unit 4 chapter 5 relations and functions · unit 4 chapter 5 – relations and functions 9 (iv)...

Unit 4 Chapter 5 – Relations and Functions 1

Unit 4 Chapter 5 – Relations and Functions

(I) Defining a Relation

What is the relation between altitude and air pressure?

____________________________________________________________

Goals:

Defining a Relation

Represent a Relation in different ways

A Set a collection of distinct objects (ie. all of the altitude readings

or all of the atmospheric pressure readings)

Example: Altitude readings {0, 2000, 4000, 6000, 8000}


(II) Representing a Relation in different ways

Relations can be represented several ways:

(i) In a Table (as above)

(ii) As a set of ordered pairs (or points)

Example: {(Altitude, Atmospheric Pressure)}

An Element of a set is one object in the set (ie. one altitude reading or one

atmospheric pressure reading)

Example: One Atmospheric Pressure Reading 101 kPa

A Relation associates the elements of one set to the elements of another set.

 (ie. each altitude measurement is paired or associated with an

atmospheric pressure reading)

Example:  When the altitude is 0 m the atmospheric pressure

 reading is 101 kPa. (0, 101)


(iii) As an Arrow Diagram has atmospheric pressure

(iv) As a Bar Graph

when elements of one or more sets are numbers


Example: The table represents the number of days lost per worker,

by cause, by province.

Express the relation:

(i) as a bar graph (ii) as an arrow diagram

Practice Questions:

P. 262 #4, #5, #6, #10


(I) Definition of a Function

You invest $1000 and each year 5% is compounded annually.

Goals:

Definition of a Function

Identifying a Function

Develop and Apply Function Notation

Determining a Range Value Given a Domain Value

Determining a Domain Value Given a Range Value

The set of first elements (the input) of a relation is the domain

  Example: The years

The set of second elements (the output) of a relation is the range

  Example: The Money


(II) Identifying a Function

A function

can algebraically model data for the purpose of prediction

(ie. extrapolate beyond data)

cannot produce two output values (range values) for the same

input value (domain value CANNOT REPEAT.)

A function is a relation where each element in the domain is associated

  with exactly one element in the range.

 Example: 

Example: For each relation below:

(i) Determine whether the relation is a function.

(ii) Identify the domain and range for each relation

that is a function.

(a) {(-1, 2), (0, 3), (1, 4), (2, 4)}


Example: For each relation below:

(i) Determine whether the relation is a function.

(ii) Identify the domain and range for each relation

that is a function.

(b)


(III) Develop and Apply Function Notation

Using Function Notation to Determine Values

Your hourly wage as an employee of Tim Hortons is $10.25.

Your gross pay depends on the hours you work.

What is the independent variable? What is the dependent variable?

A function can be represented by an equation

based on the algebraic operation(s) occurring on the input

(independent variable) that produces the output (dependent variable).

 

Using function notation P(h) =


(IV) Determining a range value given a domain value

Use the function generated above to evaluate:

(a) P(2)  (b) P(10)

Express each function using function notation.

(a) h = 3.343r + 81.224  (b) h = 3.271r

+ 89.925


(V) Determining a Domain value given a Range value.

A monthly home phone bill consists of a $30 base fee and $0.10/min for

long distance calls within Canada. This plan is modeled by the function:

 C(x) = 0.10x + 30  where x = time in minutes

  C(x) = monthly cost.

Determine x when:

(a) C(x) = 34 (b) C(x) = 80

Practice Questions:

P.270 - 273  #4, #5, #6a, c, #7a,c  #9, #13, #14, #15, #18, #19a, b


Graphs are useful for illustrating

a relation between two variables

like the domain value (input value

- depth) and the range value

(output value - water

temperature).

Goals:

Describing and interpreting possible situations in a graph

Sketch a possible graph given a situation


(I) Describing and interpreting possible situations in a graph

Example: Interpreting Graphs

(a) During what years did the most rapid increase in large salmon returns occur?

(b) During what years was the large salmon returns stabilized?

(c) During what years did the most rapid decrease in large salmon returns occur?


Example: Interpreting a Graph

The graph below illustrates the rate (m3/s) of water flowing through Star

Brook, a tributary of Star Lake. Interpret what is happening to the stream

flow over various intervals from A - G and provide a possible reason that

would influence the change.

Section Describe what is happening


(II) Sketch a possible graph given a situation

Example: Sketching a Graph for a Given Situation

Samuel went on a bike ride. He accelerated for 5 minutes until he

reached a speed of 20 km/h, then he cycled for 30 minutes at 20 km/h.

Samuel arrived at the bottom of a hill and his speed decreased to

5 km/h over 10 minutes as he cycled up the hill. He stopped for 10

minutes when he arrived at the top of the hill.

Sketch a graph of speed as a function of time. Label each section of

the graph state what it represents.

Section Describe what is happening

Practice Questions:

P.281 – P.283  #4 - #5, #7 - #10


Example:

The data below represents the change in absolute atmospheric pressure as

altitude increases.

Goals:

Graphing data and determining whether data is discrete

or continuous

(i) Graph the data. Should the points be joined? Justify.

(ii)  Does the graph represent a function? Justify.

Altitude (ft)


Example:

The data below represents the number of salmon passing through a fish

counting fence on Northwest River, Terra Nova Park over several days

during July.

(i) Graph the data. Should the points be joined? Justify.

(ii)  Does the graph represent a function? Justify.


(I) Determining properties of graphs of relations and functions

Goals:

Determining properties of graphs of relations and functions

Graphical test for a function

Determining domain and range

REMEMBER: A function is a relation where:

each element in the domain is associated with exactly one element in

the range.(ie. Domain value cannot repeat)

 


Based on the data recorded in each graph by Joe and Alice

answer each of the questions below.

(i) Does each graph represent a relation? A function?

(ii)  How can you tell?

(iii) Which of these graphs should have the data points connected?


(II) Graphical test for a function

(A) Relations that are not functions (B) Relations that are functions

Example: Which of the graphs represents a function ?


(III) Determining domain and range

Determining Domain From Graphs of Discrete and Continuous Data

The domain is extracted from the _____ axis

REMEMBER:

The DOMAIN of a function is the set of values of the independent

variable. The domain is the X-VALUES.

The RANGE of a function is the set of values of the dependent

variable. The range is the Y-VALUES.

REMEMBER: The sets of numbers below will be used to describe the

data as discrete or continuous.

(I) For Discrete Data

The Natural Numbers N = .

The Whole Numbers W = .

The Integer Numbers I = .

(II) For Continuous Data

The Real Numbers R = .

Stating the regions represented on the domain and range axis will sometimes

require inequalities:


Example:  For each line graph below, determine the domain using:

(i) set notation and  (ii) interval notation


Example: For each line graph below, determine the range using:

 (i) set notation and   (ii) interval notation

Practice Questions:

P.294  #5, #6, #8 (Determine which graphs are functions)

Complete worksheet for determining domain and range.


Worksheet Determining Domain & Range

State the domain using set notation and interval notation for:

1. 2.

Set notation:________________ Set notation:________________

Interval notation:____________

3. 4.


Interval notation:____________ Interval notation:____________

5. 6.



-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3


State the range using set notation and interval notation for:

7. 8.


Interval notation:____________

9. 10.



- 4

- 3

- 2

- 1

0

1

- 4

- 3

- 2

- 1

0

1

- 4

- 3

- 2

- 1

0

1

- 4

- 3

- 2

- 1

0

1


continued

(I) Determining domain and range from a graph

Goals:

Determining domain and range from a graph

Determining domain and range values from

the graph of a function

NOTE:

For Domain - all points FALL DOWN to the x-axis

For Range - all points MOVE OVER to the y-axis


Example:


(II) Determining domain and range from the graph of a function

Example:

Practice Problems:

P.294 - 297 #4b, c #7, #9, #12, #15, #16, #17, #22


(I) Identifying linear relations

(A) The table below represents (B) The table below represents

the cost for a cab ride over the return on a $1000 investment

time. over time.

Time (yrs) Value ($)

0 $1000

1 $1200

2 $1440

3 $1728

Time (sec) Cost ($)

0 $4.50

10 $4.60

20 $4.70

30 $4.80

Goals:

Identifying linear relations

Representing linear relations in different ways

Determining whether an equation represents a linear relation

Determining the rate of change of a linear relation from a graph

Graph each table of values on a separate grid.

(A) (B)


1. Which graph above represents a linear relation?

2. Without the aid of a graph, how could we identify the linear relation through

a table of values?

Time (yrs) Value ($)

0 $1000

1 $1200

2 $1440

3 $1728

Time (sec) Cost ($)

0 $4.50

10 $4.60

20 $4.70

30 $4.80

Example: Identifying a linear relation.


(II) Representing linear relations in different ways

Linear relations can be represented by:

(A) Tables (B) Graphs

(C) Equations

Dependent Variable = (Rate of Change)(Independent Variable) + Initial Value

C = $0.10 t + $4.50

Independent Variable Dependent Variable

Independent Variable

Dependent Variable

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 (𝑠𝑙𝑜𝑝𝑒) =𝑟𝑖𝑠𝑒

𝑟𝑢𝑛=

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

(III) Determining when an equation represents a linear relation

Example: Graph each equation.


(iii) y = 5 (iv) x = 1

Which of the equations in (i) – (iv) are linear?

(IV) Determining the rate of change of a linear relation from a graph

Practice Problems: P.308 – 310 #3 – #5, 6a(i), (iii), (v), b, #7, #12, #17


(I) Using intercepts, rate of change, domain and range to describe

the graph of a linear function

Intercepts of a Linear Function

Goals:

Using intercepts, rate of change, domain and range to describe the

graph of a linear function

Matching a graph to a given rate of change and vertical intercept

Sketching a graph of a linear function

Solving a problem involving a linear function


(C) State the Domain: ____________ (D) Determine the rate of change.

State the Range: ____________

(II) Matching graphs to a given rate of change and vertical intercept

Example:


(III) Sketching the graph of a linear function

The graph of a linear function can be sketched by 3 methods:

(A) By table of values

(B) By two intercept method – (Attain both x & y – intercepts)


(C) By slope y – intercept method

Linear functions in the form y = mx + b are in slope y – intercept form

y = mx + b

𝑚 = 𝑠𝑙𝑜𝑝𝑒 =𝑟𝑖𝑠𝑒

𝑟𝑢𝑛

b = y – intercept or initial

value

(IV) Solving a problem involving a linear function

Example

Practice Problems: P.319 - 323 #4, #6a, d, #8 #9, #10, #11, #13, #16, #17

unit 4 chapter 5 relations and functions · unit 4 chapter 5 – relations and functions 9 (iv)...

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