Graphs of straight lines - OpenStudyassets.openstudy.com/updates/...07_graphs_of_straight_lines_112129.… · 3 Here is a matchstick puzzle using a coordinate grid. ... 9 3 3 Intercept

112 7 Graphs of straight lines

1 Starting with these twelve matches, remove twomatches so that only two squares remain.

2 These four matches make a cocktail glasscontaining a cherry. Move two matches so thatthe cherry is outside the glass.

3 Move three matches so that the fish swims in theopposite direction.

CHAPTER 7

Graphs of straight lines

In this chapter you will revise earlier work on:

• using coordinates in all four quadrants.

You will learn how to:

• plot graphs of linear functions defined implicitly or explicitly• use gradient and intercept to sketch linear graphs• recognise the equation of a linear graph by looking at its gradient

and intercept• use properties of parallel lines.

You will also be challenged to:

• investigate parallels.

Starter: Matchstick puzzles

07_Chapter07_112-129.qxd 7/3/06 9:36 am Page 112

4 Move one match to make a square.

7.1 Coordinates in all four quadrantsYou will already be familiar with the idea of using x and y coordinates like this:

These are sometimes called Cartesian coordinates, after the Frenchmathematician and philosopher René Descartes, although he was not the firstmathematician to use them.

You can extend the basic Cartesian coordinate system into four regions, orquadrants, by using negative coordinates, like this:

7 Graphs of straight lines 113

1 2 3 4O

1

2

3

y

x

P (3, 2)

P (3, 2)

Q (2, �1)

S (�2, �3)

R (�4, 3)

�4 �3 �2 �1 O 1

1

�1

�2

�3

�4

2

3

2 3 4

y

x

(3, 2) indicates 3 across in the x directionand 2 up in the y direction.

In relation to the origin:negative coordinates indicate left insteadof right (x) …

… or below instead of above (y).


EXAMPLE

Plot the points A (4, 2), B (�1, 3), C (�2, �1), D (3, �2) on a coordinate grid,and join them up in order, to form a closed shape. What shape is the result?

SOLUTION

The shape ABCD is a parallelogram.

EXERCISE 7.11 Using the diagram below, write down the coordinates of A, B, C, D and E.


A (4, 2)

D (3, �2)

B (�1, 3)

C (�2, �1)

�4 �3 �2 �1 O 1

1

�1

�2

�3

�4

2

3

2 3 4

y

x

A

D

C

B

�4 �3 �2 �1 O 1

1

�1

�2

�3

�4

2

3

4

2 3 4

y

xE

X


2 The questions refer to the diagram below:a) Which point is at (�3, 1)?b) What are the coordinates of E?c) Which point has the same x and

y coordinates?d) Which point is midway between

(�2, 5) and (4, 3)?e) What are the coordinates of H?f) Which point has the largest y coordinate?g) Which point has the smallest x coordinate?

3 Here is a matchstick puzzle using a coordinate grid.Draw a coordinate grid so that x and y may both run from �6 to 6.Now draw 16 matches with heads and tails in the following positions:

Head (3, 3) Tail (3, 1) Head (�1, 1) Tail (�1,�1)Head (3, 1) Tail (1, 1) Head (�1, 1) Tail (�1, 3)Head (1, 3) Tail (1, 1) Head (�3, 1) Tail (�1, 1)Head (1, 1) Tail (1, �1) Head (�3, �1) Tail (�1, �1)Head (1, 3) Tail (�1, 3) Head (�5, 1) Tail (�3, 1)Head (1, 1) Tail (�1, 1) Head (�5, �1) Tail (�3, �1)Head (1, 3) Tail (3, 3) Head (�5, 1) Tail (�5, �1)Head (1, �1) Tail (�1, �1) Head (�3, 1) Tail (�3, �1)

Now you are ready to solve the puzzle! Your matches should make five squares. The puzzle is to movetwo matches so the matches form exactly four squares.

4 Follow these instructions carefully.Draw a coordinate grid so that x and y both run from �5 to 5. Now draw line segments as follows:

From (�5, 5) to (5, 5) From (�5, 5) to (�5, �4)

From (�5, �5) to (5, �5) From (5, �5) to (5, 4)

From (3, 5) to (3, �1) From (4, 4) to (4, �4)

From (2, 4) to (2, 0) From (�4, 4) to (�4, 0)

From (�4, 4) to (2, 4) From (�4, 0) to (2, 0)

From (�4, �1) to (�1, �1) From (�1, �1) to (�1, �4)

From (�5, �4) to (�1, �4) From (0, 0) to (0, �5)

From (3, �1) to (1, �1) From (1, �4) to (4, �4)

From (1, �1) to (1, �4)

You should find that you have made a maze puzzle. Enter the maze at thetop right corner, and find a route through to exit at the bottom left corner.


�6 �4 �2 O

�2

�4

�6

2

4

6

2 4 6 8

y

x

E

K J

N

A

B

H

G

M

L

F

D

C

I


7.2 Graphs of linear functionsExpressions such as 3x � 5 and 4 � 2x are called linear expressions. They mustnot contain any terms such as x2, x3 or 1/x. Linear expressions are always of theform ax � b, where a and b are numbers. Although a and b often take positivewhole number values, this is not always the case – they may be fractional,negative or even zero.

A relation of the form y � ax � b is called a linear function. Linear functionsare so-called because, when you plot their graphs, the result is a straight line.

EXAMPLE

Plot the graph of y � 2x � 3 for values of x from �5 to 5.

SOLUTION

When x � �5, y � 2 � (�5) � 3 � �10 � 3 � �7.

When x � 0, y � 2 � (0) � 3 � 0 � 3 � 3.

When x � 5, y � 2 � (5) � 3 � 10 � 3 � 13.


Use the formula to work out matching x and yvalues for a low value of x (�5), a middle value(0) and a high value (5).

After plotting the points, use a see-through ruler to check that the points form a straight line.

In fact you need only two points to define astraight line …

… but the third point acts as a check.

It is convenient to store these values in atable, like this.

x �5 0 5

y �7 3 13

�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x


Strictly speaking, the line y � 2x � 3 is infinitely long, since it extendsindefinitely in both directions. The portion of this line cut off between x � �5and x � 5 is more correctly known as a line segment.

EXERCISE 7.2For questions 1 to 8 you are given a linear function and an incomplete table of values. Work out the missingvalues to complete the table, and then plot the graph of the corresponding line segment. You may use eithergraph paper or squared paper.

1 y � 2x � 1 2 y � x � 4 3 y � 3x � 1

4 y � 2x � 3 5 y � �12�x � 4 6 y � x � 1

7 x � y � 10 8 2x � y � 5

9 Draw up a set of coordinate axes so that x can run from �10 to 10 and y from �25 to 25.a) Calculate the coordinates of three points that lie on the line y � 2x. Hence plot the line y � 2x on your

coordinate axes.b) Now calculate the coordinates of three points that lie on the line y � 2x � 1. Plot the line y � 2x � 1 on

the same set of coordinate axes.c) Look at your two graphs. What do you notice?


�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

Now join them with a single neat line.

Note that the line continues slightly beyondthe end-points at (5, 13) and (�5, �7).

x �4 0 4

y �7 1

x �5 0 5

y �1

x �4 0 5

y

x �2 0 4

y

x �6 0 4

y

x �5 0 5

y

x 0 5 10

y

x �2 0 4

y

X


10 Draw up a set of coordinate axes so that x can run from 0 to 10 and y from �5 to 10.a) Calculate the coordinates of three points that lie on the line x � y � 8. Hence plot the line x � y � 8 on

your coordinate axes.b) Now calculate the coordinates of three points that lie on the line x � y � 5. Plot the line x � y � 5 on

the same set of coordinate axes.c) Look at your two graphs. What do you notice?

7.3 Gradient and intercept of linear functionsIn question 9 of the previous exercise you were asked to plot the graph of y � 2x. Your graph should have looked like this:

You can measure the gradient of the line by constructing a triangle underneathit – the exact size of the triangle is unimportant – and measuring the horizontaland vertical changes. These are sometimes referred to as ‘rise’ and ‘run’. Youcan choose any two points on the line.


�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

Run � 4 � 1 � 3

Rise � 8 � 2 � 6

It is dangerous just to count squares.

You must read the values off the graphcarefully: the x and y axes may havedifferent scales, as here.


Then the gradient m is defined as:

gradient m � �r

r

i

u

s

n

e� � �

6

3� � 2

The diagram below shows a family of three graphs, all with gradient 2. They aredistinguished by the fact that each one crosses the y axis at a different position –this point is known as the intercept (or y intercept, to give it its full name).

EXAMPLE

A straight line passes through the points (0, 5) and (3, 14). Find its gradient mand intercept c.

SOLUTION

Gradient � �r

r

i

u

s

n

e� � �

9

3� � 3

Intercept � 5

Thus m � 3 and c � 5


�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

intercept 6

intercept 0

intercept –4

O 1

2

4

6

8

10

12

14

2 3 4 5

y

x

Run � 3 � 0 � 3

Rise � 14 � 5 � 9

You do not need the graph to be plottedaccurately: a sketch to show which numbersare being used is enough.


Some linear graphs have negative gradients. This simply means that the graphslopes down, as you move to the right, not up.

EXAMPLE

A straight line passes through the points (1, 6) and (3, 2). Find its gradient mand intercept c.

Gradient � �r

r

i

u

s

n

e� � �

�

2

4� � �2.

By extending the line segment to the left the intercept may be read from the y axis:

Intercept � 8.

Thus m � �2 and c � 8

EXERCISE 7.3Find the gradient m and the intercept c for each of the lines marked in questions 1 to 8 below.

1 2

O 1

1

2

3

4

5

6

7

8

2 3 4 5

y

xO 1

1

2

3

4

5

6

7

8

2 3

y

x


O 1

1

2

3

4

5

6

7

8

2 3 4 5

y

x

Run � 3 � 1 � 2

Rise � 2 � 6 � �4

X


3 4

5 6

7 8

O 1

1

2

3

4

5

6

7

8

2 3 4 5 6 7 8 9

y

xO 2

1

2

3

4

5

6

7

8

4 6 8 10 12 14 16

y

x

O 1

1

2

3

4

5

6

2 3 4 5 6 7 8 9

y

x

O 1

1

2

3

4

5

6

7

8

2 3 4 5 6 7 8 9

y

x

(4, 5)

(8, 8)

O 1

1

2

3

4

5

6

2 3 4 5 6 7 8 9

y

x

(0, 1)

(8, 5)

O 1

1

2

3

4

5

6

7

2 3 4

y

x



7.4 Equations and graphsLook again at this graph of y � 2x � 3 that was used on page 117.

Notice that the line has gradient 2, and intercept 3, and these also happen to bethe values of the two coefficients that appear in the equation of the line.

This illustrates an important general result:

The graph of the function y � mx � c has gradient m and intercept c.

You can use this principle to help sketch graphs of linear functions.

EXAMPLE

Sketch the graph corresponding to the function y � 3x � 1.

SOLUTION

The intercept is c � 1, so the graph must cross the y axis at (0, 1).The gradient is m � 3, so the graph rises by 3 units for each 1 unit to the right.Thus the graph will look like this:


�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

5

10

Gradient: 10 � 5 � 2Intercept 3

�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x

y � 3x � 1

gradient intercept

Note: the equation mustbe in the form y � …


You can use this idea the other way round, to find the equation of a givenstraight-line graph.

EXAMPLE

Find the equation of this straight line:

SOLUTION

The intercept is c � 4.

The gradient is �1

5

4

�

�

0

4� � �

1

5

0� � 2.

Thus the equation of the line is y � 2x � 4

EXERCISE 7.41 to 8 Write down the equations of the straight lines whose gradients and intercepts you found in

Exercise 7.3, questions 1 to 8.

9 The diagram shows the graph corresponding to a linear function of x.a) Write down the coordinates of the points P and Q on

the line.b) Find the gradient and intercept of the line.c) Hence write down the equation of the straight line.

10 The diagram shows the graph of a linear function of x.a) Find the gradient and intercept of the line.b) Hence write down the equation of the straight line.


O

(0, 4)

y

x

(5, 14)

O 1

1

2

3

4

5

6

2 3 4 5 6 7 8 9 10

y

x

P

Q

O 1

1

2

3

4

5

6

2 3 4 5 6 7 8 9 10

y

x

X


7.5 Parallel linesHere are the graphs of y � 2x � 4, y � 2x and y � 2x � 6 from page 119.

All three graphs have the same gradient, namely 2.

Geometrically, this means that all three lines are parallel.

In general, two lines will be parallel if, and only if, their gradients are equal.

EXAMPLE

Find the equation of the line passing through (5, 13) that is parallel to the line y � 2x � 4.

SOLUTION

Suppose the required line has equation y � mx � c.

Since it is parallel to y � 2x � 4 then the gradient must be 2, that is, m � 2.

Thus the required line has equation y � 2x � c.

Since it passes through the point (5, 13), we may substitute x � 5 and y � 13 to obtain:

13 � (2 � 5) � c

13 � 10 � c

c � 3

Thus the required line has equation y � 2x � 3


�5 �4 �3 �2 �1O 1

2

�2

�4

�6

�8

4

6

8

10

12

14

2 3 4 5

y

x


EXAMPLE

Investigate whether any of these lines is parallel to any of the others:

A y � 3x � 5B y � 2x � 5C y � x � 5D x � y � 9E 2y � 4x � 7

SOLUTION

Clearly neither A nor B nor C is parallel to another since they have gradients of3, 2, 1 respectively.

Equations D and E need to be rearranged to make y the subject before anyfurther comparison is possible.

D may be written as y � x � 9, which has gradient 1, so C and D are parallel.

E may be written as y � 2x � 3.5, which has gradient 2, so B and E are parallel.

EXERCISE 7.51 Rearrange each of these equations into the form y � ax � b. Then pick out the two that represent a pair of

parallel lines.a) x � y � 6 b) 2x � y � 5 � 0 c) y � 1 � �

12�x d) x � 2y � 5 � 0

2 Look at this list of equations. There are four pairs of parallel lines, and one odd one out.

y � 3x � 2y � 2x � 3y � x � 2y � 4 � 3x2y � 8x � 3y � 2x � 1x � y � 2x � y � 52y � 8x � 1

a) Pick out the four pairs of parallel lines.b) Suggest the equation of another line that is parallel to the odd one out.

3 The line y � ax � b is parallel to y � 5x � 1, and passes through the point (1, 0).a) Write down the value of a.b) Work out the value of b, and hence obtain the equation of the line.

4 The line y � mx � c is parallel to the line y � 4x � 1, and passes through the point (0, 3).a) Find the values of m and c, and write down the equation of the line.b) The line also passes through the point (3, p). Find p.

5 A line has equation �y �

3

2� � x.

a) Rearrange the equation into the form y � mx � c.b) Find the equation of the parallel line that passes through the point (2, 1).


X


REVIEW EXERCISE 7For questions 1 to 4 you are given a linear function and an incomplete table of values. Copy and complete thetable, and then plot the graph of the corresponding line segment.

1 y � x � 4 2 y � �12�x � 1

3 y � 2x � 5 4 x � y � 20

5 Work out the gradient and intercept of each of the lines shown below. Hence obtain their equations.a) b)

6 The diagram shows six lines, labelled A, B, C, D, E and F.

Match each line to one of these equations:y � x x � y � 6 y � x � 2y � 2x � 6 y � ��

12�x � 5 x � y � 6

7 Find the equation of a line parallel to y � 4x � 3 but with a y-intercept of 7.

8 Find the equation of the line parallel to y � 3x � 5 that passes through (2, 8).

O

y

x

B (6, 1)

A (1, 6)

O

A (0, 4)

y

x

B (6, 7)


x �5 0 2

y �1 6

x �2 0 6

y 0

x �6 0 6

y

x 0 8 20

y

O 1

1

2

3

4

5

6

2 3 4 5 6 7 8 9 10

y

x

D

EF

A

C

B


9 The diagram shows three points A (�1, 5), B (2, �1) and C (0, 5).

The line L is parallel to AB and passes through C. Find the equation of the line L. [Edexcel]

10 A straight line has equation y � �12� x � 1. The point P lies on the straight line. P has a y-coordinate of 5.

a) Find the x-coordinate of P.b) Write down the equation of a different straight line that is parallel to y � �

12� x � 1.

c) Rearrange y � �12� x � 1 to make x the subject. [Edexcel]

11 The line with equation x � 2y � 6 has been drawn on the grid below.

a) Rearrange the equation x � 2y � 6 to make y the subject.b) Write down the gradient of the line with equation x � 2y � 6.c) Write down the equation of the line that is parallel to the line with equation

x � 2y � 6 and passes through the point with coordinates (0, 7). [Edexcel]

12 ABCD is a rectangle. A is the point (0, 1). C is the point (0, 6).The equation of the straight line through A and B is y � 2x � 1

Find the equation of the straight line through D and C. [Edexcel]


O

C (0, 5)A (�1, 5)L

B (2, �1)

Diagram notaccurately drawn

y

x

�5 O

5

5

y

x

x � 2y � 6

O

Diagram notaccurately drawn

y

x

1 A

B

C

D

6



KEY POINTS

1 Points in a 2-D (two-dimensional) plane may be described using a coordinatesystem in which x runs from left to right and y from bottom to top. The coordinateaxes cross over at the origin. x values to the left of the origin are negative; so, too,are y values below the origin.

2 Linear functions such as y � 3x � 1 may be plotted accurately by drawing a tableof values. Although two points are sufficient to define a line, it is customary to plotthree points as this helps detect errors caused by a slip in the working.

3 The gradient of a linear function is defined as the ratio of the height gained to thehorizontal distance covered, or ‘rise over run’ for short. Graphs that go down as youmove to the right will have negative gradients.

4 The intercept (or y intercept) of a linear function tells you where it crosses the y axis.

5 A linear graph with gradient m and y intercept c will have equation y � mx � c. Thisprinciple allows you to sketch linear functions, and to recognise the equation of agiven straight line graph. In order to compare the gradients of two linear functions,it is best to rearrange them (if necessary) into the form y � mx � c.

6 Two lines will be parallel if, and only if, their gradients have the same value.

7 Finally, the methods in this chapter apply to linear functions containing both x andy terms. You will occasionally encounter linear graphs that are purely vertical(equation x � a constant) or purely horizontal (y � a constant), like these:

O

(5, 0)

y

x O

(0, 3)

The graphy � 3

The graphx � 5

y

x



Internet Challenge 7 �ParallelsUse the internet to help you answer these questions about parallels.

1 What name is given to a quadrilateral with two sets of parallel sides?

2 What name is given to a quadrilateral with only one set of parallel sides?

3 What is a parallelepiped? How do you draw one?

4 Which iconic rock group recorded the album ‘Parallel Lines’ in 1978?

5 What is the 49th parallel?

6 What are parallel universes?

7 What is the parallel postulate?

8 Where might you find a parallel port?

9 Where might you make a parallel turn?

10 ‘Parallel lines never meet.’ True or false?

11 Is it possible for two curves to be parallel?

12 Who might choose to place things in parallel rather than in series?

13 Look at the picture at the top of this page.a) How many of the lines running from left to right are parallel? Now check your answer with a ruler

or straight edge.b) The picture is called ‘Café Wall’. Find out the location of the café that inspired this picture, and

the name of the mathematician who first described it.


Documents

Graphs of straight lines - OpenStudyassets.openstudy.com/updates/...07_graphs_of_straight_lines_112129.… · 3 Here is a matchstick puzzle using a coordinate grid. ... 9 3 3 Intercept