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© Boardworks Ltd 2006 1 of 52 A5 Functions and graphs KS3 Mathematics.

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1 © Boardworks Ltd of 52 A5 Functions and graphs KS3 Mathematics

2 © Boardworks Ltd of 52 Contents A5 Functions and graphs A A A A A A5.1 Function machines A5.5 Graphs of functions A5.3 Finding functions A5.4 Inverse functions A5.2 Tables and mapping diagrams

3 © Boardworks Ltd of 52 Finding outputs given inputs

4 © Boardworks Ltd of 52 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 = 3 x + 2. A function can can also be be written with a mapping arrow. For example, x  3 x + 2. xy × 3+ 2

5 © Boardworks Ltd of 52 Writing functions using algebra

6 © Boardworks Ltd of 52 Ordering machines Is there any difference between xy × 2+ 1 and The first function can be written as y = 2 x + 1. The second function can be written as y = 2( x + 1) or 2 x + 2. xy + 1× 2 ?

7 © Boardworks Ltd of 52 Equivalent functions Explain why xy + 1× 2 is equivalent to xy × 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.

8 © Boardworks Ltd of 52 Ordering machines Is there any difference between xy ÷ and xy + 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

9 © Boardworks Ltd of 52 Equivalent functions Explain why xy + 4 ÷ 2 is equivalent to When an addition is followed by a division then the number that is added is also divided. xy ÷ This is also true when a subtraction is followed by a division.

10 © Boardworks Ltd of 52 Equivalent function match

11 © Boardworks Ltd of 52 Contents A5 Functions and graphs A A A A A A5.2 Tables and mapping diagrams A5.5 Graphs of functions A5.1 Function machines A5.3 Finding functions A5.4 Inverse functions

12 © Boardworks Ltd of 52 Using a table We can use a table to record the inputs and outputs of a function. We can show the function y = 2 x + 5 as xy × 2+ 5 and the corresponding table as: x y , , , 1, , 7, , 1, 6, , 7, 17, , 1, 6, 4, , 7, 17, 13,

13 © Boardworks Ltd of 52 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 xy + 1× 3 and the corresponding table as: x y , , , 2, , 9, , 2, 3, , 9, 12, , 2, 3, 4, , 9, 12, 15, When the inputs are ordered the outputs form a sequence.

14 © Boardworks Ltd of 52 Recording inputs and outputs in a table

15 © Boardworks Ltd of 52 Mapping diagrams We can show functions using mapping diagrams. Inputs along the top For example, we can draw a mapping diagram of x  2 x + 1. can be mapped to outputs along the bottom

16 © Boardworks Ltd of 52 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

17 © Boardworks Ltd of 52 Mapping diagrams of x  mx What happens when we draw the mapping diagram for a function of the form x  mx, such as x  2 x, x  3 x or x  4 x, and we project the mapping arrows backwards? For example: x  2 x The lines meet at a point on the zero line.

18 © Boardworks Ltd of 52 The identity function The function x  x is called the identity function. The identity function maps any given number onto itself. x  xx  x Every number is mapped onto itself. We can show this in a mapping diagram

19 © Boardworks Ltd of 52 Contents A5 Functions and graphs A A A A A A5.3 Finding functions A5.5 Graphs of functions A5.1 Function machines A5.4 Inverse functions A5.2 Tables and mapping diagrams

20 © Boardworks Ltd of 52 Finding functions given inputs and outputs

21 © Boardworks Ltd of 52 Contents A5 Functions and graphs A A A A A A5.4 Inverse functions A5.5 Graphs of functions A5.1 Function machines A5.3 Finding functions A5.2 Tables and mapping diagrams

22 © Boardworks Ltd of 52 Think of a number

23 © Boardworks Ltd of 52 Finding inputs given outputs x ÷ 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

24 © Boardworks Ltd of 52 Finding inputs given outputs 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 ÷ – 6

25 © Boardworks Ltd of 52 Finding inputs given outputs x 24 × 5 – 11 Find the value of x for the following: 247 x = ÷ x = 4.75 ÷ 4+ 6 x 4 – 6 × 4+ 9 – 9

26 © Boardworks Ltd of 52 Finding the inverse function x 3 x + 5 × We can write x  3 x + 5 as To find the inverse of x  3 x + 5 we start with x and we perform the inverse operations in reverse order. x x – The inverse of x  3 x + 5 is x  – 5÷ 3

27 © Boardworks Ltd of 52 Finding the inverse function We can write x  x / as To find the inverse of x  x / we start with x and we perform the inverse operations in reverse order. x 4( x – 1) x + 1 ÷ x 4 The inverse of x  is x  4( x – 1) + 1 x 4 × 4– 1

28 © Boardworks Ltd of 52 Finding the inverse function x –2 x + 3 × –2 + 3 We can write x  3 – 2 x as To find the inverse of x  3 – 2 x we start with x and we perform the inverse operations in reverse order. x x – 3 –2 3 – x 2 The inverse of x  3 – 2 x is x  ÷ –2– 3 (= 3 – 2 x ) 3 – x 2 =

29 © Boardworks Ltd of 52 Functions and inverses

30 © Boardworks Ltd of 52 Contents A5 Functions and graphs A A A A A A5.5 Graphs of functions A5.1 Function machines A5.3 Finding functions A5.4 Inverse functions A5.2 Tables and mapping diagrams

31 © Boardworks Ltd of 52 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.

32 © Boardworks Ltd of 52 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 O

33 © Boardworks Ltd of 52 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 = –3 x = –10 x = 4 x = 9 O

34 © Boardworks Ltd of 52 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. Name five other points that will lie on this line. This line is called y = 1. x y y = 1 O

35 © Boardworks Ltd of 52 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 = –2 y = 5 y = –5 y = 3 O

36 © Boardworks Ltd of 52 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.

37 © Boardworks Ltd of 52 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 y = x + 3 –3–2– (–3, 0) (–2, 1)(–1, 2)(0, 3)(1, 4)(2, 5)(3, 6)

38 © Boardworks Ltd of 52 Drawing graphs of functions 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. x y = x – 2 –3–3–2–10123 y = x – 2 –5–4–3–2–101 O

39 © Boardworks Ltd of 52 Drawing graphs of functions

40 © Boardworks Ltd of 52 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 = 3 x + 4 has a gradient of 3 and crosses the y -axis at the point (0, 4).

41 © Boardworks Ltd of 52 Linear graphs with positive gradients

42 © Boardworks Ltd of 52 Investigating straight-line graphs

43 © Boardworks Ltd of 52 The gradient and the y -intercept Complete this table: equationgradient y -intercept y = 3 x + 4 y = – 5 y = 2 – 3 x 1 –2–2 3(0, 4) (0, –5) –3–3 (0, 2) y = x y = –2 x – 7 x (0, 0) (0, –7)

44 © Boardworks Ltd of 52 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 2 y + 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: 2 y + x = 4 2 y = – x + 4 y = – x y = – x

45 © Boardworks Ltd of 52 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 2 y + 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 1 2 – and the y -intercept is 2. y = – x

46 © Boardworks Ltd of 52 What is the equation? Look at this diagram: 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

47 © Boardworks Ltd of 52 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 = 3 m = 3 m Subtracting 5: 2 = m Dividing by 3: m = 2 The equation of the line is therefore y = 2 x + 5. A line with the equation y = mx + 5 passes through the point (3, 11).

48 © Boardworks Ltd of 52 Pairs

49 © Boardworks Ltd of 52 Matching statements

50 © Boardworks Ltd of 52 Exploring gradients

51 © Boardworks Ltd of 52 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. OOO

52 © Boardworks Ltd of 52 Finding the gradient from two given points If we are given any two points ( x 1, y 1 ) and ( x 2, y 2 ) on a line we can calculate the gradient of the line as follows: the gradient = change in y change in x the gradient = y 2 – y 1 x 2 – x 1 x y ( x 1, y 1 ) ( x 2, y 2 ) y 2 – y 1 Draw a right-angled triangle between the two points on the line as follows: O


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