1. www.mathsrevision.com S4 Credit www.mathsrevision.com Functions Illustrating a Function Standard Notation for a Function f(x) Graphs of linear and Quadratic Functions Exponential Function Summary of Graphs and Functions Mathematical Modelling Sketching Quadratic Functions Reciprocal Function

2. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@mathsrevision.com Starter Questions Q1.Remove the brackets (a)a (4y – 3x) =(b)(x + 5)(x - 5) = Q2.For the line y = -x + 5, find the gradient and where it cuts the y axis. Q3.Find the highest common factor for p 2 q and pq 2.

3. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand the term function. 1.To explain what a function is in terms of a diagram and formula. 2.Apply knowledge to find functions given a diagram. www.mathsrevision.com Functions S4 Credit

4. www.mathsrevision.com S4 Credit What are Functions ? Functions describe how one quantity relates to another Car Parts Assembly line Cars

5. www.mathsrevision.com S4 Credit What are Functions ? Functions describe how one quantity relates to another Dirty Washing Machine Clean OutputInput yx Function f(x) y = f(x)

6. www.mathsrevision.com S4 Credit If the first set is A and the second B then we often write f: A  B Defining a Functions Defn: A function is a relationship between two sets in which each member of the first set is connected to exactly one member in the second set. The members of set A are usually referred to as the domain of the function (basically the starting values or even x-values) while the corresponding values come from set B and are called the range of the function (these are like y-values).

7. www.mathsrevision.com S4 Credit Functions can be illustrated in a number of ways: 1) by a formula. 2) by arrow diagram. Example Suppose that f: A  B is defined by f(x) = x 2 + 3x where A = { -3, -2, -1, 0, 1}. FORMULA then f(-3) = 0,f(-2) = -2, f(-1) = -2, f(0) = 0, f(1) = 4 NB: B = {-2, 0, 4} = the range! Illustrating Functions

8. www.mathsrevision.com S4 Credit 0 -3 -2 0 1 0 -2 0 4 A B ARROW DIAGRAM f(-3) = 0 f(-2) = -2 f(-1) = -2 f(0) = 0 f(1) = 4 f(x) = x 2 + x Illustrating Functions

9. www.mathsrevision.com S4 Credit Finding the Function Find the output or input values for the functions below : 6 7 8 36 49 64 f(x) = x 2 f: 0 f: 1 f:2 -1 3 7 f(x) = 4x - 1 4 12 f(x) = 3x 5 15 6 18 Examples

10. www.mathsrevision.com S4 Credit Finding the Function Find the function f(x) for from the diagrams. 123123 149149 f(x) f: 0 f: 1 f:2 024024 f(x) 49 510 611 f(x) = x + 5f(x) = x 2 f(x) = 2x Examples

11. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 2.1 Ch10 (page195) Illustrating Functions

12. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty12 Starter Questions Q1. Q2.Find the ratio of cos 60 o Q3.75.9 x 70 Q4. Explain why the length a = 36m 30m 24m a

13. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand function notation. 1.To explain the mathematical notation when dealing with functions. 2.Be able to work with function notation. www.mathsrevision.com Function Notation S4 Credit

14. www.mathsrevision.com S4 Credit The standard way to represent a function is by a formula. Function Notation Example f(x) = x + 4 We read this as “f of x equals x + 4” or “the function of x is x + 4 f(1) = 1 + 4 = 55 is the value of f at 1 f(a) = a + 4a + 4 is the value of f at a

15. www.mathsrevision.com S4 Credit For the function h(x) = 10 – x 2. Calculate h(1), h(-3) and h(5) h(1) = 10 – 1 2 = 9 Examples h(-3) = 10 – (-3) 2 = 10 – 9 = 1 h(5) = 10 – 5 2 = 10 – 25 = -15 h(x) = 10 – x 2  Function Notation

16. www.mathsrevision.com S4 Credit For the function g(x) = x 2 + x Calculate g(0), h(3) and h(2a) h(0) = 0 2 + 0 = 0 Examples h(3) = 3 2 + 3 = 12 h(2a) = (2a) 2 +2a = 4a 2 + 2a g(x) = x 2 + x  Function Notation

17. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 3.1 & 3.2 Ch10 (page197) Function Notation

18. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions

19. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand linear and quadratic functions. 1.To explain the linear and quadratic functions. 2.Be able to graph linear and quadratic equations. www.mathsrevision.com S4 Credit Graphs of linear and Quadratic functions

20. www.mathsrevision.com S4 Credit A graph gives a picture of a function It shows the link between the numbers in the input x ( or domain ) and output y ( or range ) Graphs of linear and Quadratic functions A function of the form f(x) = ax + b is a linear function. Its graph is a straight line with equation y = ax + b y x Input (Domain) O u t p u t ( R a n g e )

21. www.mathsrevision.com S4 Credit The parabola shown here is the graph of the function f defined by f(x) = x 2 – 1 for -4  x  4 It equation is y = x 2 - 1 Graph of Quadratic Function From the graph we can see (i)Minimum value of f(x) is -1 at x = 0 (ii)Minimum turning point (TP) is (0,-1) (iii) f(x) = 0 at x = -1 and x = 1 (iv)The axis of symmetry is the line x = 0 A function of the form f(x) = ax 2 + bx +ca ≠ 0 is called a quadratic function and its graph is a parabola with equation y = ax 2 + bx + c

22. Outcome 2 x x 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 23-Nov-16Created by Mr. Lafferty Maths Dept y = 2x + 1 xyxy 0 1 1 3 3 7 -5 -31 y = 2x - 5 xyxy 01 3 Draw the graph of the functions with equations below : 4 0 4 -2 2-2 y = 2 - x 2 xyxy -202 y = x 2 xyxy -220 xx xx

23. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 4.1 & 4.2 Ch10 (page 201) Graphs of Linear and Quadratic Functions

24. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty24 Starter Questions Q1.Round to 2 significant figures Q2.Why is 2 + 4 x 2 = 10 and not 12 Q3.Solve for x (a)52.567(b)626

25. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Be able to sketch quadratic functions. 1.To show how to sketch quadratic functions. www.mathsrevision.com S4 Credit Sketching Quadratic Functions

26. www.mathsrevision.com S4 Credit Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 1 : Sketch f(x) = 15 – 2x – x 2 Step 1 : Find where the function crosses the x – axis. i.e. 15 - 2x - x 2 = 0 SAC Method (5 + x)(3 - x) = 0 x = - 5x = 3 5 3 x - x 5 + x = 03 - x = 0 (- 5, 0)(3, 0)

27. www.mathsrevision.com S4 Credit Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = -1 Step 3 : Find coordinates of Turning Point (TP) For x = -1f(-1) = 15 – 2 x (-1) – (-1) 2 = 16 Since (-5, 0) and (3,0) lie on the curve and 0 is less than 16 Turning point TP is a Maximum at (-1, 16) (-5 + 3) ÷ 2 = -1

28. www.mathsrevision.com S4 Credit 3 -5 Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 15 – 2 x 0 – 0 2 = 15 (0,15) X Y Cuts x-axis at -5 and 3 Cuts y-axis at 15 Max TP (-1,16)(-1,16) 15 Sketching Quadratic Functions Now we can sketch the curve y = 15 – 2x – x 2

29. www.mathsrevision.com S4 Credit Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 2 : Sketch f(x) = x 2 - 7x + 6 Step 1 : Find where the function crosses the x – axis. i.e. x 2 – 7x + 6 = 0 SAC Method (x - 6)(x - 1) = 0 x = 6x = 1 x x - 6 - 1 x - 6 = 0x - 1 = 0 (6, 0)(1, 0)

30. www.mathsrevision.com S4 Credit Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = 3.5 Step 3 : Find coordinates of Turning Point (TP) For x = 3.5 f(3.5) = (3.5) 2 – 7 x (3.5) + 6 = -2.25 Since (1, 0) and (6,0) lie on the curve and 0 is greater than 2.25 Turning point TP is a Minimum at (3.5, -2.25) (6 + 1) ÷ 2 =3.5

31. www.mathsrevision.com S4 Credit Cuts x - axis at 1 and 6 6 1 Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 0 2 – 7 x 0 = 6 = 6 (0,6) X Y Cuts y - axis at 6 Mini TP (3.5,-2.25)(3.5,-2.25) 6 Sketching Quadratic Functions Now we can sketch the curve y = x 2 – 7x + 6

32. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 5.1 Ch10 (page 204) Sketching Quadratic Functions

33. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions Q1. Explain why 15% of £80 is £12 Q3. Q2. Multiply out the brackets Q4.

34. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Know the main points of the reciprocal function. 1.To show what the reciprocal function looks like. 2.Be able to sketch the reciprocal function. www.mathsrevision.com S4 Credit The Reciprocal Function

35. www.mathsrevision.com S4 Credit The function of the form is the simplest form of a reciprocal function. The Reciprocal Function The graph of the function is called a hyperbola and it divided into two branches. The equation of the graph is y is inversely proportional to x

36. www.mathsrevision.com S4 Credit The Reciprocal Function Note that x CANNOT take the value 0. The graph never touches the x or y axis. The axes are said to be asymptotes to the graph The graph has two lines of symmetry at 45 0 to the axes y x

37. 1 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 1/x xyxy -100.1 10 -0.1 1 23-Nov-16Created by Mr. Lafferty Maths Dept -0.1 100.1 Draw the graph of the function with equations below : x x x -1-10 1 y = 1/(-10) = - 0.1 y = 1/(-1) = - 1 y = 1 /(-0.1) = - 10 y = 1 / 0.1 = 10 y = 1 / 1 = 1 y = 1 / 10 = 0.1

38. 1 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 5/x xyxy -10 1 10 -5 5 23-Nov-16Created by Mr. Lafferty Maths Dept -0.5 50.5 Draw the graph of the function with equations below : x x x -1-5 1 y = 5/(-10) = - 0.1 y = 5/(-5) = - 1 y = 5/(-1) = - 5 y = 5 / 1 = 5 y = 5 / 5 = 1 y = 5 / 10 = 0.5

39. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 6.1 Ch10 (page 206) Reciprocal Function

40. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions Q1. True or false 2a (a – c + 4ab) =2a 2 -2ac + 8ab Q3. Q2. Find the missing angle Q4. 22 o

41. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Know the main points of the exponential function. 1.To show what the exponential function looks like. 2.Be able to sketch the exponential function. www.mathsrevision.com S4 Credit Exponential Function

42. www.mathsrevision.com S4 Credit A function in the form f(x) = a x where a > 0, a ≠ 1 is called an exponential function to base a. Exponential Functions Consider f(x) = 2 x x-3-2-101 2 3 f(x) 11 / 8 ¼ ½ 1 2 4 8 Exponential (to the power of) Graphs

43. 1 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 23-Nov-16Created by Mr. Lafferty Maths Dept Draw the graph of the function with equation below : x x x x = -3y = 1/8 x = -2y = 1/4 x = -1y = 1/2 x = 0y = 1 x = 1y = 2 x = 2y = 4 x y = 2 x x = 3y = 8

44. www.mathsrevision.com S4 Credit The graph is like y = 2 x (0,1) (1,2) Major Points (i) y = 2 x passes through the points (0,1) & (1,2) (ii) As x  ∞ y  ∞ however as x  ∞ y  0. (iii) The graph shows a GROWTH function. Graph

45. www.mathsrevision.com S4 Credit Remember We can calculate exponential (power) value on the calculator. Exponential Button on the Calculator Button looks like yxyx Examples Calculate the following 2 5 = 3 -2 = = 5y x 2 = -y x 3 8 32 1/9 0.111

46. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 7.1 Ch10 (page 208) Exponential Function

47. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions 39 o

48. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Know the main points of the various graphs in this chapter. 1.To summarise graphs covered in this chapter. 2.Be able to identify function and related graphs. www.mathsrevision.com S4 Credit Summary of Graphs & Functions

49. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Summary of Graphs & Functions y x x Y y x f(x) = ax + b g(x) = ax 2 + bx + c h(x) = a / x k(x) = a x Linear Quadratic Reciprocal Exponential Y x

50. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 8.1 Ch10 (page 209) Summary of Graphs & Functions

51. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr Lafferty Maths Dept Starter Questions

52. 23-Nov-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand mathematical models using functions 1.To show how we can use functions to model real-life situations. 2.Solving problems using mathematical models. www.mathsrevision.com S4 Credit Mathematical Models

53. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Mathematical Models In real-life scientists look for connections between two or more quantities. (A) They collect data, using experiments, surveys etc... (B) They organise the data using tables and graphs. (C) They analyse the data, by matching it with graphs like the ones you have studied so far. (D) Use the results to predict for results

54. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Mathematical Models Collect data Organise data Analyse data Make Predictions TablesGraphs SurveyExperiments

55. www.mathsrevision.com S4 Credit McLaren are testing a new Formula 1 car. Data was collected and organised into the table below: Example Mathematical Models By plotting the data on a graph and analysing the result is there a connection between the variables time and distance ?

56. x y x x x x = 0y = 0 x = 0.5y = 2 x = 1.0y = 8 x = 1.5y = 18 x = 2.0y = 32 x = 2.5y = 50 x y = ax 2 Does it look like part of a graph we know ? To find a pick a point of the graph and sub into equation. (1,8) y = ax 2 8 = a x (1) 2 y = 8x 2 Pick another point to double check ! (2,32)32 = 8 x (2) 2 32 = 8 x 432 = 32 x Y y = ax 2

57. www.mathsrevision.com S4 Credit We can now use the equation to predict other values. Example Mathematical Models Use the equation to calculate the following : y = 8x 2 distance when time x = 10 time when distance y = 200 y = 8x 2 y = 8 x (10) 2 y = 8 x 10 y = 800 m y = 8x 2 200 = 8x 2 x 2 = 25 x = 5 seconds

58. www.mathsrevision.com S4 Credit 23-Nov-16Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 9.1 Ch10 (page 210) Mathematical Models