04 new wa maths sb8 txt

Number and algebra

4Functionsand graphs

9780170194495118

Contents4.1 Plotting points and lines4.2 Linear graphs4.3 Connecting algebra and geometryChapter summaryChapter review

Prior learning

Chapter 4

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Parent guide

Chapter 4

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Curriculum guide

Chapter 4

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nAustralian Curriculum statementsLinear and non-linear relationshipsPlot linear relationships on the Cartesian plane withand without the use of digital technologies.

9780170194495 119

NELSON WA MATHSfor the A u s t r a l i a n C u r r i c u l um8

The graph of a function can tell you a great deal. The way the graph changes tells you about howthe function changes. The study of functions and their graphs helps scientists to accuratelydescribe the world. This allows engineers to accurately predict how bridges, cars, trains, and otherstructures and machines will behave when they are built. In this chapter you will begin to learnhow algebra and geometry are related through graphs of functions.

nMathematical literacyThe mathematical words below have special meanings that you will learn in this chapter. It isimportant that you learn to spell them and gradually learn what they mean in mathematics. Youmay find the glossary or online mathematical dictionary useful for this purpose.

Cartesian planecoordinatesdependentfunctiongradientindependent

inputinterceptlinear functionlinear graphmidpointorigin

outputparabolaquadrantriserunsatisfies

step functiontable of valuestravel graphvariablex-axisy-axis

4.1 Plotting points and linesYou have already learnt something about the Cartesian plane.

Important!

The Cartesian planeThe Cartesian plane has twoaxes at right angles. It is alsocalled a number lattice.The horizontal axis is called thex-axis; the vertical axis is calledthe y-axis. The axes cross at theorigin.The position of a point on theCartesian plane is determined byits coordinates.The x-coordinate gives thehorizontal position of the pointfrom the origin, and they-coordinate is the verticalposition of the point from theorigin. Coordinates are shown in parentheses (round brackets) with a comma between them.The x-coordinate is always shown first and the y-coordinate last. The origin is the point (0, 0).Each quarter of the Cartesian plane is called a quadrant.In the diagram, Point A is written as (3, 2), B is (�2, 2), C is (�3, �1) and D is (4, �3).

−1−2−3−4 1 2 3 4

−1

−2

−3

1

2

3

0

y-axis

Originx-axis

D

AB

C

First quadrantSecond quadrant

Fourth quadrantThird quadrant

Maths dictionary

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Puzzle sheet

Coordinates codepuzzle

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Functions and graphs

Example 1

Plot each of the following points on Cartesian axes and state its quadrant or axis.A(�1, �3), B(0, �8), C(1, �2), D(6, 0), E(1, 3), F(0, 5), G(�2, 2), H(�7, 0)

SolutionDraw and label axes to cater for thebiggest number.

1 2 3 4 5 6 7

1234567

0−1−2−3−4−5−6−7

−2−3−4−5−6−7

y

x

To plot A(�1, �3) go down from �1 onthe x-axis.

Then go left from �3 on the y-axis.

Place a point at their intersection andlabel it.

1

−3

−1

(−1, −3)

Do the same for all the other points.

1 2 3 4 5 6 7

1234567

0−1−2−3−4−5−6−7

−2−3−4−5−6−7

(−7, 0)

(−2, 2)

(0, 5)

(1, 3)

(6, 0)

(1, −2)

(0, −8)

(−1, −3)

x

y

State the quadrant or axis of each point. • A(�1, �3) is in the third quadrant• B(0, �8) is on the y-axis• C(1, �2) is in the fourth quadrant• D(6, 0) is on the x-axis• E(1, 3) is in the first quadrant• F(0, 5) is on the x-axis• G(�2, 2) is in the second quadrant• H(�7, 0) is on the y-axis

Worksheet

The number plane

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Worksheet

Plotting points andlines

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Worksheet

Number planereview

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Technology Plotting pointsYou can use digital technology to plotpoints. The spreadsheet ‘Points plotter’on the NelsonNet website for this bookautomatically plots points. Othertechnologies may also be available atyour school or demonstrated by yourteacher.Sometimes you will be required to plotpoints shown in a table. These couldform a pattern such as a straight orcurved line, or could just be a group ofpoints.

Example 2

Plot the points in the table below. What shape is formed?

x �4 �3 �2 �1 0 1 2 3 4y �2 �1 0 1 2 3 4 5 6

SolutionUse graph paper.

Plot the points, starting with (�4, �2).

1 2 3 4 x

y

123456

0−1−2

−2−3−4

The points appear to form a straight line. Checkthis with a ruler and use it to draw in the line.

Continue the line past the first and last points.

1 2 3 4 x

y

123456

0−1−2

−2−3−4

Write the answer. The shape is a straight line passingthrough the axes at (�2, 0) and (0, 2).

Technology worksheet

Excel: Plotting points

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Example 3

Plot the line given by the points in the table below. What shape is formed?

x 16 9 4 1 0 1 4 9 16y �4 �3 �2 �1 0 1 2 3 4

SolutionSince the x-values go from0 to 16 and the y-values gofrom �4 to 4, use differentscales on the graph. Try tomake the graph roughlysquare.

Plot the points, startingwith (16, �4).

2 4 6

1234

0

−2

−1

−3−4

8 10 12 14 16 18 x

y

Since the points seem tomake a curve, join them inorder with a smoothcurve. Continue the curvepast the first and lastpoints. 2 4 6

1234

0

−2

−1

−3−4

8 10 12 14 16 18 x

y

Write the answer. The shape of this line is a symmetrical curve.

The curve in Example 3 is called a parabola. It is a roughly U-shaped curve symmetrical about acentral line. The curve passes through the line of symmetry at a central point. Although it isroughly U-shaped, the distance between the curve and the line of symmetry increases as you getfurther away from the central point.

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Example 4

The following data shows the water consumption of a family over ten days. Plot the graphand find out on what days the family had guests staying.

Date 5/4 6/4 7/4 8/4 9/4 10/4 11/5 12/4 13/4 14/4 15/4Wateruse (L) 560 512 527 511 756 800 752 453 445 519 475

SolutionPlot the data.

65 7 8 9Date (April)

Daily water consumption

Con

sum

ptio

n (L

)

10 11 12 13 14

1000

200300400500600700800

Look for a pattern. On April 5�8 and 12�15, the water consumptionwas around 500 L per day, but on April 9�11 it wasclose to 800 L per day.

Write a conclusion with reasons. The increased water consumption from 9 April to11 April suggests that the family had guests in thatperiod.

Exercise 4.1 Plotting points and lines

1 Plot each of the following points on Cartesian axes.A(2, 3), B(�4, 5), C(4, 7), D(�8, 6), E(8, 5), F(�6, �5), G(�4, �1),H(�5, �5), I(�5, �6), J(7, 3), K(8, 7), L(6, �3), M(2, �5), N(6, 2), P(5, �1),Q(2, �6), R(�5, �2), S(�6, 4), T(2, 6), U(�6, 3), V(�3, 0), W(�4, 4),X(7, �2), Y(8, �1), Z(1, �5)

2 Plot each of the following points on Cartesian axes and state its quadrant or axis.A(�5, 1), B(�6, �3), C(6, 6), D(2, �3), E(�6, �7), F(3, �5), G(2, 2), H(7, 6), I(1, 5),J(�1, �5), K(�4, 2), L(0, 4), M(�3, �5), N(7, 3), P(�3, �8), Q(�4, 0), R(7, �7), S(�3, 1),T(�2, 5), U(6, �3)

3 Plot the points in the table below. What shape is formed?

x �4 �2 0 2 4 6y �7 �4 �1 2 5 8

Understanding

Extra questions

Exercise 4.1

MAT08NAEQ00011See Example 1

Fluency

See Example 2

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12



x �4 �3 �2 �1 0 1 2 3 4y 12 10 8 6 4 2 0 �2 �4


x �4 �3 �2 �1 0 1 2 3 4 5 6y �7 �6 �5 �4 �3 �2 �1 0 1 2 3


x �4 �3 �2 �1 0 1 2 3 4 5 6y 9 8 7 6 5 4 3 2 1 0 �1


x �12 �10 �8 �6 �4 �2 0 2 4y �2 �1 0 1 2 3 4 5 6


x �10 �8 �6 �4 �2 0 2y 9 6 3 0 �3 �6 �9

9 Without plotting, state the quadrant or axis of each of the following points.A(7, �3), B(5, 6), C(4, 8), D(�3, �1), E(1, 2), F(6, 3), G(�5, 2), H(8, �6), I(�5, 7), J(7, �8),K(6, �7), L(2, �7), M(3, �7), N(3, 0), P(�5, �7), Q(1, 5), R(�2, 7), S(5, �7), T(�2, �7),U(0, �5), V(�5, 3), W(�1, 3), X(�1, 6), Y(5, �5), Z(�1, 2)


x �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7 8y 46 33 22 13 6 1 �2 �3 �2 1 6 13 22 33 46


x �5 �4 �3 0 3 4 5 4 3 0 �3 �4 �5y 0 3 4 5 4 3 0 �3 �4 �5 �4 �3 0


x �2 �1 0 1 2 3 4 3 2 1 0 �1 �2y 1 0 �1 �2 �1 0 1 2 3 4 3 2 1

13 The table below shows the rain recorded on the Gold Coast in the first 10 days of February.Plot the points and comment on the results.

Date in February, d 1 2 3 4 5 6 7 8 9 10Recorded rain, r (mm) 0.6 4 0.2 0 0 0 2.6 0.8 7.6 0.6

14 The table below shows the average height of 10 seedlings for a fortnight from when they firstappeared. Plot the points and comment on the results.

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14Height (mm) 1 5 15 50 90 120 140 155 165 175 180 185 188 190

Problem solving

See Example 3

Worked solutions

Exercise 4.1

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See Example 4

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15 The table below shows the costs of tyres in dollars and the average distances the tyres lasted.Plot the points and comment on the results.

Tyre cost in dollars 200 225 193 147 122 190 228 308 333 362Distance, ‘000 km 54 27 37 46 29 48 22 49 59 36

16 The table below shows the average weight in kg of fruit produced by tomato plants whendifferent amounts of fertiliser (in mL) are added after flowering. Plot the results and decideon how much fertiliser should be used if the cost of the fertiliser is not important. Is this theway that tomato growers would make the decision?

Fertiliser (mL) 0 50 100 150 200 250 300 350 400 450 500Tomatoes (kg) 4 9 11 12.5 13.5 14.5 14.3 14 13.4 12.8 12

17 A flood gauge by the side of a creek shows its water level. The creek is considered to be inflood if the level is over 2.5 m. The readings at hourly intervals on a particularly wet day wereas shown below. Plot the points and state the flood time. What kind of flood was it?

Time 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900Flood level (m) 0.8 0.8 0.9 2 6 7 4 1.5 1 0.9 0.85 0.8

18 After the air conditioner is switched on in a hot room (41�C), the room temperature starts todrop as shown by the information below. Plot the information to estimate how long it will bebefore the temperature drops below 30�C. When will the temperature fall below 0�C?

Time (minutes) 0 1 2 3 4 5 6 7Temperature (�C) 41 40.2 39.4 38.6 37.8 37 36.2 35.4

4.2 Linear graphsYou have already done some work on graphs with straight lines in previous years.A travel graph shows the distance travelled by something over time. When the speed is constant,you get a straight line, as shown in the following example.

Example 5

A snail moves at 8 cm/minute. It moves forward for 3 minutes, then stays still for 2 minuteswhile it eats a lettuce leaf. Make a table of the distance it has moved from its starting pointand draw a travel graph of its movement.

SolutionShow the time t for t ¼ 0 to 5minutes and the distance as d cm.

Time, t (minutes) 0 1 2 3 4 5Distance, d (cm)

It moves 8 cm every minute for thefirst 3 minutes.

Time, t (minutes) 0 1 2 3 4 5Distance, d (cm) 0 8 16 24

It doesn’t change its position in thenext two minutes.

Time, t (minutes) 0 1 2 3 4 5Distance, d (cm) 0 8 16 24 24 24

Worked solutions

Example 4.1

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Reasoning

Worked solutions

Example 4.1

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Animated example

Travel graphs

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Draw the graph with time on thebottom and distance up the side.

10 2 3 4Time (minutes)

Snail movement

Dis

tanc

e (c

m)

5 t

d

50

101520

Notice that when the snail is moving forward, the graph slopes upwards. When the snail isstationary, the graph is flat. What would the graph look like if it moved back for the next minute?A travel graph can also tell you about what happened during a trip.

Example 6

This graph shows John and Henry going to the shop fromJohn’s house and then going to Fred’s house.

20 4 6 8Time (minutes)

Shop trip

Dis

tanc

e (m

)

10 12

1000

200300400500600a How long did it take them to get to the shop?

b What does the flat part show?c How long did they spend at the shop?d How long did it take them to get back from the shop to

Fred’s house?e How far was it from John’s house to the shop?f Find their speeds going to and from the shop. When did

they travel faster?g Who lived further from the shop?

Solutiona They left at 0 and got there at 3 minutes. They took 3 minutes to get to the shop.

b They are not moving when it’s flat. It shows them not moving; they are at theshop.

c They got there at 3 and left at 5 minutes. They spent 2 minutes at the shop.

d They left at 5 and got there at 1112

minutes. It took them 612

minutes to get to Fred’s.

e They were at John’s house at 200 m and atthe shop at 650 m. John’s house is 450 m from the shop.

f They took 3 minutes to travel 450 m to theshop. Work out the speed to the shop. Speed to shop ¼ Distance

Time

¼ 450 m3 minutes

¼ 150 m/minute

Work out the speed from the shop. Speed to shop ¼ DistanceTime

¼ 650 m6 1

2 minutes¼ 100 m/minute

Worksheet

The hare and thetortoise

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Which was faster? They were faster on the way to the shop.

g Compare the distances. John lives 450 m and Fred lives 650 mfrom the shop, so Fred is further from theshop.

In the example above, you should be able to see that the graph is steeper when they weretravelling faster. This was on the way to the shop.

Important!

Linear graphsA number rule is called a function in algebra. One number is changed into another one bythe rule. The number put into the rule (usually on the right-hand side) is called the input orindependent variable. The number that is produced by the rule is called the output ordependent variable.A function can be plotted by putting different values into the function and calculating theoutputs. This is usually put into a table called a table of values.A rule that produces a straight line is called a linear function and the graph is called a lineargraph. Linear functions are of the form y ¼ ?x � ??, where ? and ?? are constants.

Example 7

a Make a table of values for the function g ¼ 3n � 2 from n ¼ �2 to 4.b Plot the function.c Describe the function.

Solutiona Draw up the table with spaces for the n

values from �2 to 4.n �2 �1 0 1 2 3 4g

Work out the value of g by substitutingn ¼ �2. g ¼ 3n � 2

¼ 33�2� 2

Simplify. ¼ �6 � 2

Calculate the answer. ¼ �8

You can use your calculator.

Enter as: 3 2 2 . -83×-2–2

Put the answer into the table and workout the rest.

n �2 �1 0 1 2 3 4g �8 �5 �2 1 4 7 10

Video tutorial

Graphing linearequations

MAT08NAVT10023

Puzzle sheet

Matching linearequations

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Puzzle sheet

Graphing functions

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Worksheet

Tables of values

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Worksheet

Graphing linearequations

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b Plot the points onto a Cartesian grid andjoin them up by drawing a straight linewith a ruler.

1 2 3 4 5 n

g

2468

10

0−1−4−6−8

−2−3−4−5

c Describe the graph of the function. It is a linear function passing throughthe axes at (0, �2) and ð23 , 0Þ.

Technology Linear functionsYou can use digital technology to plot linear functions.The spreadsheet ‘Linear function plotter’ on the NelsonNet website automatically plots points.Other technologies may also be available on your school intranet (LAN) or demonstrated by yourteacher.

You can use linear functions and graphs to model many real-life situations.


Excel: Graphingstraight lines: Finding

intercepts

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Example 8

A 5-bedroom holiday house that sleeps up to 12 people is available for rent for $320 pernight for two people plus $20 for each additional person. Construct an algebraic model ofthe rent per night and plot the function. Then find the rent for 3 days for 5 people.

SolutionWrite the rent for 2people. Rent for 2 people is $320.

If 5 people stay, thereare 3 extra. For n people, there are (n � 2) extra people.

Write the additionalrent for n people. Extra rent for n people ¼ 20 3 (n � 2)

Write the total rent rfor n people. r ¼ 320 þ 20(n � 2)

Complete a table ofvalues for n from2 to 12.

n 2 3 4 5 6 7 8 9 10 11 12r 320 340 360 380 400 420 440 460 480 500 520

Plot the graph.

Since n is only from 2 to12, do not draw the linepast these limits.

65 7 821 3 4 9People (n)

Holiday house rental

Ren

t (r)

10 11 12

3000

320340360380400420440460480500520

Use the graph to findthe rent for 5 people forthree days.

The rent for 5 people is $380/night. For threedays, the rent ¼ 3 3 $380 ¼ $1140

Write the answer. The rent for 3 days for 5 people is $1140.

A step function has only flat parts on its graph, but the values usually get bigger at particularvalues of x. Like number lines, values at the end of the ‘steps’ that are not included are shown byan open, uncoloured circle. Values at the end that are included are shown by a coloured-in circle.

TLF learning object

Exploring linearequations (L6553)

MAT08NAIN00004

Weblink

Barbie Bungee

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Example 9

A fruit shop gives a discount to customers who buy larger amounts of fruit. Apples are$6.50/kg for up to 1 kg, $5.50/kg for amounts up to another 3 kg, $4.50/kg for amounts upto another 4 kg and $4.00/kg for amounts over 8 kg. Draw a graph of the unit price functionand describe it.

SolutionDraw the graph.

The unit price is $6.50 from 0 kg up to,but not including, $1. The line is flat forthis part and is open at the $1 end.

From 1 kg up to 4 kg it is at $5.50 and isopen at the 4 kg end. It is closed at the 1kg end.

Keep going in a similar way.65 7 821 3 4 9

Amount bought in kg

Unit price of apples

Cos

t in

$/kg

10 11 12

10

23456

Describe the graph. The graph looks like uneven steps going down.

In Example 9, it is cheaper to buy just over 1 kg than to buy 1 kg.

Exercise 4.2 Linear graphs

1 A car moves away from the lights in the left-hand lane at about 3 m/sec. After about 5 secondsthe lanes merge and the car increases its speed to 5 m/s for 2 seconds before speeding up to10 m/s. Draw a graph of its progress for 10 secondsand find how long it takes to travel 40 m.

2 A blue-tongued skink moves across a wide concretepath at 20 cm/s to a spot in the sun. It has to move adistance of 1.3 m across the path. Only 3 seconds afterit settles, it hears someone coming along the path andmoves back off the path, taking only 2 seconds to getoff the path. Draw a graph of its movement and findhow long it took to find the right spot.

3 The travel graph on the right shows Jemimawalking to school.a How far is it from her house to the school?

b How long does she wait at her friend’shousebefore continuing to school?

c Give an explanation of the shape of thegraph.

0 10 20 30 40Time in minutes

Walking to school

Dis

tanc

e fr

om h

ome

(km

)

0

500

1000

1500

Understanding

Extra questions

Exercise 4.2

MAT08NAEQ00012

See Example 5

Fluency

See Example 6

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4 The travel graph on the right showsthe journeys of a motorbike and a carbetween two locations on the highway.a How long was the trip for the

motorbike?

b How long was the trip for the car?

c What time did they pass eachother?

d How far apart are the places thatthey started from?

5 a Make a table of the function y ¼ 2x þ 3 from to �5 to 3.b Plot the function. c Describe the function.

6 a Make a table of the function y ¼ 4 � 2x from to �4 to 4.b Plot the function. c Describe the function.

7 a Make a table of the function y ¼ 3x � 4 from to �2 to 3.b Plot the function. c Describe the function.

8 a Make a table of the function y ¼ �x � �2 from to �5 to 5.b Plot the function. c Describe the function.

9 It costs $60 to hire a sailboard for up to 2 hours and an extra $20 an hour after that. Constructan algebraic model of the cost of hire up to 10 hours and plot the function. Find how long itcan be hired if you have $120.

10 Pine bark costs $35/m3, with a delivery charge of $45 to suburbs within the delivery area ofthe landscape garden supplier. Construct an algebraic model of the delivered cost of pine barkup to 6 m3 and plot the functions. Find the cost for 4.5 m3 of pine bark with delivery.

11 A courier of legal documents and small parcels based in the CBD charges for delivery basedon the distance from the GPO. For distances up to 2 km, the charge is $2/km. The chargeincreases for further distances by $0.50/km for every extra 2 km or part thereof, to a maximumcharge of $5/km for the part over 12 km. The courier does not deliver to addresses furtherthan 20 km from the city. Draw a graph of the charge and describe it. Find the cost ofdelivering an item to a location 9 km from the city.

12 A produce merchant sells oats as feed for cattle at a cost of $400/tonne for up to 3 tonnes. Forextra quantities up to another 4 tonnes, the price is $350/tonne, for extra quantities over 7tonnes the price is $300/tonne. Draw a graph of the price/tonne and describe it. Find the costof 9 tonnes of oats.

13 Explain how you would draw a graph of the total price per item for items delivered by thecourier in question 11 to different distances from the city centre. Draw the graph and describe it.

14 Explain how you would draw a graph of the total price of different amounts of oats boughtfrom the grain merchant in question 12. Draw the graph and describe it.

Gympie-Bundaberg trip

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 140Time in minutes

Dis

tanc

e in

km

Motorbike

Car

See Example 7

Problem solving

See Example 8

Worked solutions

Exercise 4.2

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See Example 9

Worked solutions

Exercise 4.2

MAT08NAWS00012

Reasoning

Worked solutions

Exercise 4.2

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4.3 Connecting algebra and geometry

Investigate: Linear rules and graphs: Part I

Use graph paper for this investigation.On one sheet of graph paper, plot the graphs of the following linear functions (rules)from �5 to 5.

y ¼ xþ 3

y ¼ xþ 5

y ¼ x� 1

y ¼ x� 4

y ¼ x

Think of the last three functions as y ¼ x þ �1, y ¼ x þ �4, and y ¼ x þ 0. Write the rulesfor all five functions along their lines.Use a different sheet of graph paper to plot the following linear functions from �5 to 5.

y ¼ 2xþ 1

y ¼ 2xþ 4

y ¼ 2x� 3

y ¼ 2x� 1

y ¼ 2x

Think of the last three functions as y ¼ 2x þ �3, y ¼ 2x þ �1, and y ¼ 2x þ 0. Write therules for all five functions along their lines.Use another sheet of graph paper to plot the following linear functions from �5 to 5.

y ¼ 5� x

y ¼ 3� x

y ¼ 1� x

y ¼ �2� x

y ¼ �x

Think of these functions as y ¼ �x þ 5, y ¼ �x þ 3, y ¼ �x þ 1, y ¼ �x þ �2, andy ¼ �x þ 0. Write the rules for all five functions along their lines.Make a prediction about each of the lines for the functions y ¼ �2x þ 3, y ¼ 3x � 4 andy ¼ 1.5x þ 2. Check your prediction for each line.

Video tutorial

Linear functions

MAT08NAVT00004

Teacher notes

Computer technology:Linear functions

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Important!

InterceptsThe x-intercept of a straight line is the part of the x-axis from the origin to the point wherethe line crosses the axis. The point itself or the x-coordinate of the point are also taken asbeing the intercept.The y-intercept is taken as the y-coordinate of the intersection (or the point) of the line and they-axis.

Technology Linear functionsYou can use digital technology to find the intercepts of linear functions.Use the spreadsheet ‘Linear function plotter’ on the NelsonNet website or some other digitaltechnology to find the intercepts of the functions given by your teacher.

Since linear functions are always straight lines, you only need to plot two points to find the line.However, it is a good idea to plot three as a check for calculation errors.

Example 10

Draw the line y ¼ 2x � 3 and find the intercepts.

SolutionIt is a linear function, so draw up a short table of values. x �3 1 4

y �9 �1 5


Excel: Graphingstraight lines: Changing

m- and c- values

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Plot the points and draw the line.

1 2 3 4 x

y

246

0−1−4−6−8

−2−3−4

−10

Use the graph to find the intercepts. The x-intercept is 112 and the

y-intercept is �3.

The intercepts in Example 10 could also be written as 112 ,0

� �and (0, �3).

Investigate: Linear rules and graphs: Part II

Use graph paper for this investigation.

Graph AOn a sheet of graph paper, plot the graph of the linear function y ¼ 2x� 3 from �10 to 10.

Draw a right-angled triangle in redon the line from x ¼ 2 to x ¼ 5. Itshould look like this diagram.

5 10

5

10

0

−5

−5−10

−10

x

y

Teacher notes

Computer technology:Functions and graphs

MAT08NATN00023

TLF learning object

EagleCat: Linear graph(L10090)

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Important!

Gradient of a LineFor two points on a straight line in a Cartesian plane, the rise is the vertical distance betweenthe points and the run is the horizontal distance between the points. The gradient of astraight line is calculated as gradient ¼ rise

run for any two points on the line. The usual symbolfor gradient is m.

The lengths of the sides of thistriangle have special names. Wecall the horizontal side the run.The vertical side is called the rise.Measure the rise and the run.Divide the rise by the run. Whatdo you get?Now draw a new right-angledtriangle in another colour on theline from x ¼ �1 to x ¼ 6. Itshould now look like this diagram.Measure the rise and the run onthe new triangle. Divide the riseby the run. What do you get?Now draw a new right-angledtriangle in a third colour on theline from x ¼ �3 to x ¼ 5.Measure the rise and the run onthe third triangle. Divide the riseby the run. What do you get?

5

9

10x

y

5

10

0

−5

−5−10

−10

Draw a fourth triangle on the line anywhere you want, find the rise and the run, and divide again.

Graph BUsing a new sheet of graph paper, plot the graph of the linear function y ¼ �2x þ 4 from�10 to 10.Draw a right-angled triangle in red on the line from x ¼ þ1 to x ¼ 5. Measure the rise andthe run. Check your answers for the rise and the run with your teacher before dividing.Draw right-angled triangles from �4 to 6, �2 to 3 and one other of your choice and findthe result from rise divided by run. What do you find for these?

Graph CWith a third piece of graph paper, plot the graph of the linear function y¼ xþ 3 from �10 to 10.Draw right-angled triangles from �3 to 7, �2 to 4, 1 to 8 and one other of your choice andfind the result for rise divided by run. What do you find for these?Your teacher might want you to draw some more lines.Look at your pieces of graph paper and compare the results of rise divided by run with therules for the functions. What do you find?

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Example 11

Plot the line y ¼ 5 � 2x and find the gradient.

SolutionIt is a linear function, so draw up a short table ofvalues.

x �3 1 4y 11 3 �3

Plot the points and draw the graph.

Draw a right-angled triangle on some convenientpoints on the graph.

1 2 3 4 x

y

2468

101214

0−1−4

−2−3−4

Read off the rise and the run from the triangle. Rise ¼ �8 and run ¼ 4

Work out the gradient. Gradient ¼ riserun

¼�84¼ �2

Write the answer. The gradient of y ¼ 5 � 2x is �2.

In geometry, you can find the midpoint of a line by bisecting the line segment. You can also usealgebra to find a midpoint.

Investigate: Midpoint of a line segment

Use graph paper for this investigation.Plot the points (1, 4) and (9, 10) on your graph paper.Draw a line segment between the points and find thepoint halfway along it.Write down this point. It is the midpoint of the linesegment between the points (1, 4) and (9, 10).Use the same method to find the midpoints of theline segments between the points:(�4, 5) and (1, �8)(�4, �6) and (4, �2)(2, �5) and (10, 10)Now look at each pair of points and the midpoint ofthe line segment.What pattern do you find?

5 104 93 82 71 6 x

y

5

10

4

9

3

8

2

7

1

6

0

Animated example

Linear graphs

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Technology

GeoGebra: Equationsof a line

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Important!

Midpoint of a Line SegmentThe values of the coordinates of the midpoint of a line segment between two points are theaverages of the coordinates of the points.

Example 12

Find the midpoint of the line segment between (�5, 1) and (3, 8).

Solution

Find the average of the x-coordinates. x-coordinate ¼�5þ 3

2

¼�22

Find the average of the y-coordinates. y-coordinate ¼ 1þ 82

92¼ 4 1

2

Write the answer. The midpoint of (�5, 1) and (3, 8) is �1, 4 12

� �:

Notice that in the answer to Example 12, the midpoint of the line segment is just shortened to themidpoint of the points.

Example 13

(3, 5) is the midpoint of (6, �3) and another point. Find the other point.

SolutionChoose variables for the unknowns. Let the other point have coordinates (a, b).

Use the rule for the midpoint to writean equation for the x-coordinate.

6þ a2¼ 3

Solve to find the x-coordinate. a ¼ 0

Use the rule for the midpoint to writean equation for the y-coordinate.

�3þ b2¼ 5

Solve to find the y-coordinate. b ¼ 13

Write the answer. The other point is (0, 13).

You may already have noticed that the rules of functions that are not linear do not make straightlines. Some of the curved lines they make are quite interesting shapes. In some cases, you mayneed to find values of x from values of y instead; or you may need to find values of y from valuesof x as well.

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Example 14

Use values of x from �4 to 4 to plot the function y ¼ 12

x 2:

SolutionMake a table of values.

x �4 �3 �2 �1 0 1 2 3 4y 8 4.5 2 0.5 0 0.5 2 4.5 8

Use brackets on your calculator tofind (�4)2.

Enter as 0.5 4 . 80.5×(-4)2

Plot the points and join them witha smooth line.

1 2 3 4 x

y

12345678

0−1−2−3−4

You saw earlier in this chapter that the shape in Example 14 is called a parabola.

Example 15

Use values of x between �4 and 4 to plot the line made by 3x2 þ 4y2 ¼ 48.

SolutionRearrange the equation by usinginverse operations or backtrackingto find the value of y.

3x 2 þ 4y 2 ¼ 48

4y 2 ¼ 48� 3x 2

y 2 ¼ 48� 3x 2

4

y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi48� 3x 2

4

r

Make a table of values, using yourcalculator to find approximatesquare roots.

x �4 �3 �2 �1 0 1 2 3 4y 0 2.3 3 3.4 3.5 3.4 3 2.3 0

Use brackets in your calculator.

Enter as: 48 3 3 4 .

2.291287847√((48–3×(-3)2)÷4)

Scientific calculatorexercise


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Plot the points and join them with asmooth line.

1 2 3 4 x

y

1234

0−1−2−3−4

In Example 15, there are actually two answers for y for each value of x. What shape do you get ifyou use both possible values of y?

Exercise 4.3 Connecting algebra and geometry

1 Plot each of the following lines and find the intercepts.

a y ¼ 2x þ 4 b y ¼ 3x � 6 c y ¼ �x þ 5 d y ¼ 2x � 6e y ¼ x � 3 f y ¼ �2x þ 6 g y ¼ �3x � 9 h y ¼ 4x � 4

2 Plot each of the following functions and find the gradients.

a y ¼ x � 1 b y ¼ �2x þ 8 c y ¼ 3x � 2 d y ¼ �x � 3e y ¼ 4 � 3x f y ¼ 1

2 x � 6 g y ¼ 14 x þ 2 h y ¼ �3x þ 1

3 Plot each of the following pairs of points and find their midpoints.

a (�3, 1), (8, 7) b (�4, �1), (�6, 0) c (�9, �4), (�4, �9) d (7, �1), (5, 7)e (7, �5), (1, �8) f (2, �7), (�5, �4) g (�9, �3), (7, 6) h (4, �6), (�4, 1)

4 Find the midpoints of the following pairs of points without plotting.

a (�9, �1), (�9, �7) b (7, �6), (�6, 3) c (2, 8), (�3, 9) d (�5, �1), (�9, �3)e (5, �4), (�1, �9) f (�6, 0), (3, �1) g (�4, 5), (1, 2) h (4, �8), (2, 8)

5 Use values of x from �4 to 4 to plot the line y ¼ x2 .

6 Use values of x from �3 to 3 to plot the line y ¼ 2x2

7 Use values of y from �4 to 4 to plot the line x ¼ y2.

8 Use values of y from �4 to 4 to plot the line x ¼ 12 y2.

9 Use values of x from �3 to 3 to plot the line x2 þ y2 ¼ 9.

10 Use values of x from �4 to 4 to plot the line x2 þ 2y2 ¼ 16.

11 �3,�12

� �is the midpoint of (�9, 6) and another point. Find the other point.

12 (1, �2) is the midpoint of (2, �8) and another point. Find the other point.

13 (1, �1) is the midpoint of (a, 2) and (�5, b). Find a and b.

14 1 12

�, �4 1

2Þ is the midpoint of (a, �2) and (�5, b). Find a and b.

15 Explain how you could find the gradient of a line between two points without drawing the linesegment.

16 Explain how you can tell if a rule is going to give a vertical parabola or a horizontalhalf-parabola.

Understanding

Extra questions

Exercise 4.3

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Fluency

See Example 11

See Example 12

See Example 14

See Example 15

Worked solutions

Exercise 4.3

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Problem solvingSee Example 13

Worked solutions

Exercise 4.3

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Reasoning

Worked solutions

Exercise 4.3

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Chapter 4 summary

• The Cartesian plane has two axes at right angles. It is also called a number lattice.

• The horizontal axis is called the x-axis. The vertical axis is called the y-axis. The axes cross atthe origin.

• Every point on the Cartesian plane is opposite a number on each axis. The points are namedusing numbers called coordinates.

• Coordinates are shown in parentheses (round brackets) with a comma between them. Theorigin is the point (0, 0). The x-coordinate is always shown first and the y-coordinate last.

• Each quarter of the Cartesian plane is called a quadrant.• A parabola is a roughly U-shaped curve that is

symmetrical about a central line.

• A travel graph shows the distance travelled bysomething over time. Something moving at a constantspeed produces a straight line and a stationary objectproduces a horizontal line.

• A mathematical rule that changes numbers into newones is called a function. The starting number is calledthe input or independent variable. The numberproduced is called the output or dependent variable.

• The rule for a function is usually written with the output variable on the left-hand side of theequals sign and the rule on the right in terms of the input variable.

• A function can be plotted by putting different values into the function and calculating theoutputs. This is usually put into a table called a table of values.

• A rule that produces a straight line is called a linear function and the graph is called a lineargraph. Linear functions are of the form y ¼ ?x � ??, where ? and ?? are constants.

• The x-intercept of a straight line is the part of the x-axis where the line crosses the axis.

• The y-intercept is the y-coordinate (or the point) of the intersection between the line and they-axis.

• For two points on a straight line in the Cartesian plane, the rise is the vertical distance betweenthe points and the run is the horizontal distance between the points. The gradient of a straightline is calculated as Gradient ¼ rise

run for any two points on the line. The usual symbol for thegradient is m.

• The values of the coordinates of the midpoint of a line segment between two points are theaverages of the coordinates of the points. The midpoint of a line segment is often stated asbeing the midpoint of the points at its ends.

0 xx-axis

1st quadrant

Origin

4th quadrant

2nd quadrant

3rd quadrant

y-ax

is

y

Quiz


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Puzzle sheet

Linear functionscrossword

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Chapter 4 review

1 Plot each of the following points on Cartesian axes and state its quadrant or axis.A(�3, 1), B(2, �4), C(�5, �2), D(4, 0), E(�5, 6), F(5, 2), G(0, �2)

2 Saeed is late for the bus and starts running down the street to the bus stop at 3 m/s. He turnsthe corner and after a minute he sees that the bus has not come and there are lots of peoplewaiting. So he slows to a walk, travelling at 1 m/s for half a minute. Then he sees the buscoming and resumes running at 2 m/s for another minute to catch the bus. Draw a travelgraph of his movement and find how far he ran altogether.

3 Plot each of the following lines and find the intercepts.a y ¼ x – 6 b y ¼ 2x þ 7 c y ¼ �2x þ 4 d y ¼ �3x – 6

4 Plot the points in the following table. What shape is formed?

x �3 �2 �1 0 1 2 3y 8 6 4 2 0 �2 �4

5 Without plotting, state the quadrant or axis of each of the following points.A(�5, �1), B(2, 1), C(�3, 0), D(�4, 1), E(5, �3), F(0, 6), G(�3, �5), H(7, �1)

6 a Make a table of the function y ¼ �2x þ 5 from �3 to 5.b Plot the function. c Describe the function.

7 Calculate the gradient of each of these functions.a y ¼ 2x � 1 b y ¼ 4 � 3x c y ¼ 1

2x þ 2 d y ¼ �x � 5

8 Plot each of the following pairs of points and find their midpoints.

a (6, 1), (2, 9) b (�3, 4), (�7, �3)

9 Find the midpoints of the following pairs of points without plotting.

a (�5, �3), (�1, 5) b (2, �6), (�4, 9)

10 Use values of x from �4 to 4 to plot the line y ¼ x2 – 2.

11 Use values of y from �3 to 3 to plot the line x ¼ y2 þ 1.

12 Use values of x from �4 to 4 to plot the line 3x2 þ y2 ¼ 48.

13 Plot the points in the following table. What shape is formed?

x �3 �2 �1 0 1 2 3y �6 �1 2 3 2 �1 �6

14 It costs $50 to hire a cement mixer for a day and an extra $30 for each day after that.Construct an algebraic model of the cost of hire for up to 6 days and plot the function.Find how long it can be hired if you have $160.

15 A fruit and vegetable shop sells bananas at $4.90/kg loose. It has 3 kg bags of bananas for $12and sells 10 kg boxes of bananas for $33 a box. Assume that the unit price for 3 kg applies toany quantity over 3 kg and the unit price for 10 kg applies to any quantity over 10 kg. Draw agraph of the unit cost (cost/kg) of bananas from 0–15 kg and describe it. What is the cost of9 kg of bananas?

Understanding

See Example 1

See Example 5

See Example 10

Fluency

See Example 2

See Example 7

See Example 11

See Example 12

See Example 14

See Example 15

Problem solving

See Example 8

See Example 9

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16 (�3, �12) is the midpoint of (�9, 6) and another point. Find the other point.

17 The table below shows the amount of weekly practice that different swimming squads haveand the average distance three members of the squad can swim in 10 minutes. Plot the pointsand comment on the effectiveness of training.

Training (hrs) 5 9 12 15 20Distance (m) 247 310 336 376 396

18 Explain how you would draw a graph of the total cost of different quantities of bananas inquestion 15 if you had to buy a mixture of loose, bags and boxes to make up in-betweenamounts and pay the corresponding prices. Draw the graph, describe it and comment.

19 Show that all functions with x2 in their rule do not have the same shape.

See Example 13

Reasoning

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Chapter 4 review

04 new wa maths sb8 txt

Documents

cartesian axes

cartesian plane withand

quadrant b0

quadrant d6

thecartesian plane

horizontal axis

vertical axis

graphs of functions