Part 2 Graphing lines -...

Mathematics Stage 5

PAS5.1.2 Coordinate geometry

Part 2 Graphing lines

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Extracts from Mathematics Syllabus Years 7-10 © Board of Studies 2003 Unit overviewpp iii-vPart 1 p.4, 5 Part 2p. 3, Part 3 p. 4.

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Part 2 Graphing lines 1

Contents – Part 2

Introduction – Part 2..........................................................3

Indicators ...................................................................................3

Preliminary quiz.................................................................5

Vertical lines......................................................................9

Gradients of vertical lines..........................................................9

Graphing vertical lines ............................................................11

Horizontal lines................................................................17

Gradients of horizontal lines ...................................................17

Graphing horizontal lines ........................................................18

Intercepts .......................................................................23

Graphing lines .................................................................29

How many points?...................................................................31

Suggested answers – Part 2 ...........................................41

Exercises – Part 2 ...........................................................49

2 PAS5.1.2 Coordinate geometry


Introduction – Part 2

Each line on the number plane can be described using an equation.

In this part you will explore these equations and how they relate to

the graph.

Indicators

By the end of part 2, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• understanding terms such as algebra, substitute, coordinate, horizontal,

vertical, oblique, equation, intercept, linear and intersect

• using coordinates to graph vertical and horizontal lines

• identifying the x-axis as the line y = 0

• identifying the y-axis as the line x = 0

• identifying the x- and y-intercepts of graphs

• constructing tables of values for a variety of linear equations

• graphing a variety of linear equations on the number plane.

By the end of part 2, you will have been given the opportunity to work

mathematically by:

• describing vertical and horizontal lines and their properties

• explaining why the axes have equations.Source: Adapted from outcomes of the Mathematics Years 7–10 syllabus

<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.



Preliminary quiz

Before you start this part, use this preliminary quiz to revise some skills you

will need.

Activity – Preliminary quiz

Try these.

1 Plot these points on the number plane below and label each with its

capital letter.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

A (–1, 5)

B (4, –4)

C (0, 0)

D (5, 0)

E (0, –3)

2 If k = 10 find the value of these expressions.

a k + 5 _______________________________________________

___________________________________________________

b 5k _________________________________________________

___________________________________________________

c 8 −k

2 ______________________________________________

___________________________________________________


3 Write the coordinates of the points shown on the number plane below.

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3

A

B

C

Dx

y

A _________

B _________

C _________

D _________

4 Use the diagram below to answer the following questions.

Remember that to name a line you use two capital letters for points on

that line.

A

B

C

D

E F

G

H

I

a What is the name of the vertical line? _____________________

b What is the name of the horizontal line? ___________________

c Where do EF and GH intersect? _________________________

5 Complete the following sentences.

a The x-coordinate for (5, 7) is ___________________________ .

b The y-coordinate for (–2, 0) is __________________________ .


6 Evaluate these number sentences. (Evaluate means find the answer.)

a 6 – –2 ______________________________________________

___________________________________________________

b−4 +10

2 ____________________________________________

___________________________________________________

7 Solve these equations by finding the number that the pronumeral stands

for.

a x + 7 = 9

___________________________________________________

___________________________________________________

b 5y = 30

___________________________________________________

___________________________________________________

c 7 − y = 9

___________________________________________________

___________________________________________________

dx

3= 10

___________________________________________________

___________________________________________________

Check your response by going to the suggested answers section.



Vertical lines

Lines can be described as either:

• horizontal (straight across)

• vertical (straight up and down)

• oblique (at an angle, not vertical or horizontal).

In this section you will explore aspects of vertical lines. The first feature

that will be discussed about vertical lines is their gradient.

Gradients of vertical lines

On the number plane, oblique lines are said to have a positive gradient

if they go up the graph from left to right, or a negative gradient if they

go down.

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3

Positive gradients Negative gradients

x

y

x

y

But what about vertical lines?

Work though the following task to explore this question.


For this task, you will need a pencil and a ruler.

Use the vertical line shown on the graph below to complete this activity.

0

1

2

3

4

5

–1

1 2 3 4 5 6–1 x

y

Form earlier work you know that gradient = (+ or –)rise

run.

The first thing to consider is whether the slope is positive or negative. Does

the line go up or down the graph as it moves from left to right?

Since the line doesn’t move left to right, the first problem that arises in

finding the gradient of a vertical line is that you cannot decide whether it is

positive or negative.

Ok, put this problem aside for the moment and move on.

Next you need to calculate the number part of the gradient. Use the two

points marked on the vertical line above to find the rise and the run.

rise = __________ run = __________

The rise is easy. You need to move 4 units from one point to the other.

But what is the run? Strictly speaking, the run is zero because you do not

need to move across at all to get from one point to the other.

Therefore, gradient = 4

0

Oops, another big problem! You cannot divide by zero. Even on a

calculator the answer comes up as an error.


Summarising what you have found:

• the gradient is neither positive nor negative

• the run is zero so the number for the gradient cannot be calculated.

So the gradient of a vertical line cannot be calculated using the

usual technique.

In fact, vertical lines are said to have an undefined gradient.

This means that you cannot give a number for their gradient.

The next section explores other aspects of vertical lines.

Graphing vertical lines

The points that make up a vertical line have something in common.

Find what that is by completing the following activity.

For this activity you will need a pen and ruler.

Use the diagram below to list the coordinates of all the points marked on the

vertical line.

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3

A

B

C

D

x

y

0

A ____________

B ____________

C ____________

D ____________

What have all the points got in common?

___________________________________________________________

___________________________________________________________


You should have found that all the points on a vertical line have the same x-

coordinate of 3. In fact, any vertical line you draw will have the same

property, that is, all the points will share the same x-coordinate.

This means that to describe which vertical line you want to draw, all you

have to do is describe the x-coordinate.

Algebra is a tool in mathematics that is used to write general rules without

using words.

To describe the vertical line in this

diagram using English you can say

‘the line where all the points have

an x-coordinate of 3’.

To describe this line using algebra

you can just write ‘the line x = 3’.0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y

This equation, x = 3, is said to be the equation for the line. It is really

saying that the y-coordinate can be anything as long as the x-coordinate is 3.

When you are not talking about number planes and equations for lines,

using x = 3 might mean something very different.

Look at the following example.

Follow through the steps in this example. Do your own working in the

margin if you wish.

a Graph the line x = 1.

b Does the point (4, 1) lie on this line?


Solution

a To draw the line you need to recognise the equation as a

vertical line then use a ruler to draw it at the correct place

on the graph.

For this example, the line goes through 1 on the x-axis.

The graph below shows the line.

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y

b The point (4, 1) does not lie on the line. The x-coordinate

(the first one) is not 1.

You can also see this easily if you find the point on the

graph above.

Continue to explore vertical lines by completing the following activity.

Activity – Vertical lines

Try these.

1 How do you know that (15, 9), (15, 0) and (15, –2) all lie on the same

vertical line?

_______________________________________________________

_______________________________________________________

_______________________________________________________


2 a Plot these points on the number plane and draw a line to show that

they all lie on the same vertical line.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y (–1, 6)

(–1, 3)

(–1, 1)

(–1, –2)

(–1, –5)

b Complete the equation for this line. x = __________

3 Write the equation for each of these vertical lines.

a

0

1

2

3

–1

1 2 3 4 5 6–1 x

y

b

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

1

–7–8–9 x

y


4 Graph these vertical lines on the number plane provided.

Write the equation along each line on the graph.

a x = 2

b x = –3

c (Harder) x = 41

2

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y


You have seen that the points on vertical lines have one thing in common:

they all have the same first coordinate.

You have also seen that you can describe each vertical line using a simple

equation: x = a where a is a number.

But there is one special vertical line that has not been discussed and that is

the y-axis. This line also has an equation to describe it, x = 0. So even the

vertical axis can be described using algebra.

Continue to explore your understanding of vertical lines on the number

plane by completing this exercise.

Go to the exercises section and complete Exercise 2.1 – Vertical lines.



Horizontal lines

In this section you will explore features of horizontal lines on the

number plane.

Gradients of horizontal lines

What is the gradient of a horizontal line?

Explore this question by completing the following activity.

For this activity you will need a pen.

Use the graph of the horizontal lines below to complete this activity.

0

1

2

3

4

5

–1

1 2 3 4 5 6–1 x

y

Answer these three questions before reading further.

a Is the gradient positive, negative or neither? ___________________

b What is the rise between the two points shown? ________________

c What is the run between the two points shown? _________________


Your answers to these questions should have been.

a The gradient is neither positive nor negative (because the horizontal line

does not go up or down the graph)

b The rise is 0 (because you do not move up or down to travel from one

point to another)

c The run is 5.

Using this information, you can determine the gradient of this

horizontal line.

gradient = (+ or −) rise

run

= (+ or −) 0

5= (+ or −) 0

Since zero is neither positive nor negative, the problem of not being able to

decide on the sign of the gradient is solved.

Horizontal lines have a gradient of zero because they are perfectly flat.

And this is true for all horizontal lines.

Graphing horizontal lines

You have already seen that vertical lines can be described using equations

like x = 4 or x = –7. This means that all the points on a vertical line have the

same x coordinate.

In this section you will consider if a similar equation can be written for

horizontal lines.


For this activity you will need a pen and ruler.

Plot all these points and join them with a straight line.

0

1

2

3

4

5

–1

1 2 3 4 5 6–1–2–3–4–5–6 x

y (5, 3)

(–1, 3)

(–5, 3)

(0, 3)

(2, 3)

You should have graphed a horizontal line.

What do all the points have in common?

___________________________________________________________

___________________________________________________________

What do you think the equation of this line might be? Use the equation

of vertical lines to help you.

Each point on a horizontal line has the same y-coordinate (second number).

Therefore to describe which horizontal line you want to graph you simply

have to say what the y-coordinate is.

To describe the horizontal

line shown in this graph you

can say ‘all the points with a

y-coordinate of –2’.

Or you can use the algebraic

equation y = –2.

–3

0

1

2

–1

–2

1 2 3 4 5 6–1–2–3–4–5–6 x

y

The following example shows how to graph horizontal lines and how to

write their equations.



margin if you wish.

Use the number plane below for these tasks.

0

1

2

3

4

5

–1

–2

1 2 3 4 5 6–1–2–3–4–5–6 x

y

a What is the equation of the horizontal line shown on the

number plane?

b Graph the horizontal line y = 4.

Solution

a All the points on the line have a y-coordinate of 1 so the

equation is y = 1. (You can also see that it cuts the y-axis at 1.)

b You can plot some points that have a y-coordinate of 4 like

(1, 4), (3, 4) and (–2, 4). Then you would draw the line

through them.

Or, if you know where it will go already, you don’t need to

plot points. Just graph the line.

The answer is shown below.

y = 4

2

0

1

2

3

4

5

–1

1 2 3 4 5 6–1–2–3–4–5–6 x

y


Practise graphing horizontal lines and finding their equations by completing

this activity.

Activity – Horizontal lines

Try these.

1 Without graphing these points, explain how you know they will all lie

on the same horizontal line.

(5, 6), (–1, 6), (20, 6) and −31

5, 6

⎛⎝⎜

⎞⎠⎟

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 Write the equation for each line shown on the graph below.

0 1 2 3–1–2–3–4–5–6

–1

–2

–3

–4

–5

–6

1

–7

–7–8–9

2

3

4

A B

C D

E F

x

y

The equation of AB is _____________________________________

The equation of CD is _____________________________________

The equation of EF is _____________________________________


3 The x-axis is a horizontal line so you can describe it using an equation.

What would its equation be? _______________________________

4 Graph and label these lines on the number plane below.

MN is y = 5 and PQ is y = −1

2

0

1

2

3

4

5

–1

–2

1 2 3 4 5 6–1–2–3 x

y

6


Now you know how to describe horizontal and vertical lines in both English

and using the language of algebra. You have seen the patterns on the graph

and what all the points have in common.

The following website provides you with more practice plotting points and

writing equations for both vertical and horizontal lines.

Access an interactive site dealing with vertical and horizontal lines on the

number plane by visiting the CLI webpage

<http://www.cli.nsw.edu.au/Kto12>. Select Mathematics then Stage 5.2 and

follow the links to resources for this unit Patterns and algebra, PAS5.1.2

Coordinate geometry, Part 2.

Show your understanding of this work by completing the following exercise.

Go to the exercises section and complete Exercise 2.2 – Horizontal lines.


Intercepts

In English, the word intercept means ‘to take or seize on the way from one

place to another’, like when you intercept a messenger who is taking a note

to someone else.

In mathematics, the term intercept is used to describe the place where a

curve or line meets the axes. The place where it cuts the x-axis is called the

x-intercept, and not surprisingly the place where it cuts the y-axis is called

the y-intercept.

two x-intercepts

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

–5

1

–7–8–9

y-intercept

x

y

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

–5

1

–7–8–9

y-intercept

x-intercept

x

y

Sometimes the intercept cannot be seen on the graph because the number

plane is not drawn large enough. Sometimes curves or lines don’t have both

types of intercepts because they don’t actually meet the axes.

0

1

2

3

–1

–2

1 2 3 4 5 6–1–2–3–4–5–6

The y-intercept does exist butit is off the graph.

0

1

2

3

4

5

6

–1

1 2 3 4–1–2–3

This curve never meets the x-axisso there are no x-intercepts.

x

y

x

y

The following example shows how to describe intercepts on a

number plane.



margin if you wish.

What are the intercepts of the lines shown on the graph below?

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

yC

D

B

A

Solution

For the line AB, the x-intercept is –3 and the y-intercept is 4.

You could also write the answer as:

The intercepts for AB are x = –3 and y = 4.

The line CD goes through the origin (0, 0). Therefore it cuts both

axes at the same point. The answer to this question would be:

The x- and y-intercepts are both 0.

You normally talk about points on a number plane by writing their two

coordinates. But with these special intercepts you just have to use one

number for each because you already know they are on an axis.

Intercepts can also be used to describe where a line is.

Look at the following example.



margin if you wish.

a Graph the line that has an x-intercept of 5 and a y-intercept of 2.

b Use the intercepts to find the gradient of the line.

Solution

a Put a dot on the x-axis at 5 and a dot on the y-axis at 2.

Use a ruler to carefully draw a line through the two points.

Your line should go to the edges of the grid.

The answer is shown in the graph below.

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y

b To find the gradient you can draw a right angled triangle

between any two points on the line and use the rule:

gradient =rise

run

If you use the intercepts, you can see that the axes and the line

already form a right-angled triangle. The diagram below shows

you the triangle and the lengths of the rise and the run.


0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y

The gradient isnegative becausethe line slopes down.

2

5

gradient = −2

5

Using the intercepts to find the gradient is only useful when the intercepts

are whole numbers.

Now it is your turn to identify and use intercepts.

Activity – Intercepts

Try these.

1 Use the following graph to complete these sentences.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

yC

D

B

A

a The intercepts of AB are x = __________ and y = __________.

b For the line CD, the x-intercept is __________ and the y-intercept

is __________.


2 a Graph the line with an x-intercept of –2 and y-intercept of –4.

0

1

2

3

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

b (Harder) Use the intercepts to calculate the gradient of your line.

___________________________________________________

___________________________________________________

___________________________________________________

3 (Harder) What type of straight line will not have a y-intercept?

_______________________________________________________


Continue to practise using intercepts by completing the following exercise.

Go to the exercises section and complete Exercise 2.3 – Intercepts.



Graphing lines

All lines on the number plane can be described by an algebraic equation.

This equation describes how to find points on the line. It does this by

describing the rule that links the two coordinates (the x and y numbers).

For example, the equation y = 2x says that the y-coordinate is double the

x-coordinate. Some of the points that fit this pattern are (3, 6), (5, 10) and

(–2, –4). If you graphed these three points you would find that you could

draw a straight line through them, as in the diagram below.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6

7

8

9

10

11

x

y

In fact, you can pick any other point on this line, and you would find that

the y-coordinate is double the x-coordinate. Try it!

The easiest way to find a point that fits a pattern is to substitute an x value

into the equation and work out the y value.


You can collect your x and y values into a neat table like those below for theequation y = 2x . The table can be horizontal with the x values going

across, or vertical with the x values going down.

x 3 5 –2 x y

y 6 10 –4 3 6

5 10

–2 –4

Both these tables show the three points from the earlier graph: (3, 6), (5, 10)

and (–2, –4).

In this section you will complete tables of coordinates for a variety of

equations, then graph the pattern. All the patterns in this section will form a

straight line when graphed. Patterns or equations that form straight lines

when graphed are called linear.

You have already practised drawing horizontal and vertical lines from

their equations.

y = 5

0

1

2

3

4

5

6

–1

1 2 3–1–2–3 x

y

horizontal line

0

1

2

3

–1

1 2 3 4 5 6–1 x

y

x =

5

vertical line

To graph an oblique straight line from its equation you need to work out

some points that fit the pattern, plot them and then join them. But how

many points do you need to plot?


How many points?

How many points do you need to plot before you know which line to draw

for an equation?

There are an infinite number of straight lines that you can draw through any

single point so only plotting one point will not be enough.

Once you plot two points, there is only one straight line that can be drawn

through them both. So you really only need to know two points to draw

the line.

However, it is best to plot three points just to make sure you haven’t made a

mistake. If the three points are all in the one line, then you are probably

right. If they are not in a straight line, then you know that you have to check

your working.

Look at this example.


margin if you wish.

Graph the line y = 6 − x .


Solution

Draw up a table and select three x values that will be simple to

put into the equation. The ones chosen here are 1, 3 and 6.

It is a good idea to spread your x numbers out so that the dots

are spread across the number plane. It is easier to draw the

correct line through dots that are spread out.

x 1 3 6

y

Work out the y values using the equation y = 6 − x .

You can do this in your head but the working is shown here to

help you understand.

When x =1

y = 6 −1

= 5

When x = 3

y = 6 − 3

= 3

When x = 6

y = 6 − 6

= 0

And so the completed table looks like this:

x 1 3 6

y 5 3 0

Plot the points from the table on the number plane and join

them with a straight line. The points are (1, 5), (3, 3) and (6,0).

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y


You can select other values for x in the example above, but you would still

graph the same line. Try some other values yourself and plot them on the

number plane above.

Now it is your turn to graph a straight line in the following activity.

Activity – Graphing lines

Try these.

1 Complete the table given then graph y =x

2 on the number plane below.

There is space below the graph for your working.

Remember that x

2 means x ÷ 2 .

x 4 0 –2

y

0

1

2

3

4

–1

–2

–3

–4

1 2 3 4 5 6–1–2–3–4–5–6 x

y



Sometimes you have to solve an algebraic equation to find the y value.

You can do this using the guess and check method, or by working

backwards. The following example shows you both methods.


margin if you wish.

a Complete the following table for the equation x + y = 5.

x 4 –3 –1

y

b Use the table to graph the line x + y = 5.

Solution

When you substitute the x values you get an algebraic equation

that needs to be solved.

You can solve them in your head or you might need to use other

methods. Each student below explains a different way they

solved each one.

When x = 4 the equation is 4 + y = 5.

This says 4 plus what gives 5.

I just knew the answer is y = 1.


I put x = –3 into the equation and got

–3 + y = 5.

I guessed y = 2 and checked in my head.

–3 + 2 = –1

That didn’t work. I needed a much biggernumber.

So I tried y = 8.

–3 + 8 = 5

y = 8 worked.

I put x = –1 into the equation and got

–1 + y = 5.

I had no idea what to guess so I decidedto work backwards.

The opposite of –1 is +1 so I added 1 toboth sides of the equal sign.

–1+1 + y = 5 + 1

The answer is y = 6.

The completed table is:

x 4 –3 –1

y 1 8 6


b The graph is shown below. Notice that the line extends all

the way across the grid.

0

1

2

3

–1

–2

1 2 3 4 5 6–1–2–3–4–5–6

4

5

6

7

8

9

x

y

Now you know how to graph all types of lines: vertical, horizontal and

oblique. You know about equations, tables of values, gradients and

intercepts. All these ideas are included in the following activity.


Try these.

2 a Complete the table below for the equation y = x +1 .

x –5 0 4

y


b Graph y = x +1 on the number plane below.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

c What are the intercepts of this line?

The x-intercept is __________.

The y-intercept is __________.

d Calculate the gradient of the line y = x +1 by drawing a

right-angled triangle between two of your points.

___________________________________________________

___________________________________________________

___________________________________________________


3 a Graph each of these lines on the number plane below.

x = 2

y = –3

y − x = 5

Write the equation along each line to identify it. Space is provided

below the graph for any working needed.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

b What is the gradient of the line y = –3 on the graph above?

___________________________________________________


c (Harder) Use the intercepts for x + y = 5 to find its gradient?

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

4 (Harder) Colin tried to graph the line x − y = 1 .

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

Oops!

x

y

a Explain how Colin knew he had made a mistake.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


b Correct the table and then graph the correct line.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

x

y

5 2 0


Combine all your knowledge about graphing straight lines to complete the

following activity.

Go to the exercises section and complete Exercise 4.5 – Graphing lines.


Suggested answers – Part 2

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are very

different or if you do not understand an answer, contact your teacher.

Activity – Preliminary quiz

1 Compare your points with the ones shown below. Did you remember to

label each point with its capital letter?

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

E

C D

B

A

2 For each question, you needed to substitute (replace) the pronumeral k

with the number 10 then work out the answer. You can do the working

in your head but the full solution is shown to help you understand.

a k + 5 = 10 + 5

= 15


b 5k = 5 ×10

= 50Remember to putthe × sign in.

c 8 −k

2= 8 −

10

2= 8 − 5

= 3

A fraction line meansdivide.

Then order of operationsays do the division first.

3 The coordinates are the numbers that show the position. The first

number tells you how far to go across and the second number tells you

how far to go up or down.

The coordinates are A (3, 6), B (–2, 5), C (3, –1) and D (–2, 0).

4 a CD (vertical means straight up and down)

b EF (horizontal means straight across)

c I (intersect means to cross)

5 a 5 b 0

6 The working is shown to help you to understand.

a 6 − −2 = 6 + 2

= 8

b −4 +10

2=

6

2= 3

7 There are many methods for solving equations. You can just know the

answer, you can guess the answer and check it by putting it into the

equation, or you can work backwards by using opposite operations.

The answers are given below.

a x = 2 b y = 6 c y = –2 d x = 30


Activity – Vertical lines

1 You should have written something about the x-coordinates all being

the same number, 15. (You could also plot them on a graph.)

2 a

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

b x = –1

3 a x = 5 b x = –8

4 The equation of each line is written along it.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

x =

–3

x =

2

x =

41 2


Activity – Horizontal lines

1 You should have written something about all the points having the same

y-coordinate, 6.

2 AB is y = 2.

CD is y = –1.

EF is y = –6.

3 y = 0 (all the points on the x-axis have a y-coordinate of zero)

4

0

1

2

3

4

5

6

–1

–2

1 2 3 4 5 6–1–2–3 x

y

M N

P Q

Activity – Intercepts

1 a x = 3 and y = –1

b The x-intercept is –2 and the y-intercept is −21

2

2 a

0

1

2

3

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y


b The diagram below shows the right-angled triangle to be used.

0

1

2

3

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

2

4

gradient =rise

run

= −4

2= −2

The gradient isnegative becausethe line goes downthe graph.

3 A vertical line, but not the y-axis y = 0.


1

x 4 0 –2

y 2 0 –1

Your graph should look like this no matter which points you chose

to use.

0

1

2

3

4

–1

–2

–3

–4

1 2 3 4 5 6–1–2–3–4–5–6 x

y


2 a

x –5 0 4

y –4 1 5

b

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

c The x-intercept is –1.

The y-intercept is 1.

d You could use any two points on the line. The gradient is 1.

3 The answers to parts a are shown on the graph.

y = –3

x =

2y – x

= 5

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y


b The gradient is 0. (All horizontal lines have a gradient of zero.)

c The x-intercept is –5 so the run is 5. The y-intercept is 5 so the rise

is also 5. The gradient is positive so:

gradient =5

5= 1

4 a You should have written something about the dots not being in a

straight line.

b The last y value was wrong.

x 5 2 0

y 4 1 –1

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y



Exercises – Part 2

Exercises 2.1 to 2.4 Name ___________________________

Teacher ___________________________

Exercise 2.1 – Vertical lines

1 Write the equation for each of these vertical lines.

a

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

1

–7–8–9 x

y

b

0

1

2

3

–1

1 2 3 4 5 6–1 x

y

c (Harder)

0

1

2

3

4

5

6

–1

1 2 3 4–1–2–3 x

y


2 Do the points (6, 3), (5, 2) and (4, 1) all lie on the same vertical line?

Explain your reasons.

_______________________________________________________

_______________________________________________________

_______________________________________________________

3 Graph these vertical lines on the number plane provided.

a x = 2

b x = –4

c (Harder) x = −1

2

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

5 What is another name for the line whose equation is x = 0?

_______________________________________________________


Exercise 2.2 – Horizontal lines

1 Write the equation of each line shown on the graph below.

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

–5

–6

1

–7

–8

–9

–7–8–9

A B

C D

FEx

y

The equation of AB is _____________________________________

(Harder) The equation of CD is ____________________________

(Harder) The equation of EF is _____________________________

2 Graph and label these lines on the number plane below.

JK is x = –3 and LM is x = 21

2

0 1 2 3–1–2–3–4–5–6

2

3

–1

–2

–3

–4

–5

1

–7–8–9 x

y


3 (Harder) If you draw a line through the two points (1, 7) and (1, 10)

will the line be horizontal, vertical or oblique (neither horizontal nor

vertical)? Explain your reasons.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Exercise 2.3 – Intercepts

1 Use the following graph to complete these sentences.

C

A

B

D

0

1

2

3

4

5

6

–1

–2

–3

1 2 3 4 5 6–1–2–3–4–5–6 x

y

a The intercepts of AB are x = __________ and y = __________.

b For the line CD, the x-intercept is __________ and the y-intercept

is __________.

2 Complete the following sentence.

If a straight line has an x-intercept of 0 then it must have a y-intercept of

__________.


3 a Graph the line with an x-intercept of 4 and y-intercept of –5.

0

1

2

3

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

b Use the intercepts to calculate the gradient of your line.

___________________________________________________

___________________________________________________

___________________________________________________

4 What type of straight line will not have an x-intercept?

_______________________________________________________


Exercise 2.4 – Graphing lines

1 Rachel was asked to graph the line y = x − 2 . Her table of values is

shown below.

x

y

I wonder if theseare right.

a Plot her points and explain why you know she has made a mistake.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

___________________________________________________

___________________________________________________

___________________________________________________

b Correct the error in the table. Space is given here for any working.

Plot the correct points and draw the line on the number

plane above.

___________________________________________________

___________________________________________________


2 a Complete the table below for the line y =x

3.

Space is provided for working if needed.

x 6 3 –6

y

___________________________________________________

___________________________________________________

___________________________________________________

b Use the table to graph the line on the number plane below.

0

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

1 2 3 4 5 6–1–2–3–4–5–6 x

y

c Is the line vertical, horizontal or oblique? __________________

d Calculate the gradient by drawing a right-angled triangle between

two points on the line.

___________________________________________________

___________________________________________________

___________________________________________________

e What are the x and y intercepts of the line?

___________________________________________________

___________________________________________________


3 a (Harder) Graph the lines x = 4 and x + y = 6 on the number plane

below. Space is provided below the graph for any working and any

tables you may want to draw.

0

1

2

3

4

5

6

–1

–2

–3

8

7

1 2 3 4 5 6–1–2–3–4–5–6 x

y9

b Write the coordinates of the point where these two lines intersect

(cross). _____________________________________________

___________________________________________________

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