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Think!

For the formula

y

=

:

Z

What is the value of

y

if

x

=

0?

Z

What is the value of

x

if

y

=

0?

Z

What happens to the value of

y

as

x

gets very large?

Z

What happens to the value of

y

as

x

gets very small and close to 0?

12x

------

11_NCM10EX2SB_TXT.fm Page 456 Monday, September 12, 2005 4:49 PM

graphs graphs

Z

cubic

An algebraic expression in which the highest power of the variable is 3. For example 2

x

3

+

4

x

−

1.

Z

parabola

The graph of a quadratic equation.

Z

exponential

An algebraic expression in which the variable is a power (exponent). For example 2

x

.

Z

asymptote

A line to which a graph gets closer but never touches.

Z

independent variable

A variable whose values do not depend on any other variable. For example, in

C

=

2

π

r

,

r

is the independent variable.

Z

hyperbola

The graph of a hyperbolic equation

which may take the form y

=

.kx--

PATTERNS AND ALGEBRA

In this chapter you will:

Z

interpret distance–time graphs made up of straight line segments and when the speed is variable

Z

tell a story shown by a graph, by describing how one quantity varies with the other

Z

analyse the relationship between variables as they change over time

Z

interpret graphs, making sensible statements about the rate of increase or decrease, the initial and final points, constant relationships as denoted by straight lines, and variable relationships as denoted by curved lines

Z

sketch a graph from a simple description given a variable rate of change

Z

generate simple quadratic relationships, compile tables of values and graph equations of the form

y

=

ax

2

and

y

=

ax

2

+

c

Z

find the

x

-intercept and

y

-intercept for the graph of

y

=

ax

2

+

bx

+

c

Z

graph a range of parabolas

Z

find the equation of the axis of symmetry of a parabola and the coordinates of the vertex

Z

identify and use features of parabolas and their equations to assist in sketching quadratic relationships

Z

graph equations of the form

y

=

ax

3

+

d

and describe the effect of different values of

a

and

d

Z

generate simple hyperbolic relationships, compile tables of values and graph equations of the form

y

=

Z

sketch, compare and describe key features of simple exponential curves such as

y

=

2

x

,

y

=

−

2

x

,

y

=

2

−

x

, or

y

=

−

2

−

x

Z

establish the equation of a circle, with centre the origin and radius

r

, and graph equations of the form

x

2

+

y

2

=

r

2

Z

identify a variety of graphs from their equations.

kx--

Wordbank

Z

dependent variable

A variable whose value depends on another variable, for example, in

y

=

3

x

−

2,

y

is the dependent variable because its value depends on the value of

x

.

Z

initial

At the start.

Z

x

-intercept

The value(s) at which a graph intersects the

x

-axis.

Z

y

-intercept

The value(s) at which a graph intersects the

y

-axis.

Z

coefficient

The constant term in front of a variable. For example, in

y

=

3

x

2

+

4

x

−

6, the coefficient of

x

2

is 3.

Z

quadratic

An algmebraic expression in which the highest power of the variable is 2. For example 3

x

2

−

x

+

7.


458

N EW CENTURY MATHS 10 : S TAGES 5 .2/5 .3

Describing change

When we try to apply mathematics to the real world, we often produce a mathematical description of some kind to help solve practical problems. This technique is called

mathematical modelling

. This mathematical description may be a

formula

that expresses the relationship between two variables, or a

table of values

which may be compiled by observation, or it may be a

graph

which illustrates the relationship.

For example, consider a train that is travelling at an average speed of 65 km/h. This information relates distance and time.

This relationship can be represented by:• the formula D = 65t, where D is the distance travelled (in km) and t is the time (in hours)• a table of values which can be constructed

to show this relationship

1 a If y = 5x − 2, find the value of y if:i x = 0 ii x = 3 iii x = 5 iv x = −2

b If h = 2t2 − 3, find the value of h if:i t = 1 ii t = 7 iii t = 0 iv t = −4

c If p = find the value of p if:

i m = 2 ii m = −4 iii m = 5 iv m = −1d If y = x2 + 4x − 7, find the value of y if:

i x = 1 ii x = −1 iii x = 2 iv x = −3e If Q = 3a, find the value of Q if:

i a = 0 ii a = −1 iii a = 4 iv a = −2

2 Copy and complete each of the following tables:

a P = 2n b L =

c y = x3 d y = x2

3 Find the gradient, y-intercept and the equation of each line below, in the form y = mx + b:a b c

n −2 −1 0 1 4 B 1 4 10 32 80

P L

x −3 −1 0 1 2 x −3 0 1 4 9

y y

8m----,

40B------

0

2

−1

y

x

2

10

y

x 2−1

0

y

x

Start up

Worksheet 11-01

Brainstarters 11

t (in hours) 0 1 2 3 4

D (in km) 0 65 130 195 260


GRAPHS 459 CHAPTER 11

• a graph which illustrates the relationship

In more complex cases, more than two variables may be involved. Computers can be used to process elaborate mathematical models and hasten the discovery of solutions.

In previous chapters, graphs have been used to display numerical data such as temperature, rainfall, sales figures and ocean depths. The types of graphs that have been used to represent this data are column, line, bar, picture and sector graphs, or variations of these types.

In this chapter we will again use graphs to show how two quantities can be related.

Distance–time graphsThe relationship between the two quantities of distance and time for a moving object can be shown on a distance–time graph.

40

0

80

120

160

200

240

280

1 2 3 4

Speed of train

Time (hours)

Dis

tanc

e (k

m)

Skillsheet 11-01

Reading scales

Example 1

This distance–time graph shows the journey of a cyclist on a training ride.a How many hours did the training ride take?b What was the total distance covered?c What is the cyclist’s speed in the first hour?d The cyclist’s speed increases at B. How is this shown by

the graph?e The graph is horizontal from C to D. What is:

i the cyclist’s speed from C to D? ii the gradient of CD?f Calculate the speed of the cyclist from:

i B to C ii D to EWhat do you notice?

g Calculate the gradient of the graph from:i B to C ii D to EWhat do you notice?

h Explain the meaning of your results in parts f and g.

Solutiona The ride took 6 hours.b The total distance covered was 100 km (that is, 50 km away from the start and 50 km back).c The cyclist’s speed in the first hour is:

Speed =

= 10 km/h

20

0

40

60

1 2 3 4 5 6

Journey of a cyclist

Time (h)

Dis

tanc

e (k

m)

A B

C D

E

101

------


460 NEW CENTURY MATHS 10 : S TAGES 5 .2/5 .3

Features of distance–time graphs• The gradient of the line shows the speed of the object.• If the gradient of the line is positive, the object is moving away from a fixed point.• If the gradient of the line is negative, the object is moving back towards the fixed point.• The steeper the graph, the greater the speed.

d The graph becomes steeper after B.e i From C to D, the speed is 0 km/h.

ii Gradient CD = 0

f i Speed from B to C =

= 20 km/hii Speed from D to E =

= 20 km/hg i Gradient BC = (line slopes up)

= 20ii Gradient DE = − (line slopes down)

= −20h Although the speed values are the same, the gradient values are opposite in sign because,

from D to E, the cyclist is returning to the starting point.

20

0

40

60

1 2 3 4 5 6

Journey of a cyclist

Time (h)

Dis

tanc

e (k

m)

A B

C D

E2

4050

212

402

------

50

212---

----

402

------

50

212---

----

1 The distance–time graphs on the right are for Zaid and Nooreen as they walk to school.a What is the walking speed of:

i Zaid?ii Nooreen?

b Write the gradient of the graph for:i Zaid

ii Nooreenc Which feature of the graphs determines

whether Zaid or Nooreen is walking faster?d Zaid lives 1.8 km from the school and Nooreen lives

900 m from the school. How long will it take each person to walk to school?

2 This distance–time graph shows Megan’s journey by car.a Is Megan travelling at the same (or constant)

speed throughout the journey? Give reasons.b When is the gradient of Megan’s distance–time

graph least? What is her speed then?

Zaid

Nooreen

200

0

400

600

800

1 2 3 4 5

Walking to school

Time (min)

Dis

tanc

e (m

)

1

0

2

3

4

1 2 3 4 5 6 7

Megan’s car journey

Time (h)

Dis

tanc

e (k

m)

Exercise 11-01



Distance–time graphs with variable speedAs seen from Exercise 11–01, if the speed of an object is constant, the distance–time graph will be a straight line. However many real-life situations are more complex and involve variable speed or a rate of change that is not constant.

3 This distance–time graph shows the journey of two solar-powered cars.a Calculate the speed of:

i car A ii car Bb Write the gradient of the graph for:

i car A ii car Bc Is the speed value equal to the gradient value for:

i car A? ii car B?d The graph for car A slopes up, while the graph for

car B slopes down. Explain why.

4 This graph shows a cyclist’s journey on a training ride.a Describe the journey of the cyclist, giving the

speeds at each stage.b Do any intervals of the graph indicate the cyclist is

travelling at the same speed? Give reasons.c The gradient of the interval EF is −30 but the speed

at this stage is 30 km/h. What does the negative gradient indicate?

d i What is the speed of the cyclist when the gradient of the graph is 10?

ii At what stage of the journey does this occur?

5 Kate and Colleen are downhill skiers. Here is the distance–time graph for their 1200 m trip down the mountain.

a What were the speeds of the two skiers?b Who reached the base of the mountain first and how many minutes did it take her?c How many minutes difference was there between when the first skier reached the bottom

of the mountain and when the second skier arrived?d How far had Colleen gone after 6 minutes?e How far ahead was Kate after 6 minutes?f If Colleen and Kate were skiing down the mountain, why do the graphs go up?

10

0

20

30

40

50

1 2 3 4

Journey of two cars

Time (h)

Dis

tanc

e (k

m) Arac

Brac

10

0

20

30

1 2 3 4 5

Cyclist’s training ride

Time (h)

Dis

tanc

e (k

m)

A F

BC

D E

200

0

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

Skiing down a mountain

Time (min)

Dis

tanc

e (k

m)

Kate

Colleen

Example 1

STAGE5.3



For rates of change that are not constant, such as variable speed, the graph will be a curved line.

constant rate of change (constant gradient)

variable rate of change (changing gradient)

Example 2

Write a brief description of the journey that each of the following distance–time graphs might represent.a b c

Solutiona A person leaving home starts the journey slowly (at A, the graph is

not very steep), then reaches maximum speed (at B, the graph is the steepest) before slowing down and stopping (at C, graph gradually becomes horizontal).

b A person, going home, starts the journey at a very fast speed (at A, the graph is very steep) before slowing down and stopping briefly (at B, the graph is horizontal). The person then speeds up again (at C, the graph becomes very steep) before slowing down again at arriving at home (at D).

c A person leaves home, starting slowly and then speeding up (at A) before reaching a constant speed (at B, the graph is a straight line) and continuing on the journey.

t

d

t

d

t

d

t

d

A

B

C

t

dA

B

C

D

t

d

A

B

STAGE5.3



Example 3

Draw distance–time graphs to represent each of the following:a running at a decreasing speedb running towards home at a constant speed, for 2 minutes, before slowing down and stopping

100 m from home after a further 1 minute.

Solutiona The speed of the runner is gradually decreasing, so the steepness (or

gradient) of the curve must be decreasing.

b The graph is a straight line at A, indicating a constant speed. After 2 minutes (at B) the gradient of the graph gradually decreases, indicating that speed is decreasing until stopping after the third minute (at C) when the graph is a horizontal line.

t

d

Time (min)21 43

100 mHome

Dis

tanc

e A

BC

1 Write a brief description of the journey that each of the following distance–time graphs might represent.a b c

2 Match each statement below with the correct part of this distance–time graph:a the speed is decreasing on the way homeb the person is not movingc the speed is constant and then increasesd the person returns home and increases speed very rapidlye the person gradually increases speed after going very

slowlyf the person slows down

t

d

t

d

t

d

tHome

d

A B

C DE

F

G

H

Exercise 11-02Example 2



3 Copy the axes shown into your book and use them to construct a distance–time graph to represent the following information about a sprinter completing a 100 m race:• the sprinter covers 5 m in 2 seconds and is at the 20 m

mark after 4 seconds as the running speed increases• the sprinter runs at a constant speed until the 80 m mark

is reached, 9 seconds after the start• the sprinter slows down over the last 20 m, completing

the 100 m in 11 seconds• the sprinter stops after running a further 20 m

4 Using the axes shown on the right, draw a distance–time graph for each of the following:a starting at home and running away from home at

a decreasing speedb starting at home and running away from home at

an increasing speedc starting away from home and running toward

home at an increasing speedd starting away from home and running toward

home at a decreasing speed

5 The axes below have three graphs drawn on them. Three stories that could be matched to the graphs are:

i Jade rode her bicycle homeii Cameron walked home

iii Kiet ran home

a Match the stories and graphs.b Explain in words why you made each match.c What could have caused the level section at about 4:25pm in graph C?d Describe how the rate of travel changes in graph A.

Time (seconds)

A sprinter's race

420

86

20

Dis

tanc

e (m

etre

s)

40

60

80

100

120

10 12

tHome

Awayfrom

home

d

Time (pm)

Dis

tanc

e fr

om s

hop

Home

Shop4:00 4:05 4:10 4:15 4:20 4:25 4:30 4:35 4:40

A B C

Example 3

STAGE5.3



6 Abrar leaves home and drives to the shops, arriving there after 1 hour. He shops for 2 hours and then drives home, which only takes half and hour.A distance–time graph has been drawn to represent the information above.Is the graph an accurate representation of the information?

7 Match each of the descriptions below to one of the graphs that follow:a the speed increases at an increasing rate and then stopsb the speed increases slowly, slows down to a stop, then increases to a constant ratec the speed increases at an increasing rate, slows down and stopsd the speed starts high, decreases, then stopse the speed increases to a maximum speed, then slows downf the speed increases, then slows down and stops, then begins to increase again before

stopping for an instant before increasing to a constant rateA B C

D E F

Time (hours)210 43

Shops

Home

t

d

t

d

t

d

t

d

t

d

t

d

Time before and time after1 Examine these examples:

a What is the time 4 hours and 25 minutes after 6:30pm?6:30pm + 4 hours = 10:30pmCount: ‘6:30, 7:30, 8:30, 9:30, 10:30’10:30pm + 25 minutes = 10:55pm

b What is the time 7 hours and 40 minutes after 11:45am?11:45am + 7 hours = 6:45pmCount: ‘11:45, 12:45, 1:45, 2:45, 3:45, 4:45, 5:45, 6:45’6:45pm + 40 minutes = 6:45pm + 15 minutes + 25 minutes

= 7:00pm + 25 minutes= 7:25pm

Skillbank 11ASkillTest 11-01

Time before and after



or

c What is the time 10 hours and 15 minutes after 1850 hours?1850 hours + 10 hours = 0450 hours (next day)Count: ‘1850, 1950, 2050, 2150, 2250, 2350, 0050, 0150, 0250, 0350, 0450’0450 hours + 15 minutes = 0450 hours + 10 minutes + 5 minutes

= 0500 hours + 5 minutes= 0505 hours

or

2 Now find the time of day:a 3 hours 20 minutes after 9:05am b 5 hours 40 minutes after 7:30pmc 4 hours 35 minutes after 6:15pm d 11 hours 10 minutes after 11:45ame 2 hours 45 minutes after 0325 hours f 7 hours 5 minutes after 1705 hoursg 8 hours 30 minutes after 12:40am h 4 hours 55 minutes after 10:20pmi 6 hours 25 minutes after 0435 hours j 2 hours 15 minutes after 2050 hoursk 9 hours 50 minutes after 2:30pm l 3 hours 10 minutes after 8:25am

3 Examine these examples:a What is the time 3 hours and 15 minutes before 11:20am?

11:20am − 3 hours = 8:20am Count back: ‘11:20, 10:20, 9:20, 8:20’8:20am − 15 minutes = 8:05am

b What is the time 2 hours and 40 minutes before 7:20pm?7:20pm − 2 hours = 5:20pm Count back: ‘7:20, 6:20, 5:20’5:20pm − 40 minutes = 5:20pm − 20 minutes − 20 minutes

= 5:00pm − 20 minutes= 4:40pm

or

c What is the time 8 hours and 45 minutes before 1115 hours?1115 hours − 8 hours = 0315 hoursCount back: ‘1115, 1015, 0915, 0815, 0715, 0615, 0515, 0415, 0315’ (or 11 − 8 = 3).0315 − 45 minutes = 0315 hours − 15 minutes − 30 minutes

= 0300 hours − 30 minutes= 0230 hours

or

11:45am 12:00noon

15 minutes 7 hours 25 minutes = 7 hours 40 minutes

7:00pm 7:25pm

1850 hours 1900 hours



4:40pm 5:00pm


7:00pm 7:20pm






Graphs and other rates of changeGraphs have already been used to compare the quantities of distance and time. Graphs can be used to compare or relate any two quantities.

VariablesWhen we look at how two quantities are related, the quantities are referred to as variables because they change or vary. The numbers they take are called their values. For example, the time it takes a typist to type a certain number of words is a variable, and so is the number of words that have been typed. If it takes 2 minutes to type 120 words, then the variables have taken on values.

In situations where there are two variables, the variables are either independent or dependent. For example, the independent variable x may have any value, but the dependent variable y takes values that depend on x. In the example of the typist already given, time is the independent variable and the number of words typed is the dependent variable.

The variables are graphed on different axes. The independent variable is shown on the horizontal axis and the dependent variable on the vertical axis.

4 Now find the time of day:a 1 hour 15 minutes before 7:20pm b 4 hours 40 minutes before 11:20amc 3 hours 20 minutes before 3:30pm d 5 hours 35 minutes before 8:25ame 2 hours 10 minutes before 1455 hours f 3 hours 45 minutes before 0740 hoursg 5 hours 25 minutes before 4:15am h 9 hours 30 minutes before 9:45pmi 4 hours 20 minutes before 2005 hours j 2 hours 15 minutes before 0615 hoursk 3 hours 55 minutes before 5:30pm l 4 hours 40 minutes before 12:00 noon

Worksheet11-02

Graphs of change

Example 4

The graph below shows the number of students in a school playground during the day.

a Identify the independent variable and the dependent variable.b Write a description for the graph.c Is the rate of change of students entering and leaving the playground constant?

Solutiona The number of students in the playground will depend on the time of day (for example

change of classes, recess and lunchtime). So the independent variable is time, and the dependent variable is the number of students.

TimeA D E G H K N Q R S

Num

ber

of s

tude

nts

Students in the playground

L M

B C F I J O P



b Students begin to arrive at school (A to B), then the bell rings for Period 1 and students go to class (C to D). Period 1 ends and students leave their classrooms, crossing the playground to go to their Period 2 classrooms (E to G). Period 2 finishes and students go to recess (H to I). Recess finishes and students go to Period 3 (J to K). They change rooms for Period 4 (L to M), then they go to lunch (N to O). Lunch finishes and students go to Period 5 (P to Q). Period 5 ends and students leave their classrooms, collect their belongings and go home (R to S).

c Rate of change of number of students is not constant. Constant change would be shown as a straight line.

Sketch a graph to show the noise level of a classroom during a lesson.

SolutionOne possible graph has been sketched below. The variables are time (the independent variable) and noise level (the dependent variable).

The students enter the classroom and the noise level increases. They settle down and work (noise level is low).There may be a classroom discussion (the noise level increases) and then the class settles down again. Towards the end of the period the noise level increases. The period ends and the students leave (the noise level decreases to zero).

Water is poured into the container shown at a constant rate. Draw a graph to show the relationship between the level of water in the container (height) and the time taken to fill the container.

Solution

Since the container is wider at B than at A, the rate of increase in the level of water will slow down (or decrease). As the container narrows (at C), the rate of increase in the level of water increases (the graph becomes steeper). At D, the container is cylindrical and so the rate of increase in the level of water is constant (the graph is a straight line).

Example 5N

oise

leve

l

Time

Example 6

B

CD

AB C

D

A

Hei

ght

Time



1 For each of the following graphs:i identify the independent variable and the dependent variable

ii write a description of the graph.a b

c d

2 For each of the following graphs:i indicate the independent variable and the dependent variable

ii describe the constant rate of changeiii calculate the gradient.

a b

c d

Tem

pera

ture

Time (days)

Hei

ght o

f tid

e

Time

Vol

ume

of p

etro

l

Distance

Hei

ght

(cm

)

Age (years)

4

0

8

12

16

20

2 4 6 8 10 12Number of people

Cos

t ($)

c

n

30

0

60

90

120

5 10 15 20 25Time (s)

Spee

d (m

/s)

s

t

5

0

10

15

20

25

10 20 30 40 50Quantity

Prof

it ($

)

p

q

20

0

40

60

80

5 10 15 20 25Time (min)

Wat

er le

vel (

cm)

H

t




3 For each of the following:i identify the independent variable and the dependent variable from those

given in bracketsii draw a graph to describe the situation or event.

a temperature change during the day (temperature, time)b mood swings of a spectator during a football game (mood, time)c noise at a rock concert (sound level, time)d the area of a circle as the radius increases (A, r)e the cost of petrol (cost, litres)f the masses of adults of various heights (mass, height)g the height to which a tennis ball bounces when it has been dropped onto a concrete

playground (height, time)h the water level in a bath that is being filled and then empties (depth, time)i the level of skill achieved in playing a musical instrument and the number of practice

sessions (skill, number of sessions)j people’s heights and shoe sizes (height, shoe size)k the thirst level of a person and the temperature (thirst, temperature)

4 Water is poured into each of the following containers at a constant rate. Draw a graph to show the relationship between the level of water (H) and the time taken to fill each container.

a b c

d e f

5 Draw graphs that model the following situations. Use the variables given in brackets.

a The distance travelled by a person who runs at a constant speed. (time, distance travelled)b The distance travelled by a person who walks at a constant speed (slower than the person

in part a. (time, distance travelled)c The distance travelled by a person who first walks at a constant speed and then runs at a

(faster) constant speed. (time, distance travelled)d The distance travelled by a person who first runs at a constant speed and then walks at a

(slower) constant speed. (time, distance travelled)e The level of petrol in a car on a long trip. It stops once but does not refuel. (time, litres)f The level of petrol in a car on a long trip. It stops once and refuels. (distance, litres)g The distance covered by a car that is slowing down at a constant rate. (distance

travelled, speed)h The distance covered by a car that is gradually slowing down. (distance travelled, speed)i The water level in a pool with people swimming in it on a hot summer day. (time, height)j The height of a child’s head above ground level as the child climbs up the ladder of a slide,

sits and then slides down. (time, height)

Example 5

Example 6



Interpreting graphsWhen interpreting graphs, we need to consider the rate of increase or decrease (shown by the steepness or gradient of the graph), the initial point and final point, and whether the relationship between the variables is constant (the graph is a straight line) or variable (the graph is a curved line).

k The distance from a shop of a person walking at a steady speed up and down the street past that shop. (time, distance from shop)

l The noise level at a party. (time, sound level)

Worksheet11-02

Graphs of change

Example 7

A plane flies from Sydney to Canberra. Choose the graph that matches the situation:

A B C D

SolutionThe correct solution is B.

A is incorrect because the initial point of the graph indicates that the plane is already flying.

C is incorrect for the same reason as A.

D is incorrect because it shows the plane’s height increasing for most of the journey, and then decreasing very rapidly.

1 A cup of tea sits on the kitchen bench cooling. At first it loses heat quickly but, as time passes, it loses heat more slowly until it is at room temperature. Which of the graphs below best illustrates this?

A B C D

SolutionThe correct solution is B. The temperature decreases rapidly initially, but then this rate of decrease slows (the temperature decreases at a decreasing rate).

A is incorrect, because it shows the temperature decreases to a minimum and then increases.

C is incorrect because it shows the temperature decrease is slow initially and then the temperature decreases at an increasing rate.

D is incorrect because it shows that the temperature begins to decrease slowly. The rate of temperature decreases then increases, before slowing again.

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Example 8

Tem

pera

ture

Time

Tem

pera

ture

Time

Tem

pera

ture

Time

Tem

pera

ture

Time

STAGE5.3



2 Nooreen climbs a hill at a steady rate and then starts to run down the hill. Choose the graph of this situation from the graphs below:

A B C D

SolutionD is the correct solution because the speed is constant (the graph is horizontal) and then increases as Nooreen begins to run down the hill.

Time

Spee

d

Time

Spee

d

Time

Spee

d

Time

Spee

d

1 Each of the following graphs tells a story:

i ii iii

iv v vi

vii viii ix

Match each of the stories below with one of the graphs. (The variables are given in brackets.)a People were using the drink-vending machine until the machine broke down. (number of

cans in machine, time)b The bath was half full of water all day. (depth of water, time)c The cost of a mobile phone call is charged at a constant rate. (rate, time)d People were playing a DVD movie and changed the disc halfway through. (amount of

movie seen, time)e Interest rates rapidly rise at a constant rate as inflation increases, then steadily fall at a

constant rate and stay at a constant low. (rate, time)


STAGE5.3



f A taxi fare includes the hire charge plus a constant amount per kilometre. (cost, kilometres travelled)

g The car is consuming petrol at a steady rate until it runs out of fuel. (litres in tank, time)h A runner jogs at a steady rate then stops and rests. (distance travelled, time)i A runner jogs at a steady rate then walks home (at a slower rate). (distance travelled, time)

2 Select the graph that best describes the given situation each time:a the height of water in a cylindrical bucket being filled from a tap

A B C

b the change in height of a person over that person’s lifetime

A B C

c the speed of a bus that stops three times before it reaches its destination

A B C

d the speed of a car as it goes around a sharp corner

A B C

e the share each person receives when a group of people win Lotto

A B C

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

TimeH

eigh

tTime

Spee

d

Distance

Spee

d

Distance

Spee

d

Distance

Spee

d

Time

Spee

d

Time

Spee

d

Time

Am

ount

($)

Group size

Am

ount

($)

Group size

Am

ount

($)

Group size

Example 8



f the speed of a runner going up a steep hill from a standing start

A B C

g the height reached by a person jumping on a trampoline

A B C

3 Select the graph that best describes the height of the water as it is poured at a constant rate into the given container each time.

a A B C

b A B C

c A B C

d A B C

Spee

d

Distance

Spee

d

Distance

Spee

d

Distance

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

STAGE5.3



e A B C

4 Eight graphs and five different-shaped containers are shown below.Match a graph to each container as it is filled at a constant rate.a b c d e

A B C D

E F G H

5 Match each of the descriptions below to one of the graphs that follow:a As one variable increases at a constant rate, the other variable increases at a constant rate.b As one variable increases at a constant rate, the other variable decreases at a constant rate.c As one variable increases at a constant rate, the other variable increases at an

increasing rate.d As one variable increases at a constant rate, the other variable decreases at a

decreasing rate.e As one variable increases at a constant rate, the other variable increases at a

decreasing rate.f As one variable increases at a constant rate, the other variable decreases at an

increasing rate.

A B C

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

T

H

T

H

T

H

T

H

T

H

T

H

T

H

T

H



D E F

6 Use the following statements to label the parts of each graph below:A increasing at a constant rate B decreasing at a constant rateC increasing at an increasing rate D increasing at a decreasing rateE decreasing at an increasing rate F decreasing at a decreasing rate

a b

7 A B C

D E F

Select the graph above that best describes:a a car accelerating until it reaches a constant speedb a car stopped at traffic lightsc a bus travelling at a constant speed before stopping for passengersd a car travelling at a constant speede a train slowing down at a constant rate until it stopsf a rocket launched into space

8 Draw speed–time graphs to show each of the following:a a plane takes off and reaches its normal flying heightb a tennis ball was thrown into the air and was caught when it fellc a car travels at a constant speed before slowing and then stoppingd a bus leaves the station and makes three stops before returning to the depote a runner begins at a steady pace, increases his speed and then gradually slows

down before stoppingf a cyclist rides down a hill, gradually increasing speed before hitting a pothole

and falling off

i

ii iii

iv

i

iiiii

iv

Time

Spee

d

Time

Spee

d

TimeSp

eed

Time

Spee

d

Time

Spee

d

Time

Spee

d

STAGE5.3



9 The diagrams below represent cycling tracks. For each track, draw a speed–time graph for a cyclist in the tenth lap of a race.a b c

10 The diagram below shows the road which goes from Serena’s house to her school.She travels to school in her father’s car each day. They drive at 60 km/h on straight sections of the road, slowing down for the corners.Copy the speed–distance axes into your book and draw a graph to show Serena’s journey to school.

Start Start Start

Distance from home

Spee

d

Serena’shouse

School

Time differences1 Examine these examples:

a What is the time difference between 11:40am and 6:15pm?From 11:40am to 5:40pm = 6 hoursCount: ‘11:40, 12:40, 1:40, 2:40, 3:40, 4:40, 5:40’From 5:40am to 6:00pm = 20 minutesFrom 6:00pm to 6:15pm = 15 minutes6 hours + 20 minutes + 15 minutes = 6 hours 35 minutes

or

b What is the time difference between 8:30pm and 1:20am?From 8:30pm to 12:30am = 4 hours Count: ‘8:30, 9:30, 10:30, 11:30, 12:30’From 12:30am to 1:00am = 30 minutesFrom 1:00am to 1:20am = 20 minutes4 hours + 30 minutes + 20 minutes = 4 hours 50 minutes

or

11:40am 12:00noon


6:00pm 6:15pm

8:30pm 9:00pm


1:00am 1:20am

Skillbank 11BSkillTest 11-02

Time differences



Graphs of equationsThe graph of an equation can be drawn by completing a table of values and plotting points.

The straight lineThe general form of a linear equation is ax + by + c = 0, where a, b and c are integers, and a is positive. The gradient intercept form of a linear equation is y = mx + b, where m is the gradient and b is the y-intercept. The graph of a linear equation is a straight line.

The graphs of straight lines can be drawn by plotting points from a table of values, but they can also be drawn using the x-intercept and y-intercept.

The parabolaEquations, such as y = x2, y = 3x2 − 4 and y = x2 − 4x + 1 are examples of quadratic relationships, in which the highest power of x is 2.

c What is the time difference between 1645 hours and 2320 hours?From 1645 hours to 2245 hours = 6 hours (22 − 16 = 6)From 2245 hours to 2300 hours = 15 minutesFrom 2300 hours to 2320 hours = 20 minutes6 hours + 15 minutes + 20 minutes = 6 hours 35 minutes

or

2 Now find the time difference between:a 11:10am and 7:40pmb 6:20pm and 12:00 midnightc 4:45pm and 8:10pmd 2:30am and 10:55ame 1:05pm and 12:30amf 9:35am and 11:15amg 0425 hours and 0935 hoursh 1440 hours and 2025 hoursi 7:55am and 3:50pmj 2:45pm and 10:10pm




1 Draw the graphs of the following equations on separate sets of axes:a y = 2x − 6 b 3x + y − 12 = 0c 4x − y = 8 d y = 2x + 3e 3x + 2y − 1 = 0 f x − 2y = 5

Exercise 11-05

SkillBuilders10-01 to 10-06Linear equations

CAS 11-01

Linear graphs

Worksheet 11-03

Graphing parabolas



The parabola y = x2The equation y = x2 is a quadratic relationship. The graph of the equation y = x2 is a smooth curve called a parabola.

Features of y = x2 are as follows:• The axis of symmetry is the y-axis. It is also referred to as the axis

of the parabola.• The vertex is (0, 0). The vertex is also called the turning point

because the parabola changes direction at this point.• The graph of y = x2 is said to be concave up.

Working mathematicallyApplying strategies, reasoning and communicating: Graphing quadratics1 a Copy and complete this table of values for y = x2, where values

of x are taken from −4 to 4:

b On grid paper, draw a set of axes, similar to those shown, using a scale of 1 cm to 1 unit.

c Plot the points from the table of values onto the grid paper.d Draw a smooth, continuous curve through all the points.e Compare your graph with those of other students.

2 List as many features as you can about the graph of y = x2 that you have drawn. Compare your list with those of other students.

x −4 −3 −2 −1 −0.5 0 0.5 1 2 3 4

y

2 4−2

2

4

6

8

10

12

14

16

−20−4

y

x

0

y

x

y = x243

2

1

−1 1−2 2

Working mathematicallyApplying strategies, reasoning and communicating: Which way is up?1 Using a scale of 1 cm = 1 unit on both the x-axis and the y-axis, draw the graph of y = x2

for −3 � x � 3. (Use only values of x from −3 to 3, inclusive.)

2 a On the same set of axes, draw the graph of the reflection of y = x2, when it is reflected in the x-axis.

b Use the reflection of y = x2 to copy and complete this table:

c Use the graph of the reflection and the values in the table above to determine the equation of the reflection of y = x2.

d Compare your graph and equation with those of other groups.

3 a List the features that are the same for both parabolas.b List the features that are different for both parabolas.

x −3 −2 −1 0 1 2 3

y −4



The parabola y = −x2The graph of the equation of y = −x2 is a parabola that is concave down (or is an upside-down parabola).

Features of y = −x2 are as follows:• The axis of symmetry is the y-axis.• The vertex or turning point is (0, 0).• The graph is concave down.• It is a reflection of y = x2 in the x-axis.

The parabola y = ax2

0

y

x−2 −1 1 2−1−2−3−4

y = −x2

Graphing quadratic relationships: y = ax2In this task we are going to investigate quadratic graphs.

1 Using your spreadsheet application, copy and complete the table below:

2 What formulas will appear in cells B2 to D2? Enter these formulas and Fill Down.

3 Select the table of values and graph the three quadratic graphs on the same set of axes. Remember, the x-values should be on the horizontal axis and the y-values should be on the vertical axis.

4 Compare the graphs of y = 2x2 and y = with that of y = x2. What do you notice?

5 Using your spreadsheet application, set up a table of values and graph the following set of quadratics:

y = −x2 y = −2x2 y = −

A B C D

1 x value y=x^2 y=2*x^2 y=1/2*x^22 −3

3 −24 −15 0

6 1

7 2

8 3

12---x2

12---x2

Using technology

Spreadsheet

For the graph of y = ax2, the size of a (the coefficient of x2) determines whether the parabola is ‘wide’ or ‘narrow’.



As the size of a in the rule at the bottom of the facing page increases, the graph of the parabola becomes ‘narrower’ and, as the size of a decreases, the graph of the parabola ‘widens’. The parabolas below illustrate this.

1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

10

−10

y

x

y = 2x2

y = 4x2

y = x 2

y = x21_2

y = x21_4

y = x21_9

−1−2−3−4−5−6

y = −2x2

y = −4x2

y = −x2

y = − x21_2

y = − x21_4

y = − x21_9

1 2 3 4 5 6−1

1

−10

0−2−3−4−5−6

−9−8−7−6−5−4−3−2−1

y

x

1 Which parabola in each pair will be ‘narrower’ when compared with the other?a y = x2 or y = 3x2 b y = 3x2 or y = 2x2

c y = or y = 0.5x2 d y = −x2 or y = −2x2

e y = −3x2 or y = − f y = − or y = −10x2

2 For each of the following, draw the graph of y = x2. Then:i draw a sketch of the given equation

ii find the coordinates of the turning pointiii find the y-coordinate of the point on the parabola where x = 3.a y = 4x2 b y = −2x2 c y = d y = 6x2 e y = − f y = 2x2

g y = −3x2 h y = 3x2 i y =

14---x2

15---x2 1

10------x2

13---x2

12---x2

14---x2

Exercise 11-06

SkillBuilder

13-07The effect of ‘a ’

CAS 11-02

Parabolas

Working mathematicallyQuestioning and reasoning: Going up or going down?You will need: 1 cm graph paper, or a graphics calculator, or a suitable software program

1 a Using a scale of 1 cm = 1 unit on both the x-axis and the y-axis, draw the graph of y = x2 for −3 � x � 3.

b Copy and complete this table of values for y = x2 + 2:

c Plot the points from the table on the set of axes from part a. Draw a smooth curve through all the points.

x −3 −2 −1 0 1 2 3

y



The parabola y = ax2 + c

d i What is the equation of the axis of symmetry of y = x2 + 2?ii Write the coordinates of the vertex of the graph of y = x2 + 2.

iii What is the minimum value for y = x2 + 2?iv What properties do the graphs of y = x2 and y = x2 + 2 have in common?v How are the two graphs different?

e Describe how the graph of y = x2 + 2 could be drawn using the graph of y = x2.

2 Repeat the procedure from Question 1 for the parabolas:a y = x2 + 4 b y = x2 − 1 c y = x2 − 4

3 Write a sentence to describe the effect of the constant term (which is added to, or subtracted from, x2) on the shape and position of the parabola y = x2.

4 a On the same set of axes, draw these graphs for −3 � x � 3. (Label your graphs clearly.)i y = −x2

ii y = −x2 + 2iii y = −x2 − 3iv y = −x2 − 1

b For each of the parabolas in part a:i what is the equation of the axis of symmetry?

ii what are the coordinates of the vertex?iii what is the maximum value?

c What do you notice about the shape and position of each of the parabolas in part a? (Compare your results with those of other students.)

The parabola y = ax2 + c (or wide, narrow, up or down)1 a Use a graphics calculator or a spreadsheet to draw the graphs of the following equations

on the same set of axes:i y = x2 ii y = 2x2 + 1 iii y = 2x2 − 3

b How are the graphs of y = 2x2 + 1 and y = 2x2 − 3 different from the graph of y = x2?c What is the y-intercept of each of the parabolas drawn in part a?

2 a Use a graphics calculator or spreadsheet to draw the graphs of each of the following equations on the same set of axes:

i y = x2 ii y = + 3 iii y = − 4

b How are the graphs of y = and y = − 4 different from the graph of y = x2?c What is the y-intercept of each of the parabolas drawn in part a?

12---x2 1

5---x2

12---x2 1

5---x2

Using technology

Worksheet 11-04

Graphing curves (graphics calculator)

Spreadsheet

The effect of the constant term c on the graph of y = ax2 + c is to move the parabola up or down.

(The constant term c is also the y-intercept of the graph y = ax2 + c.)

Worksheet 11-03

Graphing parabolas



Example 9

Draw the graphs of the following:a i y = x2 b i y = −x2

ii y = x2 − 4 ii y = −x2 − 4iii y = x2 + 2 iii y = −x2 + 5

(Show the coordinates of the vertex of each parabola on the graphs.)

Solutiona b

For each of the following, state:i whether the parabola is wider or narrower than the graph of y = x2

ii whether the parabola has moved up or down when compared to the graph of y = x2iii the y-intercept.

a y = 3x2 − 1 b y = + 2

Solutiona i The coefficient of x2 is 3, while the coefficient of x2 in y = x2 is 1.

∴ The parabola will be narrower than y = x2.ii The constant term is −1.

∴ The parabola has moved down.iii The y-intercept is −1.

b i The coefficient of x2 is .∴ The parabola will be wider.

ii The constant term is 2.∴ The parabola has moved up.

iii The y-intercept is 2.

1 2 3−1

2

4

6

7

1

3

5

0−2−3

−2−3−4

−1

y

x

y = x2 − 4

y = x2 + 2y = x2

(0, 2)

(0, 0)

(0, −4)

1 2 3−1

2

4

1

3

5

0−2−3

−2−3−4−5−6−7

−1

y

x

y = −x2 − 4

y = −x2 + 5

y = −x2

(0, 5)

(0, 0)

(0, −4)

Example 10

13---x2

13---



Parabolas in actionWhen an object is thrown outwards from Earth’s surface, its path is a parabola. So, when an object such as a stone or ball is thrown, or when a bullet is fired from a gun, the path is parabolic and the object, under the force of gravity, will fall back to Earth’s surface. The length of the path will depend on the initial velocity. As a result of air resistance, the paths are not exactly ‘true’ or ‘perfect’ parabolas.

If a parabola is rotated about its axis, a paraboloid will be formed. Parabolic mirrors or reflectors are of this shape and are used in car headlights and spotlights. Parabolic mirrors are also used in astronomical telescopes because they bring parallel light rays into focus without any distortion. Parabolic reflectors are also used in radio astronomy and radar.

Explain, with the aid of a diagram, the property of parabolas that makes then useful as parabolic reflectors.

Parabolic paths are formed by the material being thrown from this erupting volcano.

Just for the record

1 Sketch each of the following parabolas. State the axis of symmetry and the vertex of each parabola.a y = x2 + 10 b y = −x2 + 10 c y = −x2 − 10d y = x2 − 10 e y = 6 + x2 f y = −6 + x2g y = 6 − x2 h y = −6 − x2 i y = 10 − x2j y = x2 − 3 k y = −x2 − 5 l y = 2 + x2

2 Match the following equations with the sketches below or on the facing page:i y = x2 ii y = −x2 iii y = x2 − 8

iv y = −12 − x2 v y = + x2 vi y = 8 − x2

vii y = 8 + x2 viii y = −x2 + ix y = x2 − 12x y = 12 − x2 xi y = −x2 − 8 xii y = x2 + 12

a b c

12---

12---

y

x0

8

y

x0

(0, −12)

y

x0


SkillBuilders13-08 to 13-10

Quadratics

CAS 11-03

Parabolas, changing rules



d e f

g h i

j k l

3 Find the equation of each of the following parabolas in the form y = x2 + c or y = −x2 + c (where c is a constant), given:a vertex (0, 0), concave downb concave up, turning point (0, 0)

c turning point (0, concave down

d vertex (0, 3), concave up

e y-intercept = 9, axis of symmetry x = 0, concave downf concave up, turning point (0, −8)

4 For each of the following parabolas, state:i whether it is wider, narrower or the same when compared to the graph of y = x2

ii whether it has moved up or down when compared to the graph of y = x2iii the y-intercept.

a y = 4x2 − 1 b y = + 3 c y = + 4d y = −4 + e y = −3 + x2 f y = 2x2 −

0

y

−8

x

y

x0

(0, 12)y

x0

−8

y

x0

0

y

x

(0, 12)

0.5

y

x0

y

x0

0.5

y

x0

8

0

y

x

(0, −12)

12---),

12---x2 1

5---x2

12---x2 1

2---

Example 10

Spreadsheet 11-01

The graph of y = ax2 + c



5 Match the following graphs with the equations below:

a b c

d e f

g h i

j k l

i y = 5x2 ii y = 2x2 + 1iii y = − 1 iv y =

v y = 2x2 − 1 vi y = −5x2

vii y = − + 1 viii y = −

ix y = −2x2 − 1 x y = + 1

xi y = −2x2 + 1 xii y = − − 1

0

y

x

(2, 9)

1

0

y

x

1

(2, −1)

0

y

x

(−5, 5)

0

y

x1

(−2, 3)

0

y

x

(2, 7)

−1

0

y

x−1

(−2, −9)

(5, 125)

0

y

x

1

(−2, −7)

0

y

x (5, −5)0

y

x

(−2, −3)−10

y

x

(−5, −125)

0

y

x

(2, 1)

−10

y

x

12---x2 1

5---x2

12---x2 1

5---x2

12---x2

12---x2



6 For each of the following tables:i draw the graph of the results and join the points with a smooth curve

ii find the equation relating the two variables in the form y = ax2 + ciii compare your answers with those of students in other groups.

a

b

c

d

7 A stone is dropped from a cliff and its height (h metres) at any time (t seconds) is given by h = 80 − 4.8t2.a Draw a graph of the equation for 0 � t � 5.b What is the height of the cliff?c How far will the stone have dropped after 3 seconds?d When will the stone hit the ground?

8 A farmer wants to build a rectangular pig pen of the greatest possible area with 40 m of fencing materials.a If L and B are the length and breadth of the rectangular pig pen,

explain why L + B = 20.b If L = 6 m, find B and show that the area, A, is 84 m2.c i Copy and complete this table for the length and area of the

pig pen.

ii Graph the results from the table. Use a scale of 1 cm = 2 m on the horizontal axis (length) and a scale of 1 cm = 10 m2 on the vertical axis (area).

iii Join the points with a smooth curve.

d i Describe the graph that you drew in part c.ii What is the maximum area of the pig pen that can be

built using only 40 m of fencing?iii What are the dimensions of the pig pen that has

maximum area?

x 0 1 2 3 4 5 6

y 0 1 4 9 16 25 36

x −3 −2 −1 0 1 2 3

y 8 3 0 −1 0 3 8

x −3 −2 −1 0 1 2 3

y −9 −4 −1 0 −1 −4 −9

x −4 −3 −2 −1 0 1 2 3 4

y −13 −6 −1 2 3 2 −1 −6 −13

Pig pen B

L

L 0 1 2 … 20

A

A

L

(area)

(length)



Working mathematicallyApplying strategies and reasoning: Graphs of parabolas1 In y = ax2, what is the effect of a?2 In y = ax2 + k, what is the effect of k?3 a i Draw the graph of y = x2 for −4 � x � 4. (Use a scale of 1 cm to 1 unit on the x-axis

and y-axis.)ii Copy and complete this table of values for y = (x − 1)2:

iii Plot the points (from the table of values) on the set of axes used in part i. Draw a smooth curve through all the points.

b i What is the equation of the axis of symmetry of y = (x − 1)2?ii Write the coordinates of the vertex of the graph y = (x − 1)2.

iii What is the minimum value of y = (x − 1)2?iv How are the graphs of y = x2 and y = (x − 1)2 the same?v How are the two graphs in part iv different?

vi Describe how the graph of y = (x − 1)2 could be obtained from the graph of y = x2.Compare your answers with those of other students.

4 a Repeat the procedure from Question 3 for the parabolas:i y = (x + 1)2 ii y = (x − 2)2 iii y = (x + 2)2

b On the same set of axes draw these graphs:i y = −x2 ii y = −(x − 1)2 iii y = −(x + 1)2

iv y = −(x − 2)2 v y = −(x + 2)2c For the graph of the parabola y = (x ± h)2, describe the effect on the graph for the

different values of the constant, h.

5 a i Draw the graph y = x2 using a scale of 1 cm to 1 unit on both the x-axis and y-axis.ii Copy and complete this table of values for y = (x − 1)2 + 2:

iii Plot the points on the same set of axes used in part i. Draw a smooth curve through all the points.

b i What is the equation of the axis of symmetry of y = (x − 1)2 + 2?ii What are the coordinates of the vertex of y = (x − 1)2 + 2?

iii What properties do the graphs of y = x2 and y = (x − 1)2 + 2 have in common?iv How are the two graphs from part iii different?

c How can the graph of y = x2 be moved to the position of y = (x − 1)2 + 2?d By comparing them to y = x2, comment on the shape and position of these graphs:

i y = (x − 1)2 − 2 ii y = (x + 1)2 − 2 iii y = (x + 1)2 + 2e On the same set of axes, draw the graphs of these equations:

i y = −x2 ii y = −(x − 2)2 − 3iii What is the equation of the axis of symmetry of y = −(x − 2)2 − 3?iv What are the coordinates of the vertex of y = −(x − 2)2 − 3?v How can the graph of y = −x2 be moved to the position of y = −(x − 2)2 − 3?

f On the same set of axes, sketch the graphs of the equations y = (x − 1)2 + 2 and y = 2(x − 1)2 + 2 and comment on their shape. Compare your results with those of other students.

x −4 −3 −2 −1 0 1 2 3 4

y

x −3 −2 −1 0 1 2 3

y

Worksheet 11-05

Matching parabolas

STAGE5.3



The parabola y = ax2 + bx + cThe equation y = a(x ± h)2 + k is sometimes knows as the vertex form of the quadratic relationship or parabola. This equation can be expanded and simplified to give the general form of the quadratic relationship.

g For the graph of the parabola y = (x ± h)2 + k, describe the effect on the graph for different values of the constants h and k. Compare your answers with those of other students.

6 Sketch the graph of each of the following equations:a y = (x + 5)2 + 3 b y = −(x − 1)2 − 1 c y = (x + 2)2 − 3 d y = (x − 2)2 + 4Compare your graphs with those of other students in your class.

7 Write the equation of each of the following parabolas:

a b c

Compare your equations with those of other students in your class.

8 a i Copy and complete this table of values for y = (x + 2)(x − 2):

ii Plot the points from the table of values and draw a smooth curve through all the points.

b i What are the x-intercepts of the graph you drew in part a?ii What is the y-intercept?

c Draw neat sketches of:i y = (x + 1)(x − 1) ii y = (x + 3)(x − 3) iii y = −(x + 2)(x − 2)

d Compare your results with those of other students in your class.

x −4 −3 −2 −1 0 1 2 3 4

y

3

9

0

y

x

10

y

x

(−2, −3)

4

20

y

x

(2, 4)

Worksheet11-03

Graphing parabolas

The general form of the quadratic relationship is y = ax2 + bx + c where a, b and c are real numbers and a ≠ 0. The graph of this relationship is a parabola.

Worksheet11-05

Matching parabolas

Working mathematicallyApplying strategies and reasoning: Graphing y = ax2 + bx + c1 a i Copy and complete this table of values for y = x2 − 2x − 3:

x −3 −2 −1 0 1 2 3 4 5

y



When drawing parabolas of the form y = ax2 + bx + c:• factorise y = ax2 + bx + c and solve the quadratic equation to find the x-intercepts whenever

possible• find the y-intercept of y = ax2 + bx + c• if a > 0, the parabola is concave up and, if a < 0, the parabola is concave down• sketch the parabola.

ii Plot the points and draw the graph of y = x2 − 2x − 3.iii What are the x-intercepts?iv What is the y-intercept?

b Factorise x2 − 2x − 3 and, hence, solve the equation x2 − 2x − 3 = 0.c What do you notice about the x-intercepts of y = x2 − 2x − 3 and the solution to

x2 − 2x − 3 = 0?

2 What is the y-intercept of the parabola y = ax2 + bx + c?

3 For each of the following equations:i factorise and solve the quadratic equation to find the x-intercepts

ii find the y-interceptiii sketch the parabola.

a y = x2 + 5x + 4 b y = 2x2 + 7x − 15 c y = x2 − 7x − 30(Compare your results with those of other students.)

The parabola y = ax2 + bx + c1 For each of the following parabolas, solve the quadratic equation ax2 + bx + c = 0:

a y = x2 + 3x + 2 b y = 2x2 + 1x + 15 c y = x2 − 2x − 15

2 Use a graphics calculator or spreadsheet to draw the graphs of the equations from Question 1.

3 Find the x-intercepts and y-intercept of the graphs drawn in Question 2 and compare them with the results from Question 1.

Using technology

Example 11

Graph each of the following:a y = x(2x − 3) b y = 2x2 − 2x − 3 c y = -x2 − 3x + 10

Solutiona y = x(2x − 3)

x-intercepts: x(2x − 3) = 0x = 0 and 2x − 3 = 0

∴ x = 0 and x = y-intercept: When x = 0, y = 0Also: for y = x(2x − 3)

y = 2x2 − 3x∴ a = 2 > 0

So: y = x(2x − 3) is concave up.11 20

y

x1_2

112---

STAGE5.3



b y = 2x2 − 2x − 3= (2x − 3)(x + 1)

x-intercepts: (2x − 3)(x + 1) = 0∴ x = and x = −1

y-intercept: y = −3 (Why?)Since a = 2 > 0, the parabola is concave up.

c y = −x2 − 3x + 10= −(x2 + 3x − 10)= −(x + 5)(x − 2)

x-intercepts: −(x + 5)(x − 2) = 0∴ x = −5 and x = 2

y-intercept: y = 10Since a = −1, the curve is concave down.

−3

1−1−2 1 20

y

x1_211

2---

2

10

−5 0

y

x

1 Draw the graph of each of the following parabolas by first finding the x-intercepts and the y-intercept:a y = x(x − 4) b y = 3x(x + 2) c y = (x + 3)(x − 5)d y = (x − 2)(2x − 5) e y = −(x + 1)(3x + 2) f y = x2 + 6x + 8g y = −x2 + 4x + 5 h y = 3x2 + 11x − 20 i y = −2x2 + 5x − 2


CAS 11-04

Sketching parabolas

Working mathematicallyQuestioning and reasoning: The axis of symmetry and vertex of y = ax2 + bx + c1 a The graph of the parabola y = x2 − 6x + 8 and its axis of

symmetry have been drawn.b Explain why the equation of the axis of symmetry is x = 3.c i What are the x-intercepts of the parabola

y = x2 – 6x + 8?ii What is the midpoint of the interval joining the

x-intercepts?d How can the x-intercepts be used to find the equation of

the axis of symmetry? (Use your results from parts b and c.)

e Copy and complete:The equation of the axis of symmetry can be found by finding the a_______ of the x-intercepts. 2 4

axis

of

sym

met

ry

8

0

y

x



The axis of symmetry and the vertex of a parabola

The axis of symmetry gives the x-coordinate of the vertex and this value is substituted in the equation of the parabola to find the y-coordinate of the vertex.

2 a For y = x2 − 6x + 8, what is the value of:i a (the coefficient of x2)

ii b (the coefficient of x)

b Find the value of − .

c How is the value of − related to the equation of the axis of symmetry of the parabola y = x2 − 6x + 8?

d Copy and complete:

x = − is the e_______ of the axis of ___________ of the parabola y = ax2 + bx + c.

3 The vertex of a parabola is where the axis of symmetry intersects with the parabola.a The equation of the axis of symmetry of y = x2 − 6x + 8 is x = 3. The coordinates of

the vertex are (3, −1). Explain why the y-coordinate of the vertex is −1.b The equation of the axis of symmetry of y = −x2 + 4x − 7 is x = 2. Show that the

coordinates of the vertex are (2, −3).

b2a------

b2a------

b2a------

If the equation of a parabola is in the general form y = ax2 + bx + c, the equation of its axis of symmetry is x = − .b

2a------

Worksheet 11-06

Features of a parabola

Worksheet 11-07

A page of parabolas

Example 12

Find the equation of the axis of symmetry of the parabola in the diagram on the right.

SolutionThe x-intercepts are −1 and 3.

∴ The equation of the axis of symmetry is:

x = (average of x-intercepts)

∴ x = 1

For the parabola with equation y = 2x2 − 4x + 3, find:a the equation of its axis of symmetryb the coordinates of the vertex of the parabola

−1 30

y

x−1 3+

2---------------

Example 13

STAGE5.3



Finding the vertex by completing the squareThe method of completing the square can be used to find the coordinates of the vertex of a parabola.

Solutiona For y = 2x2 − 4x + 3, a = 2, b = −4 and c = 3.

∴ The equation of the axis of symmetry is: x = −

= −

= 1∴ x = 1 is the equation of the axis of symmetry.

b The vertex lies on the axis of symmetry.∴ Substitute x = 1 in y = 2x2 − 4x + 3∴ y = 2 × 12 − 4 × 1 + 3

= 1∴ Coordinates of the vertex = (1, 1)

b2a------

-42 2×------------

Example 14

Complete the square to find the coordinates of the vertex of the parabola y = x2 + 8x + 5.

Solutiony = x2 + 8x + 5

= (x2 + 8x + 16) − 16 + 5 (completing the square on x2 + 8x)= (x + 4)2 − 11

The equation of the parabola has been expressed in vertex form.

∴ Vertex = (−4, −11) (The axis of symmetry of y = x2 + 8x + 5 is x = −4)

Sketch the parabola y = 2x2 + 5x + 1.

SolutionThe equation y = 2x2 + 5x + 1 cannot be factorised so we use the vertex and y-intercept to sketch the parabola.

The equation of the axis of symmetry is:

x = − where a = 2, and b = 5

= −

= −

Substitute x = − in y = 2x2 + 5x + 1

∴ y = 2 × + 5 × + 1

= −

∴ The vertex has coordinates .Also: y-intercept, y = 1.

Example 15

b2a------

52 2×------------

114---

0

1

y

x

(−1 , −2 )1_41_8

114---

−114---⎝ ⎠

⎛ ⎞ 2 −114---⎝ ⎠

⎛ ⎞

218---

−114--- −21

8---,⎝ ⎠

⎛ ⎞



SummaryTo sketch parabolas (or quadratic relationships) of the form y = ax2 + bx + c:• find the x-intercepts (where possible)• find the y-intercept• find the equation of the axis of symmetry and obtain the coordinates of the vertex• use the sign of a (the coefficient of x2) to determine concavity• then use the above information to sketch the parabola

1 Write the equation for the axis of symmetry of each parabola.

a b c

d e f

2 For each parabola whose equation is given below:i find the equation for the axis of symmetry of the parabola

ii find the coordinates of the vertex of the parabola

a y = x2 − 6x + 8 b y = −x2 + 6x − 9 c y = x2 − 6x + 10d y = 9 + 8x − x2 e y = 8x − 25 − x2 f y = 5x2 + 40xg y = 40x − 5x2 h y = 4x2 + 2x − 1 i y = 1 − 3x − 9x2

3 Use the method of completing the square to find the coordinates of the vertex of the parabola for each of the following equations:a y = x2 + 4x + 1 b y = −x2 + 2x + 5 c y = x2 + 5x − 4d y = x2 − 6x e y = 2 − 2x + x2 f y = 3x − x2 + 1

4 For the graph of each of the following equations find:i the x-intercepts (where possible)

ii the y-interceptiii the equation of the axis of symmetryiv the coordinates of the vertexv the concavity

Hence sketch each parabola:

a y = x2 − 6x − 40 b y = x2 − 3x c y = 2x2 + 3x + 4d y = −x2 + 6x + 5 e y = 21 − 12x − 4x2 f y = x2 − 8x + 3g y = 5x2 + 7x + 4 h y = 8x − 2x2 i y = −2x2 + 7x − 6

0 6

y

x0 5−5

y

x0 2.5

y

x

0−4 2

y

x0−1 4

y

x

0

(3, 4)

y

x

Exercise 11-09

Example 13

Example 12

Example 14

Example 15

CAS 11-04

Vertex of a parabola

STAGE5.3



The cubic curveEquations such as y = x3, y = x3 − 4 and y = 3x3 + 1 are examples of cubic relationships, in which the highest power of x is 3.

The cubic curve y = x3The equation y = x3 is a cubic relationship. The graph of the equation y = x3 is a smooth curve called a cubic.Features of y = x3 are as follows:• The graph has no axis of symmetry but it does have rotational

symmetry of 180° (called point symmetry).• The graph passes through (0, 0).• The graph increases very sharply (compared to y = x2).• When x is negative, y is also negative.

Working mathematicallyApplying strategies, reasoning and communicating: Graphing cubics1 a Copy and complete this table of values for y = x3, where values

of x are taken from −2 to 2:

b On grid paper, draw a set of axes, similar to those shown, using a scale of 1 cm to 1 unit on both the x-axis and the y-axis.

c Plot the points from the table of values onto the grid paper.d Draw a smooth, continuous curve through all the points.e Compare your graph with those of other students.

2 List as many features as you can about the graph of y = x3 that you have drawn. Compare your list with those of other students.

x −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

y

0

8

−2

6

−4

−4 2−2 4

4

−6

2

−8

y

x

y

x

y = x3

0

Working mathematicallyApplying strategies, reasoning and communicating: Graph of y = x31 Using a scale of 1 cm = 1 unit on both the x-axis and the y-axis, draw the graph of y = x3

for −3 � x � 3. (Use only values of x from −3 to 3, inclusive.)

2 a On the same set of axes, draw the graph of the reflection of y = x3, when it is reflected in the y-axis.

Worksheet11-08

Graphing cubics 1



The cubic curve y = –x3The graph of the equation of y = −x3 is a cubic.Features of y = −x3 are as follows:• It is a reflection of y = x3 in the y-axis.• The graph has no axis of symmetry, but it does have point symmetry.• The graph passes through (0, 0).• The graph decreases very sharply.

The cubic curve y = ax3

As the size of a in the rule above increases, the graph of the cubic becomes ‘narrower’ and as the size of a decreases, the graph of the cubic ‘widens’. The cubic graphs below illustrate this:

b Use the reflection of y = x3 to help you copy and complete this table:

c Use the graph of the reflection and the values in the table above to determine the equation of the reflection of y = x3.

d Compare your graph and equation with those of other groups.

3 a List the features that are the same for both cubics.b List the features that are different for both cubics.

x −3 −2 −1 0 1 2 3

y −8

0

y

x

y = −x3

For the graph of y = ax3, the size of a (the coefficient of x3) determines whether the cubic is ‘wider’ or ‘narrower’ (when computed to y = x3).

0

y

x

y = x3

y = 2x3

0

y

x

y = x3

y = x31_2

STAGE5.3



The cubic curve y = ax3 + d

The cubic y = ax3 + d (or wide, narrow, up or down)1 a Use a graphics calculator or a spreadsheet to draw the graphs of the following equations

on the same set of axes:i y = x3

ii y = 2x3 + 1iii y = 2x3 − 3

b How are the graphs of y = 2x3 + 1 and y = 2x3 − 3 different from the graph of y = x3?c What is the y-intercept of each of the cubics drawn in part a?

2 a Use a graphics calculator or spreadsheet to draw the graphs of each of the following equations on the same set of axes:

i y = x3

ii y = + 3

iii y = − 4

b How are the graphs of y = and y = − 4 different from the graph of y = x3?c What is the y-intercept of each of the cubics drawn in part a?

12---x3

15---x3

12---x3 1

5---x3

Using technology

The effect of the constant term d on the graph of y = ax3 + d is to move the cubic curve up or down.

(The constant term d is also the y-intercept of the graph y = ax3 + d.)

Example 16

For each of the following, state:i whether the cubic graph is wider or narrower than the graph of y = x3

ii whether the cubic graph has moved up or down when compared to the graph of y = x3iii the y-intercept.

a y = 3x3 − 1 b y = + 2

Solutiona i The coefficient of x3 is 3, while the coefficient of x3 in y = x3 is 1.

∴ The cubic will be narrower than y = x3.ii The constant term is −1.

∴ The cubic has moved down.iii The y-intercept is −1.

b i The coefficient of x3 is .

∴ The cubic will be wider.ii The constant term is 2.

∴ The cubic has moved up.iii The y-intercept is 2.

13---x3

13---

Worksheet11-04




1 The graph of which cubic in each pair will be ‘narrower’ when compared with the other?a y = x3 or y = 3x3 b y = 3x3 or y = 2x3

c y = or y = 0.5x3 d y = −x3 or y = −2x3

e y = −3x3 or y = − f y = − or y = −10x3

2 For each of the following, draw the graph of y = x3. Then:i draw a sketch of the given equation

ii find the y-coordinate of the point on the graph of the cubic where x = 3.

a y = 4x3 b y = −2x3 c y = x3

d y = 6x3 e y = − x3 f y = 2x3

g y = −3x3 h y = 3x3 i y = x3

3 For each of the following cubics, state:i whether its graph is wider, narrower or the same when compared to the graph of y = x3

ii whether its graph has moved up or down when compared to the graph y = x3.

a y = 4x3 − 1 b y = x3 + 3 c y = x3 + 4

d y = −4 + x3 e y = −3 + x3 f y = 2x3 −

4 Match each of the following graphs with its equation on the next page:

a b c

d e f

g h i

14---x3

15---x3 1

10------x3

13---

12---

14---

12--- 1

5---

12--- 1

2---

01

(2, 17)y

x 01

(2, −3)

y

x 0

1

(−2, −3)

y

x

0−1

(2, 15)y

x 0−1

(−2, 15)

y

x 0

(−2, −4)

y

x

01

(−2, 17)

y

x

0−1

(2, −5)

y

x 0−1

(2, 3)

y

x

Exercise 11-10

Example 16

Spreadsheet 11-02

Sketching cubics

STAGE5.3



i y = 2x3 + 1 ii y = x3 − 1 iii y = 2x3 − 1

iv y = x3 v y = − x3 + 1 vi y = −2x3 − 1

vii y = x3 + 1 viii y = −2x3 + 1 ix y = − x3 − 1

12---

12--- 1

2---

12--- 1

2---

Working mathematicallyApplying strategies, reasoning and communicating: The graph of y = 1 a Copy and complete the following table for y = :

b Explain why no y-value exists for x = 0.

2 a Plot the points using a scale of 1 cm = 1 unit on both axes.b What do you notice about the position of the points you have plotted? In which

quadrants do the points lie?

3 There are two parts or ‘branches’ to your graph. Draw a smooth curve to join the points of each branch.

4 a Use your graph to explain what happens to the y-value as x becomes very largeb Explain what happens to the y-value as x approaches 0.

5 The graph of y = has two axes of symmetry. Draw them on your graph.

6 a Complete this table of values for y = − :

b Plot the points and draw the graph.

c How does the graph of y = − compare with that of y = ?

7 Compare your results for this activity with those of students in other groups.

x −5 −4 −3 −2 −1 −0.5 −0.2 −0.1 0 0.1 0.2 0.5 1 2 3 4 5

y

x −5 −4 −3 −2 −1 −0.5 −0.2 −0.1 0 0.1 0.2 0.5 1 2 3 4 5

y

1x---

1x---

1x---

1x---

1x--- 1

x---



The hyperbola

Features of the hyperbola• The graph has two parts, called

branches.• There is no x-intercept or y-intercept.

The graph approaches, or gets very close to, the x-axis and y-axis but never intersects with them. The x-axis and y-axis are asymptotes.

• The graph has two axes of symmetry. Their equations are y = x and y = −x.

Working mathematically

Reasoning and communicating: y = 1 a On the same set of axes draw the graphs of these equations:

i y = ii y = iii y = iv y =

b Compare these graphs. What happens to the graph of y = as k increases?

2 a On separate sets of axes draw the graphs of these equations:

i y = ii y = −

b How are the two graphs related?

c Sketch the graphs of y = and y = − on the same set of axes.

3 a For y = complete this table of values:

b What happens to the y-values when the x-values get very large?

c For y = complete this table of values:

d What happens to the y-values when the x-values get very small or closer to zero?

4 Copy and complete the following:

For y = :

a the y-values become very s as the x-values become very l .b the y-values become very l as the x-values get closer to zero.

x 1 2 5 10 100 200 1000

y

x 0.0001 0.01 0.1 0.5 1 5

y

kx---

1x--- 2

x--- 5

x--- 10

x------

kx--

2x--- 2

x---

3x--- 3

x---

4x---

4x---

kx--

The general equation for the graph of a hyperbola is y = , k ≠ 0, where k is a constant.kx--

Worksheet 11-09

Graphing hyperbolas

Worksheet 11-04


0 x

y

y = x

y =

y = −x

1–x

x

y

y = −x

y = xy = − 1–x

0



• The higher the value of k, the further the hyperbola is from the x-axis and y-axis.• As x becomes very large, y gets closer to 0.• As x gets closer to 0, y becomes very large.• When k is negative, the branches of the hyperbola are in different quadrants from those they are

in when k is positive.

The hyperbola in actionIn astronomy, the orbits of some comets are hyperbolic in shape. These comets may have a sufficiently high velocity and a large enough mass, and may be too far away, to be captured by the Sun’s gravitational pull. Such comets can follow a hyperbolic orbit. The Voyager space probes also followed a hyperbolic orbit (sometimes called an ‘open orbit’), passing near a planet before leaving the solar system.

On a much smaller scale, protons bombarding an atomic nucleus may also follow a hyperbolic path. As a proton (which has a positive charge) approaches an atomic nucleus with a positive charge, it is repelled and is deflected from its original path. The trajectory followed by the proton as it nears and then departs from the nucleus is a hyperbola.

The hyperbola is also used in navigation. Write a brief report on the use of the hyperbola in the long range electronic navigation (loran) system, or on any other use of the hyperbola you find interesting.

Some comets follow a hyperbolic orbit.

Tracks produced by particles when a high speed proton strikes an atom.

Just for the record

Example 17

Draw a neat sketch of each of these hyperbolas and mark a point on the curve:

a y = b y = −

Solutiona y =

Let x = 5 (any value of x could be chosen)∴ y =

= 2A point on the curve is (5, 2).

b y = −

Let x = 2 (any value of x could be chosen)∴ y = −

= −1A point on the curve is (2, −1).

0

y

x

(5, 2)

10x

------ 2x---

10x

------

105

------

0

y

x(2, −1)

2x---

22---



The exponential curveThe graph of y = ax (where a is positive) is called the exponential curve.The variable x is called the exponent (power) of a in the expression ax.

Features of exponential graphs• The y-intercept is 1 since a0 = 1.• As x approaches a large negative number, ax approaches zero. This means the

graph of y = ax approaches the x-axis as x approaches a large negative number. The x-axis is an asymptote because the curve approaches it but never touches it.

• As x approaches a large positive number, ax becomes very large. Graphically, this means the graph of y = ax increases very rapidly.

1 On separate sets of axes, sketch the graphs of the following hyperbolas. Clearly mark a point on the curve each time.

a y = b y = − c y =

d y = e y = − f y = −

g y = h y = −

4x--- 2

x--- 20

x------

5x--- 8

x--- 16

x------

12x

------ 1x---


Working mathematicallyApplying strategies, reasoning and communicating: The graph of y = 2x

1 a Copy and complete the following table for y = 2x:

b Draw the graph y = 2x by plotting the points from the table and joining them with a smooth curve.

2 Repeat the procedure from Question 1 to draw the graph of y = 2−x.

3 Compare the graphs of y = 2x and y = 2−x. Describe:a any similarities b any differences

4 The y-intercept of any graph that has an equation y = ax (where a is a positive integer) is always 1. Explain why.

5 a Explain why we say the graph of y = 2x is an increasing function.b Is the graph of y = 2−x increasing or decreasing? Give reasons.

6 Describe what happens to the graph of y = 2x when:a x approaches a large positive numberb x approaches a large negative number.

x −3 −2 −1 0 1 2 3 4

y

CAS 11-05

Hyperbolas

Spreadsheet 11-03

The graph of

y = kx--

Worksheet 11-10

Graphing exponentials

Worksheet 11-04


0

1

y

x

y = ax

STAGE5.3



For the graph of y = 4x, the x-axis is an asymptote.

0

1

(1, 4)

y

x

y = 4x

1 a On the same set of axes draw the graphs of these equations:i y = 2x ii y = 3x iii y = 5x

b What is the y-intercept of each graph?c Describe what happens to the graph y = ax, as a increases.

2 a On the same set of axes draw the graphs of these equations:i y = 3x ii y = 3−x

b Copy and complete:i The reflection of y = 3x in the y-axis is …

ii The reflection of y = ax in the y-axis is …

3 a On the same set of axes draw the graphs of these equations:i y = 2x ii y = −2x

b On the same set of axes draw the graphs of these equations:i y = 3x ii y = −3x

c How are the pairs of graphs you have drawn related?d Copy and complete:

The reflection of y = ax in the x-axis is …

4 State whether these graphs are increasing or decreasing:

a b

c d

5 Sketch the exponential curves for the following equations showing where each curve cuts the y-axis:a y = 2x b y = −4x c y = 3−x d y = −2−xe y = 4−x f y = −5−x g y = 2−x h y = −10−x

0

1

y

x

0−1

y

x

(−2, −16)

0−1

y

x

(−2, 16)

0

1

y

x

Exercise 11-12CAS

11-07Sketching

exponentials

Spreadsheet 11-04

Exponential graphs



Exponential growth and decayWhen change can be described using an exponential equation, the change is either exponential growth or exponential decay. Examples of exponential growth are the growth of population (people and bacteria) and monetary investments. Examples of exponential decay include light a

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