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Functions

Functions. Illustrating a Function. Standard Notation for a Function f(x). Graphs of linear and Quadratic Functions. Sketching Quadratic Functions. Reciprocal Function. Exponential Function. Summary of Graphs and Functions. Mathematical Modelling. Starter Questions.

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Functions

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

  2. 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 p2q and pq2. Created by Mr. Lafferty@mathsrevision.com

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

  4. What are Functions ? Functions describe how one quantity relates to another Car Parts Cars Assembly line

  5. What are Functions ? Functions describe how one quantity relates to another Dirty Clean Washing Machine x y Function Input Output y = f(x) f(x)

  6. 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. If the first set is A and the second B then we often writef: A  B The members of set A are usually referred to as the domainof 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. Illustrating Functions 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) = x2 + 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!

  8. Illustrating Functions ARROW DIAGRAM A B f(x) = x2 + x f(-3) = 0 f(-2) = -2 f(-1) = -2 f(0) = 0 f(1) = 4 -3 0 -2 -2 -1 -2 0 0 0 1 4

  9. Finding the Function Examples Find the output or input values for the functions below : 6 7 8 36 49 64 4 12 f: 0 f: 1 f:2 -1 3 7 5 15 6 18 f(x) = x2 f(x) = 4x - 1 f(x) = 3x

  10. Finding the Function Examples Find the function f(x) for from the diagrams. f(x) 1 2 3 1 4 9 f(x) 4 9 f(x) f: 0 f: 1 f:2 0 2 4 5 10 6 11 f(x) = x + 5 f(x) = x2 f(x) = 2x

  11. Illustrating Functions Now try MIA Ex 2.1 Ch10 (page195) Created by Mr. Lafferty@www.mathsrevision.com

  12. Starter Questions Q1. Q2. Find the ratio of cos 60o Q3. 75.9 x 70 30m Explain why the length a = 36m Q4. 24m a Created by Mr. Lafferty

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

  14. Function Notation The standard way to represent a function is by a formula. 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 = 5 5 is the value of f at 1 f(a) = a + 4 a + 4 is the value of f at a

  15. Function Notation Examples For the function h(x) = 10 – x2. Calculate h(1) , h(-3) and h(5) h(x) = 10 – x2 h(1) = 10 – 12 = 9 h(-3) = 10 – (-3)2 = 10 – 9 = 1 h(5) = 10 – 52 = 10 – 25 = -15

  16. Function Notation Examples For the function g(x) = x2 + x Calculate g(0) , h(3) and h(2a) g(x) = x2 + x  h(0) = 02 + 0 = 0 h(3) = 32 + 3 = 12 h(2a) = (2a)2 +2a = 4a2 + 2a

  17. Function Notation Now try MIA Ex 3.1 & 3.2 Ch10 (page197) Created by Mr. Lafferty@www.mathsrevision.com

  18. Starter Questions Created by Mr. Lafferty Maths Dept.

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

  20. Graphs of linear and Quadratic functions y 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 ) x A function of the form f(x) = ax + b is a linear function. Output (Range) Its graph is a straight line with equation y = ax + b Input (Domain)

  21. A function of the form f(x) = ax2 + bx +c a ≠ 0 is called a quadratic function and its graph is a parabola with equation y = ax2 + bx + c Graph of Quadratic Function The parabola shown here is the graph of the function f defined by f(x) = x2 – 1 for -4  x  4 It equation is y = x2 - 1 • From the graph we can see • Minimum value of f(x) is -1 at x = 0 • Minimum turning point (TP) is (0,-1) • f(x) = 0 at x = -1 and x = 1 • The axis of symmetry is the linex = 0

  22. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Draw the graph of the functions with equations below : y = 2x - 5 y = 2x + 1 xy 3 0 1 xy 0 1 3 x x -5 -3 1 1 3 7 x x x x y = x2 y = 2 - x2 xy -2 0 2 xy -2 0 2 4 0 4 -2 2 -2 Created by Mr. Lafferty Maths Dept

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

  24. Starter Questions Q1. Round to 2 significant figures (a) 52.567 (b) 626 Q2. Why is 2 + 4 x 2 = 10 and not 12 Q3. Solve for x Created by Mr. Lafferty

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

  26. Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 1 : Sketch f(x) = 15 – 2x – x2 Step 1 : Find where the function crosses the x – axis. SAC Method i.e. 15 - 2x - x2 = 0 5 x 3 - x (5 + x)(3 - x) = 0 5 + x = 0 3 - x = 0 x = - 5 (- 5, 0) x = 3 (3, 0)

  27. Sketching Quadratic Functions Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 (-5 + 3) ÷ 2 = -1 Equation is x = -1 Step 3 : Find coordinates of Turning Point (TP) For x = -1 f(-1) = 15 – 2x(-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)

  28. Sketching Quadratic Functions Step 4 : Find where curve cuts y-axis. For x = 0 f(0) = 15 – 2x0 – 02 = 15 (0,15) Now we can sketch the curve y = 15 – 2x – x2 Y 3 Cuts x-axis at -5 and 3 -5 15 Cuts y-axis at 15 Max TP (-1,16) (-1,16) X

  29. Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 2 : Sketch f(x) = x2 - 7x + 6 Step 1 : Find where the function crosses the x – axis. SAC Method i.e. x2 – 7x + 6 = 0 x - 6 x - 1 (x - 6)(x - 1) = 0 x - 6 = 0 x - 1 = 0 x = 6 (6, 0) x = 1 (1, 0)

  30. Sketching Quadratic Functions Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 (6 + 1) ÷ 2 =3.5 Equation is x = 3.5 Step 3 : Find coordinates of Turning Point (TP) For x = 3.5 f(3.5) = (3.5)2 – 7x(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)

  31. Sketching Quadratic Functions Step 4 : Find where curve cuts y-axis. For x = 0 f(0) = 02 – 7x0 = 6 = 6 (0,6) Now we can sketch the curve y = x2 – 7x + 6 Y 6 Cuts x - axis at 1 and 6 1 6 Cuts y - axis at 6 Mini TP (3.5,-2.25) (3.5,-2.25) X

  32. Sketching Quadratic Functions Now try MIA Ex 5.1 Ch10 (page 204) Created by Mr. Lafferty@www.mathsrevision.com

  33. Starter Questions Q1. Explain why 15% of £80 is £12 Q2. Multiply out the brackets Q3. Q4. Created by Mr. Lafferty Maths Dept.

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

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

  36. The Reciprocal Function 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 450 to the axes y x Note that x CANNOT take the value 0.

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

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

  39. Reciprocal Function Now try MIA Ex 6.1 Ch10 (page 206) Created by Mr. Lafferty@www.mathsrevision.com

  40. Starter Questions Q1. True or false 2a (a – c + 4ab) =2a2 -2ac + 8ab Q2. Find the missing angle 22o Q3. Q4. Created by Mr. Lafferty Maths Dept.

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

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

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

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

  45. Exponential Button on the Calculator Remember We can calculate exponential (power) value on the calculator. yx Button looks like 0.111 Examples Calculate the following yx 5 32 2 = 25 = yx - 8 1/9 3 = 3-2 =

  46. Exponential Function Now try MIA Ex 7.1 Ch10 (page 208) Created by Mr. Lafferty@www.mathsrevision.com

  47. Starter Questions 39o Created by Mr. Lafferty Maths Dept.

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

  49. Summary of Graphs & Functions Y x y y Y Reciprocal f(x) = ax + b x x g(x) = ax2 + bx + c Quadratic Exponential h(x) = a / x x Linear k(x) = ax Created by Mr. Lafferty@www.mathsrevision.com

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

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