AS 90285

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

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. y 2 y = x 10 8 6 4 2 x -6 -4 -2 2 4 6 V e r t e x -2 Parabola • Basic form y=x2 • Vertexalways in the middlemirrors about this line

4. Vertical movements • When we add/subtract a number • For equations in the form: y= x2 + a

5. Identify the parabolas A A y = Vertex ( , ) B y = Vertex( , ) C y = Vertex ( , ) D y = Vertex ( , ) 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 C A D B 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 • Find x interceptstheses are when y=o • Find the vertexthis is always in the middle, so half way between the two x intercepts • 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) • x intercepts, set y=00=(x+3)(x-1)intercepts are at x=-3 and x=1 • VertexHalf way between x=-3 and x=1 is x=-1sub this into equation to find y valuey =(-1+3)(-1-1) =(2)(-2) =-4so the coordinate of vertex is (-1,4)

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

12. Now your turn... y=(x-4)(x+2) Find • x-intercepts • Coordinates of vertex • 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 x2 it will either make the graph steeper or flatter • For equations in the form: y= ax2

15. The blue line is y=x2 The other lines are y = ½x2 y = 2x2

16. Summary • If the number in front is BIGGER than 1e.g. 3x2 means “3 times the x value squared”makes the parabola steeper than the basic y=x2 • if the number in front is smaller than 1 e.g. ¼x2 means “one quarter of the x value squared” makes the parabola flatter than the basic y=x2

17. You need to know how to: • Graph parabolas of the form: • y=x2 + a • y=(x+a)2 • y=(x+a)(x+b) • y= ax2 • Identify the key features • x-intercepts • vertex • y-intercepts

18. The Cubic Equations with x3 as their highest power

19. y 10 5 x -4 -2 2 4 -5 -10 The basic cubic y=x3 We can plot this by filling in a table to work out values of the graph

20. Vertical movements • When we add/subtract a number • For equations in the form: y= x3 + a

21. Examples: y=x3 + 5 y=x3 -3

22. General rule: when we have a cubic in the form y=x3 + 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

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) x2 y=x(x-4)2

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

28. EXAMPLES:

29. y =(x-1)(x+4)(x+3) y = -x2 (x-2) y = (x+6)(x+5)(x+2) y = (x+3)(x-1)2 y = (x-1)(2-x)(x-4) For each of these find: #1. x-intercepts #2. y-intercept #3. is the cubic “+” or “-”

30. y A y =(x-1)(x+4)(x+3) 5 x -10 -5 5 10 -5 -10 y Intercept ( 0 , -12 ) -15

31. B y = -x2 (x-2) y 4 2 x -4 -2 2 4 y Intercept ( 0 , 0 ) -2 -4

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

33. D y y = (x+3)(x-1)2 10 8 6 4 y Intercept ( 0 , 3 ) 2 x -10 -5 5 -2 -4

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

35. Write the equations for:

36. The Circle 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) x2 + y2 = r2 With r being the radius of the circle

39. Example: x2 + y2 = 4 1st 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. y 4 2 x -4 -2 2 4 -2 -4 EXAMPLE: y2 + x2 = 4

41. Steps: • Find the radius • Plot on the axis • Join to make circle Try and draw: • x2 + y2 = 36 • x2 + y2 = 49 • x2 + y2 = 25 • x2 + y2 = 9 • x2 + y2 = 1 Then try: Ex 19.2 pg 169 Questions 2-5

42. copy this into your notes Exponential Curves y=ax 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)

43. Lets take a look at y=2x Copy and complete (substitute values into your calculator) What happens to y as x gets bigger?

45. What about for y=0.5x Copy and complete (substitute values into your calculator) What happens to y as x gets bigger?

47. copy this into your notes y y 1 x 1 x Summary for exponentials • Are always in the form: y=ax • The graph alwayscuts the y axis at y=1 Growth Curve If a is greater than 1 we get a growth curvea>1 Decay Curve If a is less than 1 (i.e. decimal or fraction) we get a decay curve0<a<1

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

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

50. Lets see what happens when we change the value of a… What happens as a increases? What always happens at x=1? Is the graph ever below the x-axis?