1. AS 90285 Mathematics 2.2 Draw straightforward non-linear graphs Level 2 3 CreditsEXTERNAL

2. To Achieve you need to: Given an equation be able to graph and identify key features of: Parabola Cubic and other polynomial functions Hyperbolae Circles Exponentials Logarithmic functions

3. Parabola Basic form y=x 2 Vertex always in the middle mirrors about this line

4. Vertical movements When we add/subtract a number For equations in the form: y= x 2 + a

5. Identify the parabolas A y = Vertex (, ) B y = Vertex (, ) C y = Vertex (, ) D y = Vertex (, ) A B C D

6. Horizontal movements When we add/subtract a number inside the brackets For equations in the form: y= (x + a) 2

7. Identify the parabolas B A C D If we add a number inside the brackets it shifts left (negative direction) If we subtract a number inside the brackets it shifts right( + direction)

8. Factorised form DO NOT expand We can easily locate x-intercepts when in brackets For equations in the form: y=(x+a)(x+b)

9. Steps to graph factorised form 1.Find x intercepts theses are when y=o 2.Find the vertex this is always in the middle, so half way between the two x intercepts 3.Plot the vertex, x intercepts and join with a smooth curve following the pattern of the basic parabola

10. EXAMPLE: y=(x+3)(x-1) 1.x intercepts, set y=0 0=(x+3)(x-1) intercepts are at x=-3 and x=1 2.Vertex Half way between x=-3 and x=1 is x=-1 sub this into equation to find y value y=(-1+3)(-1-1) =(2)(-2) =-4 so the coordinate of vertex is (-1,4)

11. Plot Key Points: x-intercepts Vertex y-intercept when x=o

12. Now your turn... y=(x-4)(x+2) Find 1.x-intercepts 2.Coordinates of vertex 3.y-intercept EXTRA: y=x(x-6)

13. KEY POINTS: x-intercepts x=-2 and x=4 Vertex (1,-9) y-intercept (0,-8)

14. Changing steepness If there is a number in front of the x 2 it will either make the graph steeper or flatter For equations in the form: y= ax 2

15. The blue line is y=x 2 The other lines are y = ½x 2 y = 2x 2 The other lines are y = ½x 2 y = 2x 2

16. Summary If the number in front is BIGGER than 1 e.g. 3x 2 means “3 times the x value squared” makes the parabola steeper than the basic y=x 2 if the number in front is smaller than 1 e.g. ¼x 2 means “one quarter of the x value squared” makes the parabola flatter than the basic y=x 2

17. You need to know how to: Graph parabolas of the form: ▫y=x 2 + a ▫y=(x+a) 2 ▫y=(x+a)(x+b) ▫y= ax 2 Identify the key features ▫x-intercepts ▫ vertex ▫ y-intercepts

18. Equations with x 3 as their highest power

19. The basic cubic y=x 3 We can plot this by filling in a table to work out values of the graph xy=x 3 -3-27 -2-8 00 11 28 327

20. Vertical movements When we add/subtract a number For equations in the form: y= x 3 + a

21. Examples: y=x 3 + 5 y=x 3 -3 Examples: y=x 3 + 5 y=x 3 -3

22. General rule: when we have a cubic in the form y=x 3 + a The graph moves up or down by a units

23. Horizontal movements When we add/subtract a number inside the brackets For equations in the form: y= (x + a) 3

24. Examples: y=(x-4) 3 y=(x+2) 3 Examples: y=(x-4) 3 y=(x+2) 3

25. General rule: when we have a cubic in the form y=(x+ a) 3 The graph moves LEFT or RIGHT by a units

26. Cubics in factorised form DO NOT expand We can easily locate x-intercepts when in brackets y=(x+a)(x+b)(x+c) NOTE: one or more of the letters could be zero e.g.y=x(x+2)(x-3) y= (x+7) x 2 y=x(x-4) 2

27. How to plot a factorised cubic #1. Find x-intercepts Found where y=0 #2. Find y-intercepts Found when x=0 #3.Is it negative or positive look at the signs in front of the x’s #1. Find x-intercepts Found where y=0 #2. Find y-intercepts Found when x=0 #3.Is it negative or positive look at the signs in front of the x’s

28. EXAMPLES:

29. For each of these find: #1. x-intercepts #2. y-intercept #3. is the cubic “+” or “-” For each of these find: #1. x-intercepts #2. y-intercept #3. is the cubic “+” or “-” A.y =(x-1)(x+4)(x+3) B.y = -x 2 (x-2) C.y = (x+6)(x+5)(x+2) D.y = (x+3)(x-1) 2 E.y = (x-1)(2-x)(x-4)

30. A y =(x-1)(x+4)(x+3)

31. B x -4-224 y -4 -2 2 4 y Intercept ( 0, 0 ) y = -x 2 (x-2)

32. C x -10-55 y -10 -5 5 10 y Intercept ( 0, 60) y = (x+6)(x+5)(x+2)

33. D x -10-55 y -4 -2 2 4 6 8 10 y Intercept ( 0, 3 ) y = (x+3)(x-1) 2

34. E x -2246810 y -4 -2 2 4 6 8 10 y Intercept ( 0, 8 ) y = (x-1)(2-x)(x-4)

35. Write the equations for:

36. This type of graph is different because: Every x-value has 2 coordinates Every y-value has 2 coordinates

37. Example: this is the graph of a circle with a radius of 5 Each point on the circle is the same distance away from (0,0)

38. General formula of a circle For circles centered at (0,0) x 2 + y 2 = r 2 With r being the radius of the circle

39. Example: x 2 + y 2 = 4 1 st we need to know the radius We find this by finding √4 √4=2 Meaning the radius is 2 We can then mark 2 units away from the origin on each of the axis and join the points with a compass…

40. EXAMPLE: y 2 + x 2 = 4 x -44 y -2 2 4 2

41. Try and draw: 1.x 2 + y 2 = 36 2.x 2 + y 2 = 49 3.x 2 + y 2 = 25 4.x 2 + y 2 = 9 5.x 2 + y 2 = 1 Then try: Ex 19.2 pg 169 Questions 2-5 Steps: 1.Find the radius 2.Plot on the axis 3.Join to make circle Steps: 1.Find the radius 2.Plot on the axis 3.Join to make circle

42. y=a x a is the base this number must always be greater than 0 a can NEVER be equal to zero (a≠0) x is called the exponent This is the variable that changes When x=0 the graph is at y=1 (because anything to the power of 0 equals 1) COPY THIS INTO YOUR NOTES

43. Lets take a look at y=2 x xy=2 x -2 1 0 1 2 3 4 5 6 What happens to y as x gets bigger? Copy and complete (substitute values into your calculator)

45. What about for y=0.5 x xy=0.5 x -2 1 0 1 2 3 4 5 6 What happens to y as x gets bigger? Copy and complete (substitute values into your calculator)

47. Summary for exponentials Are always in the form: y=a x The graph always cuts the y axis at y=1 Growth Curve If a is greater than 1 we get a growth curve a>1 Decay Curve If a is less than 1 (i.e. decimal or fraction) we get a decay curve 0<a<1 x y 1 x y 1 COPY THIS INTO YOUR NOTES

48. Lets see what happens when we change the value of a…

50. 1.What happens as a increases? 2.What always happens at x=1? 3.Is the graph ever below the x-axis?

51. y=a x x y 1 1 a (1,a) Because any number to the power of 1 stays the same e.g. 2 1 =2 5 1 = 5 65 1 = 65 Because any number to the power of 1 stays the same e.g. 2 1 =2 5 1 = 5 65 1 = 65 The graph always cuts through y=1 and goes through the coordinate (1,a) COPY THIS INTO YOUR NOTES Note: The graph never goes below the x-axis

52. Exercise 19.9 pg 187 Q1-5 (DON’T worry about doing the asymptote, domain and range, JUST DRAW THEM) 1.If you are stuck read the notes on the previous page 2.If you are still having difficulty put your hand up and ask for help

53. Movement of exponential curves

54. Example 1 y=2 x +3 What do we know about y=2 x ? 1.It is exponential so will cut the y axis at y=… 2.2 is greater than 1 so will be a ……….. Curve (Growth or decay) 3.At x=1 the graph will cut through co-ordinate (…,…) What do you think will happen if we add 3 to this graph?

55. y=2 x +3 y=2 x

56. y=2 x +3

57. y=2 x y=2 x +3

58. By adding b it moved the graph up by b The shape of the graph stayed the same

59. Example 2 y=2 (x-1) What do you think will happen this time? H INT : it moves the same way as the other graphs we have looked at

60. y=2 x y=2 (x-1)

61. y=2 x y=2 (x-1)

62. By adding b inside the bracket it moved the graph across by b The shape of the graph stayed the same NO longer cuts the y-axis at 1, will cut line x=b at 1 it is like the y-axis has moved to x=b

63. Worksheet on exponentials Think about what way it should look like before you draw it If your not sure put your hand up and ask for help

64. Worksheet Answers

67. Summary What did you notice?

68. y=log a (x) This is the inverse or opposite of the exponential graph y=a x is equivalent to x=log a (y) NOTE: the log button on your calculator is log 10 *There is a way to work out log a (x), we will learn that later on COPY THIS INTO YOUR NOTES

69. y=2 x y=log 2 (x) To graph log graphs we need to know what the exponential looks like

70. Comparing the two graphs

71. Key idea y=a x means the same as x=log a (y) So to graph we do the same as for exponentials but flip the axis around Sounds confusing but lets try one!

72. Graph: y= log 3 (x) This will be the opposite graph to y=3 x We can use a table to find points for this (Do this on GC ) Xy=3 x (x,y) -23^(-2) = 0.11111(-2,0.11) 3^(-1) = 0.33333 03^(0) = 13^(1) = 23^(2) = 33^(3) =

73. The graph of y=log 3 (x) is the mirror of y=3 x through the green line

74. So y=log 3 (x) is the reflection of y=3 x What will these be the reflection of? 1.y=log 4 (x) 2.y=log 7 (x) 3.y=log 2 (x) 4.y=log 9 (x) 5.y=log 1.5 (x)

75. On your calculator There is a way to work out y=log a (x) on your calculator so you can draw up a table of this type y=log a (x) put into your calculator [log(x)]÷[log(a)] Knowing this you can draw up a table straight from the y=log a (x) On your calculator There is a way to work out y=log a (x) on your calculator so you can draw up a table of this type y=log a (x) put into your calculator [log(x)]÷[log(a)] Knowing this you can draw up a table straight from the y=log a (x)

76. Example: y= log 5 (x) xy=log 5 (x)(x,y) 0.5 [log(0.5)]÷ [log(5)] (0.5,-0.43) 1 [log(1)]÷ [log(5)] (1, ) 2 5 10

77. Graph: 1.y=log 4 (x) 2.y=log 7 (x) 3.y=log 2 (x) 4.y=log 9 (x) 5.y=log 1.5 (x) METHOD #1 By knowing what the opposite looks like we can just graph the key points on the opposite axis METHOD #2 Draw up a table and find values to plot METHOD #3 Draw on your GC and copy the key points onto a graph METHOD #1 By knowing what the opposite looks like we can just graph the key points on the opposite axis METHOD #2 Draw up a table and find values to plot METHOD #3 Draw on your GC and copy the key points onto a graph Then try Theta: Ex 19.10 pg 189

78. NOTE: any of the letters a,b,c,d could be 0 This graph has two parts which are split by asymptotes Asymptote: is a line that the graph gets very close to but NEVER touches COPY THIS INTO YOUR NOTES

79. x y Example: Red dotted line = asymptote

80. How to draw a hyperbola in 4 easy steps: #1. find Vertical asymptote found when the bottom =0 #2. find Horizontal asymptote find when x is large i.e. substitute x=1000 in and see what it gets close to #3. Find x-intercept found when the top is equal to 0 #4. Find y-intercept find by substituting x=0 into the equation #1. find Vertical asymptote found when the bottom =0 #2. find Horizontal asymptote find when x is large i.e. substitute x=1000 in and see what it gets close to #3. Find x-intercept found when the top is equal to 0 #4. Find y-intercept find by substituting x=0 into the equation

81. Example: 1.Vertical asymptote find when bottom = 0 x-1=0 if x=1 so the line x=1 is our vertical asymptote 2.Horizontal asymptote Find when x is large, try x=1000 on your calculator put in (2x10000+5) ÷ (1000-1) = Round this to the nearest whole number so y= ……. is our horizontal asymptote

82. 3.X -intercept Found when the top = 0 2x+5=0 4.Y-intercept Found when x=0 Substitute x=0 into the equation on your calculator Type in (2x0+5)÷(0-1)= Example:

83. Draw it:

85. Activities: 1.Worksheet Do the 4 steps then graph 2.Theta 19.4 pg 178

86. Practice doing asymptotes 1.Worksheet Do the 4 steps then graph 2.Theta 19.4 pg 178

87. To plot a graph: 1.draw up a table of x-values (can do on GC) 2.for each x value work out the corresponding y-value (GC does for you) 3.plot each point 4.join points with a smooth curve. Remember: Table → Values → Axes→ Plot → Draw → Label

88. Tips to successfully sketch a graph: Key features and the shape of the graph must be shown. Remember to draw smooth curves with rounded turning points. Key features may be: → x and y intercept(s) →turning points →axis of symmetry →horizontal and vertical asymptotes →centre and radius of circle