© boardworks ltd 2004 1 of 51 a5 functions and graphs ks3 mathematics

© Boardworks Ltd 20041 of 51

A5 Functions and graphs

KS3 Mathematics


A5.1 Function machines

Contents

A5.5 Graphs of functions

A5.3 Finding functions

A5.4 Inverse functions


A5.2 Tables and mapping diagrams


Finding outputs given inputs


Introducing functions

A function is a rule which maps one number, sometimes called the input or x, onto another number, sometimes called the output or y.

A function can be illustrated using a function diagram to show the operations performed on the input.For example:

A function can be written as an equation. For example, y = 3x + 2.

A function can can also be be written with a mapping arrow. For example, x 3x + 2.

x y× 3 + 2


Writing functions using algebra


Ordering machines

Is there any difference between

x y× 2 + 1

and

The first function can be written as y = 2x + 1.

The second function can be written as y = 2(x + 1) or 2x + 2

x y+ 1 × 2 ?


Equivalent functions

Explain why

x y+ 1 × 2

is equivalent to

x y× 2 + 2

When an addition is followed by a multiplication; the number that is added is also multiplied.

This is also true when a subtraction is followed by a multiplication.


Ordering machines

Is there any difference between

x y÷ 2 + 4

and

x y+ 4 ÷ 2 ?

The first function can be written as y = + 4.x

2

The second function can be written as y = or y = + 2.x

2

x + 4

2


Equivalent functions

Explain why

x y+ 4 ÷ 2

is equivalent to

When an addition is followed by a division then the number that is added is also divided.

x y÷ 2 + 2

This is also true when a subtraction is followed by a division.


Equivalent function match



Contents







Using a table

We can use a table to record the inputs and outputs of a function.

For example,

We can show the function y = 2x + 5 as

x y× 2 + 5

and the corresponding table as

x

y

3

33

11

11

3, 1

11

1

11, 7

1

7

3, 1, 6

7

6

11, 7, 17

6

17

3, 1, 6, 4

17

4

11, 7, 17, 13

4

13

3, 1, 6, 4, 1.5

13

1.5

11, 7, 17, 13, 8

1.5

8


Using a table with ordered values

It is often useful to enter inputs into a table in numerical order.

For example,

We can show the function y = 3(x + 1) as

x y+ 1 × 3

and the corresponding table as

x

y

1

11

6

6

1, 2

6

2

6, 9

2

9

1, 2, 3

9

3

6, 9, 12

3

12

1, 2, 3, 4

12

4

6, 9, 12, 15

4

15

1, 2, 3, 4, 5

15

5

6, 9, 12, 15, 18

5

18

When the inputs are orderedthe outputs form a sequence.


Recording inputs and outputs in a table


Mapping diagrams

We can show functions using mapping diagrams.

Inputs along the top

For example, we can draw a mapping diagram of x 2x + 1.

can be mapped to outputs along the bottom.

0 2 3 4 5 6 7 8 9 101

0 2 3 4 5 6 7 8 9 101


Mapping diagrams of x x + c

What happens when we draw the mapping diagram for a function of the form x x + c, for example, x x + 1, x x + 2 or x x + 3?

x x + 2

The lines are parallel.

0 2 3 4 5 6 7 8 9 101

0 2 3 4 5 6 7 8 9 101


Mapping diagrams of x mx

What happens when we draw the mapping diagram for a function of the form x mx, for example, x 2x, x 3x or x 4x and project the mapping arrows backwards?

For example,

x 2x

0 2 3 4 5 6 7 8 9 101

0 2 3 4 5 6 7 8 9 101

The lines meet at a point on the zero line.


The identity function

The function x x is called the identity function.

The identity function maps any given number onto itself.

x x

Every number is mapped onto itself.

We can show this in a mapping diagram.

0 2 3 4 5 6 7 8 9 101

0 2 3 4 5 6 7 8 9 101



Contents







Finding functions given inputs and outputs



Contents







Think of a number


Finding inputs given outputs

x 1+ 3 ÷ 8

Suppose

How can we find the value of x?

To find the value of x we start with the output

1

To find the value of x we start with the output and we perform the inverse operations in reverse order.

5

x = 5

× 8– 3



x – 1× 3 – 7

Find the value of x for the following:

– 12

x = 2

+ 7÷ 3

4– 8

x = – 8

× 5+ 2

x 4– 2 ÷ 5 + 6

– 6



x 24÷ 5 + 11

Find the value of x for the following:

247

x = 7

– 11× 5

44.25

x = 4.75

÷ 4+ 6

x 4– 6 × 4 + 9

– 9


Finding the inverse function

x 3x + 5× 3 + 5

We can write x 3x + 5 as

To find the inverse of x 3x + 5 we start with xTo find the inverse of x 3x + 5 we start with x and we perform the inverse operations in reverse order.

xx – 53

x – 53The inverse of x 3x + 5 is x

– 5÷ 3



We can write x x/4 + 1 as

To find the inverse of x x/4 + 1 we start with xTo find the inverse of x x/4 + 1 we start with x and we perform the inverse operations in reverse order.

x4(x – 1)

x + 1÷ 4 + 1 x4

The inverse of x is x 4(x – 1)+ 1x4

× 4 – 1



x –2x + 3× –2 + 3

We can write x 3 – 2x as

To find the inverse of x 3 – 2x we start with xTo find the inverse of x 3 – 2x we start with x and we perform the inverse operations in reverse order.

xx – 3–2

3 – x2The inverse of x 3 – 2x is x

÷ –2 – 3

(= 3 – 2x)

3 – x2 =


Functions and inverses



Contents







Coordinate pairs

When we write a coordinate, for example,

Together, the x-coordinate and the y-coordinate are called a coordinate pair.

the first number is called the x-coordinate and the second number is called the y-coordinate.

(3, 5)

x-coordinate

(3, 5)

y-coordinate

(3, 5)

the first number is called the x-coordinate and the second number is called the y-coordinate.


Graphs parallel to the y-axis

What do these coordinate pairs have in common?

(2, 3), (2, 1), (2, –2), (2, 4), (2, 0) and (2, –3)?

The x-coordinate in each pair is equal to 2.

Look what happens when these points are plotted on a graph.

x

y All of the points lie on a straight line parallel to the y-axis.

Name five other points that will lie on this line.

This line is called x = 2.x = 2


Graphs parallel to the y-axis

All graphs of the form x = c,

where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0).

x

y

x = –3x = –10 x = 4 x = 9


Graphs parallel to the x-axis


(0, 1), (4, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)?

The y-coordinate in each pair is equal to 1.

Look what happens when these points are plotted on a graph.

x

y All of the points lie on a straight line parallel to the x-axis.

Name five other points that will lie on this line.

This line is called y = 1.

y = 1


Graphs parallel to the x-axis

All graphs of the form y = c,

where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c).

x

y

y = –2

y = 5

y = –5

y = 3


Drawing graphs of functions

The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function.


(1, 3), (4, 6), (–2, 0), (0, 2), (–1, 1) and (3.5, 5.5)?

In each pair, the y-coordinate is 2 more than the x-coordinate.

These coordinates are linked by the function:

y = x + 2

We can draw a graph of the function y = x + 2 by plotting points that obey this function.



Given a function, we can find coordinate points that obey the function by constructing a table of values.

Suppose we want to plot points that obey the function

y = x + 3

We can use a table as follows:

x

y = x +3

–3 –2 –1 0 1 2 3

0

(–3, 0)

1 2 3 4 5 6

(–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5) (3, 6)



to draw a graph of y = x – 2:

1) Complete a table of values:

2) Plot the points on a coordinate grid.

3) Draw a line through the points.

4) Label the line.

5) Check that other points on the line fit the rule.

y = x - 2

xy = x – 2

–3 –2 –1 0 1 2 3–5 –4 –3 –2 –1 0 1

For example,


The equation of a straight line

The general equation of a straight line can be written as:

y = mx + c

The value of m tells us the gradient of the line.

The value of c tells us where the line crosses the y-axis.

This is called the y-intercept and it has the coordinate (0, c).

For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).


Linear graphs with positive gradients


Investigating straight-line graphs


The gradient and the y-intercept

Complete this table:

equation gradient y-intercept

y = 3x + 4

y = – 5

y = 2 – 3x

1

–2

3 (0, 4)

(0, –5)

–3 (0, 2)

y = x

y = –2x – 7

x2

12

(0, 0)

(0, –7)


Rearranging equations into the form y = mx + c

Sometimes the equation of a straight line graph is not given in the form y = mx + c.

The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.

We can rearrange the equation by transforming both sides in the same way

2y + x = 4

2y = –x + 4

y =–x + 4

2

y = – x + 212


Rearranging equations into the form y = mx + c

Sometimes the equation of a straight line graph is not given in the form y = mx + c.

The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.

Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.

So the gradient of the line is 12

– and the y-intercept is 2.

y = – x + 212


What is the equation?

Look at this diagram:

C

A

B

E

G H

F

D

0 5

5

10-5

10

What is the equation of the line passing through the points

a) A and E

b) A and F

c) B and E

d) C and D

e) E and G

f) A and C?

x = 2

y = 10 – x

y = x – 2

y = 2

y = 2 – x

y = x + 6


Substituting values into equations

A line with the equation y = mx + 5 passes through the point (3, 11). What is the value of m?

To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5.

This gives us: 11 = 3m + 5

6 = 3mSubtracting 5:

2 = mDividing by 3:

m = 2

The equation of the line is therefore y = 2x + 5.


Pairs


Matching statements


Exploring gradients


Gradients of straight-line graphs

The gradient of a line is a measure of how steep a line is.

The gradient of a straight line y = mx + c is given by

m =change in ychange in x

For any two points on a straight line, (x1, y1) and (x2, y2)

m =y2 – y1

x2 – x1

© boardworks ltd 2004 1 of 51 a5 functions and graphs ks3 mathematics

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