

Do #’s 7, 12, 14, 15, 17 pp. 308-310 text in your homework booklet.

https://www.youtube.com/watch?v=oApNazdkqhE

32

§5.7 Interpreting Graphs of Linear Relations (2 classes)


Outcomes

1. Define the horizontal intercept of a graph. p. 315

2. Define the vertical intercept of a graph. p. 315

3. Given the graph of a linear function, determine its intercepts, domain, and range pp. 314-315

4. Given the equation of a linear function, sketch its graph. P. 315

5. Given the rate of change of a linear function and an intercept of its graph, identify the graph

of the linear function. P. 316

6. Solve problems involving linear functions. pp. 317-318

nDef : The first coordinate of the point where the line intersects the

horizontal axis (x-axis) is called the horizontal intercept (x-

intercept).

nDef : The second coordinate of the point where the line intersects

the vertical axis (y-axis) is called the vertical intercept (y-intercept).

E.g.: For the graph to the right, the horizontal (x) intercept is 5.1.

The vertical (y) intercept is -4.5.

E.g.: What is the horizontal intercept of the graph to the right?

The horizontal intercept is ______

What is the vertical intercept of the graph to the right?

The vertical intercept is ______

33

E.g.: Use the graph to the right to answer the following

questions.

a) What is the independent variable?

b) What is the dependent variable?

c) What is the domain?

d) What is the range?

e) What is the cost of the cell phone after 9 months?

f) What is the vertical intercept of the graph to the

right? What does it represent?

g) What is the rate of change of this linear relation?

a) The independent variable is __________________

b) The dependent variable is __________________

c) Domain | 0 11x x

d) Range | 70 500y y

e) The cost of the cell phone after 9 months is __________

f) The vertical intercept is ______. It represents the cost to purchase the cell phone.

g) 2 points on the line are 6,300 and 11,500 . The rate of change is

$ 500 300 $200$40/ mnth

11 6 mnths 5mnths

E.g.: Use the graph to the right to answer the following questions.

a) What is the independent variable?

b) What is the dependent variable?



e) What is the length of the candle after 30 minutes?

f) What is the vertical intercept of the graph to the right?

What does it represent?

g) What is the horizontal intercept of the graph to the

right? What does it represent?

h) What is the rate of change of this linear relation?

a) The independent variable is __________________

b) The dependent variable is __________________

c) Domain

d) Range

e) The length of the candle after 30 minutes is __________

f) The vertical intercept is ______. It represents the length of the candle before it is lit.

g) The horizontal intercept is ______. It represents the time when _________________________.

34

h) 2 points on the line are 50,5 and 60,4 . The rate of change is

5 4 in 1in0.1in / min

50 60 min -10min

. This is the amount by which the length of the candle

decreases each minute.

Do #’s 5, 4, p. 319 text in your homework booklet.

Sketching the Graph of a Linear Function Written in Function Notation

E.g.: Sketch the graph of 4 5f x x

Method 1: Intercept-Intercept Method

Find the x-intercept let 0f x Find the y-intercept (let x = 0)

0 4 5

0 5 4 5 5

5 4

5 4

4 4

1.25

x

x

x

x

x

The x-intercept has coordinates 1.25,0

4 0 5

0 5

5

f x

f x

f x

The y-intercept has coordinates 0,5 . Where does

the “5” show up in the equation 4 5f x x ?

We now plot both intercepts and join them to complete the graph. The fact that the line extends

completely across the grid and that there are no solid dots at the endpoints indicate the line continues

indefinitely. You can also place arrows at each end to show that the line continues indefinitely.

35

What is the rate of change for this graph? Where does this number appear in the equation?

Two points on the graph are 1,1 and 0,5 , so the rate of change is 5 1 4

40 1 -1

. This number is the

number in front of the variable (x) in the equation.

Method 2: y-intercept & Rate of Change

The “5” in 4 5f x x indicates that the y-intercept is 5. The " 4" in 4 5f x x indicates that

the rate of change is 4

4 or 1

. This indicates that when the change in x is 1, the change in y is -4.

So we can plot the y-intercept (5) and go 1 unit right and 4 units down (-4) and plot a second point. We

can then join these points to get the graph.

y-intercept Rate of Change

( ) 4 5f x x

36

Summary

Do #’s 6, 15, pp. 319-322 text in your homework booklet.

Matching a Graph to a Given Rate of Change and Vertical Intercept

E.g.: Which graph below has a vertical intercept of -5 and a rate of change of 5

2.52

?

The left graph has a vertical intercept of 3 and a rate of change of 4

0.85

, so it is NOT the correct

graph. The right graph has a vertical intercept of -5 and a rate of change of 5

2.52

, so it is the correct

graph.

Do # 8, p. 320 text in your homework booklet.

Problem Solving Involving Linear Functions

37

E.g.: The length of a baby for its first 16 months is modeled in the graph below.

a) What is the vertical intercept? What does it represent?

b) Determine the rate of change for the length of the baby? What does it represent?



e) How old is the baby when it is 22 inches long?

f) What is the length of the baby when it is 14 months old?

a) The vertical intercept is 20 inches. It represents the length of the bay when it was born.

b) Two points on the graph are 4,21 and 8,22 . The rate of change is

22 21 in 1in0.25in/mnth

8 4 mnths 4mnths

. This rate of change represents the amount by which the

baby grows each month.

c) Domain = | 0 16x x

d) Range = | 21 24x x

e) The baby is 8 months old when it is 22in long.

f) The baby is 23.5in when it is 14 months old.

Do #’s 9, 10, 11, 13, 14, 16, pp. 320-322 text in your homework booklet.

Do #’s 2, 3 b, d, 4, 5, 6 a, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, pp. 326-328 text in your homework

booklet.

math 1201 unit 5: relations & functions

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