Ch. 5 - Relations and Functions Notes AnswerKey...Foundations of Mathematics & Pre-Calculus 10...

Foundations of Mathematics & Pre-Calculus 10 Chapter 5 – Relations and Functions

Created by Ms. Lee 1 of 18 Reference: Foundations and Pre-Calculus Mathematics 10, Pearson

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Ch. 5 – Relations and Functions Notes

5.1 – REPRESENTING RELATION & 5.4 GRAPHING DATA 2

Ch. 5.1 HW: p.262 #3, 4, 9, p.286 #1 3

5.2 – PROPERTIES OF FUNCTIONS 4

Ch. 5.2 HW: p. 270 #4 – 7, 9, 14 – 19 6

5.3 – INTERPRETING AND SKETCHING GRAPHS 7

Ch. 5.3 HW: p. 281 #3 – 15 8

5.4 – GRAPHING DATA 9

5.5 – PRE‐REQUISITES: INEQUALITIES LESSON 11

5.5 – GRAPHS OF RELATIONS AND FUNCTIONS 13

Ch. 5.5 HW: p. 294 #4 – 10, 14, 16, 17 14

5.6 – PROPERTIES OF LINEAR RELATIONS 15

Ch. 5. 6 HW: p. 308 #3 – 8, 10, 12, 17 18



5.1–RepresentingRelation&5.4GraphingData Definition: A set is a collection of distinct objects. An element of a set is one object in the set. A relation associates the elements of one set with the elements of another set. Example: Representing a Relation in variety of ways A set of fruits: {apple, blueberry, orange, banana}

- Apple, blueberry, orange, and banana are _____________ in this set. A set of colour: {red, green, blue, orange, yellow}

- Red, green blue, orange, and yellow are _____________ in this set. We can represent a relation that associates fruits with their colours in several ways:

1) In words

2) As a set of ordered pairs

3) As a table Fruit Colour

4) As an arrow diagram



Example: Some relations are better represented as a bar graph Given the relation between breeds of dogs and their mean heights in a table, represent the relation as a bar graph. Breed of Dog Mean Height

(cm) Afghan hound 75 Chihuahua 20 Corgi 30 Golden Retriever 60 German Shepherd

60

Lasa Apso 20

Example: Some relations are better represented as scattered plots To rent a car for less than one week from Ace Car Rentals, the cost is $65 per day for the first three days, then $60 a day for each additional day. a) Represent the relation between the number of days car is rented and total cost as table. b) Then use a scattered plot to represent the relation. c) Should we join the points? Why or why not? a) Number of Days Car is Rented

Total Cost

b)

c)

Ch. 5.1 HW: p.262 #3, 4, 9, p.286 #1



5.2–PropertiesofFunctions Definitions: In the workplace, a person’s gross pay, P dollars, often depends on the number of hours worked, h. We say P (gross pay) is the __________________ variable because a person’s gross pay ‘depends’ on the number of hours worked. The number of hours worked, h (hours worked) is called the _________________ variable. Domain – A set of first elements of a relation. (The values of the independent variable) Range – A set of related second elements of a relation. (The values of the dependent variable) Function – A special type of relation where each element in the domain is associated with exactly one element in the range. Examples: Understanding functions 1) may have the colour

2) has this number of wheels

Domain: Range: Is this relation a function?

Domain: Range: Is this relation a function?

3) The table shows the masses, m grams, of different numbers of identical marbles, n.

Number of Marbles, n

Mass of Marbles, m (grams)

1 1.27 2 2.54 3 3.81 4 5.08 5 6.25 6 7.62

a) What is the independent variable? b) What is the dependent variable?

c) Domain:

d) Range:

e) Is this relation a function? Explain



4) There is a relationship between x and y where the values of y depends on the value of x and is defined by the equation, y = x + 3.

a) Create a table of values for the following values of x: x y -3 -2 -1 0 1 2 3

b) Graph the data.

a) What is the domain?

b) What is the range? c) Is the relation a function?

Function Notation: Any function that can be written as an equation in two variables can be written in function notation. Examples:

Equations of Functions

Equations of the functions using function notation

Purpose of function notations: It is easy to express the ‘output’ given a certain ‘input’:

y = x + 3

y = -2x + 5

V = -0.08d + 50

Examples: 1. There is a relationship between radius and area of a circle.

a) What is the relationship?

b) Write this relationship using a function notation.

c) What is the area when the radius is 5 cm?



2. Given 52 tD . a) Write D as a function of t using a function notation.

b) What is the value of D when t = -3?

c) What is the value of D when ?0t

3. For the function 53)( xxf , determine:

a) )1(f =

b) )3(f = 4. For the function 50100)( nng , determine:

a) )1(g

b) )2(g =

5. The equation V = -0.08d + 50 represents the volume, V litres, of gas remaining in a vehicle’s tank after travelling d kilometers. The gas tank is not refilled until it is empty.

a) Describe the function. Represent the relation using a function notation. b) Determine the value of V(600) c) Determine the value of d when V(d) = 26. What does this number represent?

Ch. 5.2 HW: p. 270 #4 – 7, 9, 14 – 19



5.3–InterpretingandSketchingGraphs Example 1: Describing a possible scenario for a graph. This graph shows the depth of water in a bathtub as a function of time.

Match each line segment with the scenarios below:

Possible Scenario S1 You turn off the taps and relax for a while.

S2 You want the water to be hotter so you turn on the taps to get more hot water.

S3 You put the plug back in and relax a bit more.

S4 You put the plug in the bath and turn on the taps.

S5 You feel there’s too much water in the tub so you pull out the plug to let some water

drain out.

S6 There’s enough water in the tub so you turn off the taps for 2 minutes.

S7 You pull out the plug and drain all the water in the tub.

Example 2: Sketching a graph given a situation Sketch a graph to represent this situation: You put the plug in the bath and turn on the

taps. You leave the bathroom and return to

discover that the bath has overflowed. You turn off the taps and pull out the plug to

let out some water. You put the plug back in.



Example 3: Sketching a graph given a situation Sketch a graph to represent this situation: Samuel went on a bicycle ride. He

accelerated until he reached a speed of 20 km/h, then he cycled for 30 min at approximately 20 km/h.

Samuel arrived at the bottom of a hill, and his speed decreased to approximately 5 km/h for 10 min as he cycled up the hill.

He stopped at the top of the hill for 10 min.

Example 4: Interpreting a Graph The graph shows the height of the tide in a harbor as a function of time in one day. Explain the answer to each question below. a) What is the greatest height? At what times does it occur?

b) What is the least height? At what times does it occur? c) How high is the tide at 08:00? d) When is the tide 6 m high?

Ch. 5.3 HW: p. 281 #3 – 15



5.4–GraphingData In this lesson, you will learn to graph data:

using Excel Spreadsheet manually by hand

Excel Spreadsheet Exercise: 1. For each table of values below:

a) At a constant pressure, the speed of sound in air is related to the air temperature. Enter these values on the spreadsheet

Air Temperature C

Speed of Sound )/( sm

0 331 5 334 10 337 15 340 20 343

What type of graphs should you use to represent the data? Explain whether you should join the points. Does the graph represent a function? Explain

Cut and paste the resulting graph from the spreadsheet

b) The recommended daily dose of vitamin C is related to a female’s age in years.

Enter these values on the spreadsheet

Age (year)

Does of Vitamin C Tablet (mg)

3 15 6 25 9 45 12 45 15 65 18 65 21 75

Explain whether you should join the points. What type of graphs should you use to represent the data? Does the graph represent a function? Explain

Cut and paste the resulting graph from the spreadsheet



Graphing by Hand: 2. For each table of values, plot the data and decide whether to join the points.

a) If you buy more cans of juice, the unit price gets cheaper and cheaper. Number of Cans of Juice Purchased, n

Cost, C ($)

1 2.25 2 4.40 3 6.60 4 8.00 5 9.50 6 10.80


Graph the data by hand.

b) As you climb up a mountain, it gets colder and colder. Altitude, (m)

Temperature C

610 15.0 1220 11.1 1830 7.1 2440 3.1 3050 -0.8 3660 -4.8


Graph the data by hand.



5.5–Pre-requisites:InequalitiesLessonSign Read as Example Read as

Domain Write the Inequality to

describe the domain Another way to write the domain



Multiple Choice Questions: 1) Which of the following inequalities represents:

a) 2x b) 2x c) 2x d) 2x

2) Which of the following inequalities represents:

a) 1x b) 1x c) 1x d) 1x

3) Which of the following inequalities represents:

a) 3x and 4x b) 43 x c) 3x or 4x d) both a and b



5.5–GraphsofRelationsandFunctions Recap: Function: A relation where there’s only ONE output for every valid input. Domain: a set of valid input values (x-values) of a given function Range: a set of valid output values (y-values) of a given function # Relations Function?

(yes/no) Domain/Range

1

2

3

4

5

6

8



9

10

11 x y -3 3 2 5 8 10 -3 7

12 x y 1 2 3 5 4 8 5 8

13

14

15

Ch. 5.5 HW: p. 294 #4 – 10, 14, 16, 17



5.6–PropertiesofLinearRelations Definition: Linear relation: A relation in which a constant change in the independent variable results in a constant change in the dependent variable.

Rate of change = 12

12

var

var

xx

yy

iabletindependeninchange

iabledependentinchange

Example 1: Reviewing Definitions Distance (km) Cost ($) 0 60 100 80 200 100 300 120 400 140

The constant change, _______ in distance results in the constant change, ________ in cost; therefore, the above relation is a ____________ relation.

A linear relation, when graphed, produces a __________________.

Rate of Change =

Examples 2: Determining whether a Table of Values Represents a Linear Relation

C F 0 32 5 41 10 50 15 59 20 68

The relation between temperature in degrees Celsius, C, and temperature in degrees Fahrenheit, F is given. Is it a linear relation?

t n 0 1 20 2 40 4 60 8 80 16 100 32

The relation between the number of bacteria in a culture, n, and time, t minutes is given. Is it a linear relation?



Example 3: Determining whether an Equation Represents a Linear Relation Relations Make a table of values ketch to confirm y = -2x + 4 x y

y = 3x – 2 x y

y = 5 x y

x = 3 x y



y = x2 + 1 x y

y = x x y

Example 4: Identifying a Linear Relation a) A new car is purchased for $24 000. Every year, the value of the car decreased by 15%. The value

is related to time. Create a table of values, then check to see if the relation is linear. Time (years) Value of Car ($) 0 24 000 1 20 400

b) For a service call, an electrician charges a $75 flat rate, plus $50 for each hour he works. The total

cost for service is related to time. Create a table of values, then check to see if the relation is linear. Time (h) Cost ($) 0 75 1 125



Example 5: Determining the Rate of Change of a Linear Relation from Its Graph. A water tank on a farm near Swift Current, Saskatchewan, holds 6000 L. Graph A represents the tank being filled at a constant rate. Graph B represents the tank being emptied at a constant rate. Graph A

Graph B

a) Identify the independent and dependent

variables. b) Determine the rate of change of the relation,

then describe what it represents.

a) Identify the independent and dependent variables.

b) Determine the rate of change of the relation, then describe what it represents.

Ch. 5. 6 HW: p. 308 #3 – 8, 10, 12, 17

Ch. 5 - Relations and Functions Notes AnswerKey...Foundations of Mathematics & Pre-Calculus 10...

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