Download A5 Functions and graphs

KS3 Mathematics
A5 Functions and graphs
1 of 52
© Boardworks Ltd 2006
Contents
A5 Functions and graphs
A A5.1 Function machines
A A5.2 Tables and mapping diagrams
A A5.3 Finding functions
A A5.4 Inverse functions
A A5.5 Graphs of functions
2 of 52
© Boardworks Ltd 2006
Finding outputs given inputs
3 of 52
© Boardworks Ltd 2006
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.
x
×3
+2
y
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.
4 of 52
© Boardworks Ltd 2006
Writing functions using algebra
5 of 52
© Boardworks Ltd 2006
Ordering machines
Is there any difference between
x
×2
+1
y
×2
y
and
x
+1
?
The first function can be written as y = 2x + 1.
The second function can be written as y = 2(x + 1) or 2x + 2.
6 of 52
© Boardworks Ltd 2006
Equivalent functions
Explain why
x
+1
×2
y
is equivalent to
x
×2
+2
y
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.
7 of 52
© Boardworks Ltd 2006
Ordering machines
Is there any difference between
x
÷2
+4
y
÷2
y
and
x
+4
?
x
The first function can be written as y =
+ 4.
2
x+4
x
The second function can be written as y =
or y = + 2.
2
2
8 of 52
© Boardworks Ltd 2006
Equivalent functions
Explain why
x
+4
÷2
y
is equivalent to
x
÷2
+2
y
When an addition is followed by a division then the number
that is added is also divided.
This is also true when a subtraction is followed by a division.
9 of 52
© Boardworks Ltd 2006
Equivalent function match
10 of 52
© Boardworks Ltd 2006
Contents
A5 Functions and graphs
A A5.1 Function machines
A A5.2 Tables and mapping diagrams
A A5.3 Finding functions
A A5.4 Inverse functions
A A5.5 Graphs of functions
11 of 52
© Boardworks Ltd 2006
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
×2
x1
3, 1,3,
6,1,
3,
4,3,
6,
1,
1.5
6
3
4
+5
11 y 7
11,
7, 17,
17 13,
13 8
and the corresponding table as:
12 of 52
x
3
1
6
4
1.5
y
11
7
17
13
8
© Boardworks Ltd 2006
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
×3
+1
x5
1, 1,
2, 1,
2,
3, 1,
2,
3,
4,
2
3
1
4
6 y9
6,
9, 12
12, 15
15, 18
and the corresponding table as:
13 of 52
x
1
2
3
4
5
y
6
9
12
15
18
When the
inputs are
ordered
the outputs
form a
sequence.
© Boardworks Ltd 2006
Recording inputs and outputs in a table
14 of 52
© Boardworks Ltd 2006
Mapping diagrams
We can show functions using mapping diagrams.
For example, we can draw a mapping diagram of x  2x + 1.
Inputs along the top can be mapped to outputs along the
bottom.
15 of 52
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
© Boardworks Ltd 2006
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
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
The lines are parallel.
16 of 52
© Boardworks Ltd 2006
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
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
The lines meet at a point on the zero line.
17 of 52
© Boardworks Ltd 2006
The identity function
The function x  x is called the identity function.
The identity function maps any given number onto itself.
We can show this in a mapping diagram.
xx
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Every number is mapped onto itself.
18 of 52
© Boardworks Ltd 2006
Contents
A5 Functions and graphs
A A5.1 Function machines
A A5.2 Tables and mapping diagrams
A A5.3 Finding functions
A A5.4 Inverse functions
A A5.5 Graphs of functions
19 of 52
© Boardworks Ltd 2006
Finding functions given inputs and outputs
20 of 52
© Boardworks Ltd 2006
Contents
A5 Functions and graphs
A A5.1 Function machines
A A5.2 Tables and mapping diagrams
A A5.3 Finding functions
A A5.4 Inverse functions
A A5.5 Graphs of functions
21 of 52
© Boardworks Ltd 2006
Think of a number
22 of 52
© Boardworks Ltd 2006
Finding inputs given outputs
Suppose
x
÷8
+3
1
How can we find the value of x?
To find the value of x we start with the output and we perform
the inverse operations in reverse order.
5
–3
×8
1
x=5
23 of 52
© Boardworks Ltd 2006
Finding inputs given outputs
Find the value of x for the following:
×3
x
÷3
2
–7
–1
+7
–1
x=2
x
–8
–2
÷5
+6
4
+2
×5
–6
4
x = –8
24 of 52
© Boardworks Ltd 2006
Finding inputs given outputs
Find the value of x for the following:
×5
x
– 11
÷5
7
24
+ 11
24
x=7
x
4.75
–6
×4
+9
4
+6
÷4
–9
4
x = 4.75
25 of 52
© Boardworks Ltd 2006
Finding the inverse function
We can write x  3x + 5 as
x
×3
3x + 5
+5
To find the inverse of x  3x + 5 we start with x and we
perform the inverse operations in reverse order.
x–5
3
÷3
–5
The inverse of x  3x + 5 is x 
26 of 52
x
x–5
3
© Boardworks Ltd 2006
Finding the inverse function
We can write x  x/4 + 1 as
x
÷4
x
+1
4
+1
To find the inverse of x  x/4 + 1 we start with x and we
perform the inverse operations in reverse order.
4(x – 1)
×4
–1
The inverse of x  x + 1
4
27 of 52
is
x
x  4(x – 1)
© Boardworks Ltd 2006
Finding the inverse function
We can write x  3 – 2x as
x
× –2
+3
–2x + 3
(= 3 – 2x)
To find the inverse of x  3 – 2x we start with x and we
perform the inverse operations in reverse order.
3–x
x–3
=
2
–2
÷ –2
–3
The inverse of x  3 – 2x is x 
28 of 52
x
3–x
2
© Boardworks Ltd 2006
Functions and inverses
29 of 52
© Boardworks Ltd 2006
Contents
A5 Functions and graphs
A A5.1 Function machines
A A5.2 Tables and mapping diagrams
A A5.3 Finding functions
A A5.4 Inverse functions
A A5.5 Graphs of functions
30 of 52
© Boardworks Ltd 2006
Coordinate pairs
When we write a coordinate, for example,
(3, 5)
(6,
2)
x-coordinate
y-coordinate
the first number is called the x-coordinate and the second
number is called the y-coordinate.
y-coordinate.
Together, the x-coordinate and the y-coordinate are called a
coordinate pair.
31 of 52
© Boardworks Ltd 2006
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.
All of the points lie on a straight
line parallel to the y-axis.
y
O
x
Name five other points that will lie
on this line.
This line is called x = 2.
x=2
32 of 52
© Boardworks Ltd 2006
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).
y
x
O
x = –10
33 of 52
x = –3
x=4
x=9
© Boardworks Ltd 2006
Graphs parallel to the x-axis
What do these coordinate pairs have in common?
(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.
y
y=1
O
x
Name five other points that will lie
on this line.
This line is called y = 1.
34 of 52
© Boardworks Ltd 2006
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).
y
y=5
y=3
O
x
y = –2
y = –5
35 of 52
© Boardworks Ltd 2006
Drawing graphs of functions
The x-coordinate and the y-coordinate in a coordinate
pair can be linked by a function.
What do these coordinate pairs have in common?
(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.
36 of 52
© Boardworks Ltd 2006
Drawing graphs of functions
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
–3
–2
–1
0
1
2
3
y=x+3
0
1
2
3
4
5
6
(–3, 0) (–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5) (3, 6)
37 of 52
© Boardworks Ltd 2006
Drawing graphs of functions
To draw a graph of y = x – 2:
y
1) Complete a table of values:
x
–3 –2 –1 0 1 2
y = x – 2 –5 –4 –3 –2 –1 0
y=x–2
3
1
O
x
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.
38 of 52
© Boardworks Ltd 2006
Drawing graphs of functions
39 of 52
© Boardworks Ltd 2006
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).
40 of 52
© Boardworks Ltd 2006
Linear graphs with positive gradients
41 of 52
© Boardworks Ltd 2006
Investigating straight-line graphs
42 of 52
© Boardworks Ltd 2006
The gradient and the y-intercept
Complete this table:
43 of 52
equation
gradient
y-intercept
y = 3x + 4
3
(0, 4)
x
y=
–5
2
1
2
(0, –5)
y = 2 – 3x
–3
(0, 2)
y=x
1
(0, 0)
y = –2x – 7
–2
(0, –7)
© Boardworks Ltd 2006
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
–x + 4
y=
2
y=– 1 x+2
2
44 of 52
© Boardworks Ltd 2006
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.
y=– 1 x+2
2
So the gradient of the line is – 1 and the y-intercept is 2.
2
45 of 52
© Boardworks Ltd 2006
What is the equation?
What is the equation of
the line passing through
the points
Look at this diagram:
y
10
A
G
H
5
B
-5
0
E
C
5
x=2
b) A and F?
y=x+6
c) B and E? y = x – 2
F
D
a) A and E?
x
10
d) C and D? y = 2
e) E and G? y = 2 – x
f) A and C?
46 of 52
y = 10 – x
© Boardworks Ltd 2006
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
Subtracting 5:
6 = 3m
Dividing by 3:
2=m
m=2
The equation of the line is therefore y = 2x + 5.
47 of 52
© Boardworks Ltd 2006
Pairs
48 of 52
© Boardworks Ltd 2006
Matching statements
49 of 52
© Boardworks Ltd 2006
Exploring gradients
50 of 52
© Boardworks Ltd 2006
Gradients of straight-line graphs
The gradient of a line is a measure of how steep the line is.
The gradient of a line can be positive, negative or zero if,
moving from left to right, we have:
an upwards slope
y
O
a horizontal line
y
x
Positive gradient
O
Zero gradient
a downwards slope
y
x
O
x
Negative gradient
If a line is vertical its gradient cannot be specified.
51 of 52
© Boardworks Ltd 2006
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:
change in y
the gradient =
change in x
y
(x2, y2)
y2 – y1
(x1, y1)
Draw a right-angled triangle
between the two points on
the line as follows:
y2 – y1
the gradient =
x2 – x1
52 of 52
x2 – x1
O
x
© Boardworks Ltd 2006