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KS3 Mathematics

KS3 Mathematics. A5 Functions and graphs. A5 Functions and graphs. Contents. A5.2 Tables and mapping diagrams. A5.1 Function machines. A5.3 Finding functions. A5.4 Inverse functions. A5.5 Graphs of functions. Finding outputs given inputs. Introducing functions. × 3. + 2. x. y.

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KS3 Mathematics

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  1. KS3 Mathematics A5 Functions and graphs

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

  3. Finding outputs given inputs

  4. Introducing functions × 3 + 2 x y Afunction 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 anequation. For example, y = 3x + 2. A function can can also be be written with a mapping arrow. For example, x 3x + 2.

  5. Writing functions using algebra

  6. Ordering machines × 2 + 1 + 1 × 2 x y x y ? Is there any difference between 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

  7. Equivalent functions + 1 × 2 × 2 + 2 x y x y Explain why is equivalent to 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. Ordering machines x x x + 4 2 2 2 ÷ 2 + 4 + 4 ÷ 2 x y x y ? The first function can be written as y = + 4. The second function can be written as y = or y = + 2. Is there any difference between and

  9. Equivalent functions + 4 ÷ 2 ÷ 2 + 2 x y x y Explain why is equivalent to 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.

  10. Equivalent function match

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

  12. Using a table × 2 + 5 x y x y 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 3 3, 1, 6, 4, 1.5 3, 1, 6, 4 3, 1, 6 3, 1 11, 7 11, 7, 17, 13, 8 11 11, 7, 17 11, 7, 17, 13 and the corresponding table as 3 3 1 1 6 6 4 4 1.5 1.5 11 11 7 7 17 17 13 13 8

  13. Using a table with ordered values + 1 × 3 x y x y 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 1 1, 2, 3, 4, 5 1, 2, 3, 4 1, 2, 3 1, 2 6 6, 9, 12, 15, 18 6, 9, 12, 15 6, 9 6, 9, 12 When the inputs are ordered and the corresponding table as 1 1 2 2 3 3 4 4 5 5 the outputs form a sequence. 6 6 9 9 12 12 15 15 18

  14. Recording inputs and outputs in a table

  15. Mapping diagrams 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 We can show functions using mapping diagrams. For example, we can draw a mapping diagram of x2x + 1. can be mapped to outputs along the bottom. Inputs along the top

  16. Mapping diagrams of x x + c 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 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.

  17. Mapping diagrams of x mx 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 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 The lines meet at a point on the zero line.

  18. The identity function 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 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 Every number is mapped onto itself.

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

  20. Finding functions given inputs and outputs

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

  22. Think of a number

  23. Finding inputs given outputs + 3 ÷ 8 1 x – 3 × 8 Suppose How can we find the value of x? To find the value of x we start with the output To find the value of x we start with the output and we perform the inverse operations in reverse order. 5 1 x = 5

  24. Finding inputs given outputs × 3 – 7 – 1 x – 2 ÷ 5 + 6 4 x ÷ 3 + 2 × 5 + 7 – 6 Find the value of x for the following: 2 – 1 x = 2 – 8 4 x = – 8

  25. Finding inputs given outputs ÷ 5 + 11 24 x – 6 × 4 + 9 4 x × 5 + 6 ÷ 4 – 11 – 9 Find the value of x for the following: 7 24 x = 7 4.25 4 x = 4.75

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

  27. Finding the inverse function ÷ 4 + 1 x x x + 1 + 1 4 4 × 4 – 1 The inverse of x is x 4(x – 1) We can write xx/4 + 1 as To find the inverse of xx/4 + 1 we start with x 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) x

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

  29. Functions and inverses

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

  31. Coordinate pairs When we write a coordinate, for example, (3, 5) (3, 5) (3, 5) x-coordinate y-coordinate the first number is called the x-coordinateand the second number is called the y-coordinate. the first number is called the x-coordinateand the second number is called the y-coordinate. Together, the x-coordinate and the y-coordinate are called a coordinate pair.

  32. Graphs parallel to the y-axis y x 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. Name five other points that will lie on this line. This line is called x = 2. x = 2

  33. Graphs parallel to the y-axis y x 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 = –10 x = –3 x = 4 x = 9

  34. Graphs parallel to the x-axis y x What do these coordinate pairs have in common? (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. All of the points lie on a straight line parallel to the x-axis. y = 1 Name five other points that will lie on this line. This line is called y = 1.

  35. Graphs parallel to the x-axis y x 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 = 5 y = 3 y = –2 y = –5

  36. 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. Drawing graphs of functions x –3 –2 –1 0 1 2 3 y = x +3 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: 0 1 2 3 4 5 6 (–3, 0) (–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5) (3, 6)

  38. Drawing graphs of functions x –3 –2 –1 0 1 2 3 y = x – 2 For example, to draw a graph of y = x– 2: y = x - 2 1) Complete a table of values: –5 –4 –3 –2 –1 0 1 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.

  39. Drawing graphs of functions

  40. The equation of a straight line y = mx + c The general equation of a straight line can be written as: 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-interceptand 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).

  41. Linear graphswith positive gradients

  42. Investigating straight-line graphs

  43. The gradient and the y-intercept 1 x 2 2 Complete this table: 3 (0, 4) (0, –5) (0, 2) –3 (0, 0) y = x y = –2x– 7 (0, –7)

  44. Rearranging equations into 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. 1 2 –x + 4 y = 2 y = –x + 2 Sometimes the equation of a straight line graph is not given in the form y = mx + c. We can rearrange the equation by transforming both sides in the same way 2y + x = 4 2y = –x + 4

  45. Rearranging equations into the form y = mx + c 1 1 y = –x + 2 2 2 – 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 2. and the y-intercept is

  46. What is the equation? 10 A G H 5 B F D C -5 0 5 10 E What is the equation of the line passing through the points Look at this diagram: a) A and E x = 2 b) A and F y = x + 6 y = x – 2 c) B and E d) C and D y = 2 e) E and G y = 2 –x f) A and C? y = 10 –x

  47. 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.

  48. Pairs

  49. Matching statements

  50. Exploring gradients

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