KS3 Mathematics

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A5 Functions and graphs

KS3 Mathematics


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AA5.1 Function machines

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.

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.x2

The second function can be written as y = or y = + 2.x2

x + 42


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


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A5.1 Function machines




Using a table

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

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.

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, such as 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, such as x 2x, x 3x or x 4x, and we 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 101The 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


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Finding functions given inputs and outputs


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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

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

44.75

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 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 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 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


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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

(6, 2)

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

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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

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Graphs parallel to the x-axis


(0, 1), (3, 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.

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.

x

y

y = 1O


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

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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.

xy = x – 2

–3 –2 –1 0 1 2 3

y

x

y = x – 2

–5 –4 –3 –2 –1 0 1O




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 = 42y = –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

x

y


Substituting values into equations

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 + 56 = 3mSubtracting 5:

2 = mDividing by 3:

m = 2

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

A line with the equation y = mx + 5 passes through the point (3, 11).


Pairs


Matching statements


Exploring gradients


Gradients of straight-line graphs

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

y

x

a horizontal line

Zero gradient

y

x

a downwards slope

Negative gradient

y

x

an upwards slope

Positive gradient

The gradient of a line can be positive, negative or zero if, moving from left to right, we have:

If a line is vertical its gradient cannot be specified.

O O O


Finding the gradient from two given points

If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows:

the gradient = change in ychange in x

the gradient = y2 – y1

x2 – x1

x

y

x2 – x1

(x1, y1)

(x2, y2)

y2 – y1

Draw a right-angled triangle between the two points on the line as follows:

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