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Functions

Quadratic Functions. Int 2. Functions. Quadratic Functions y = ax 2. Quadratics y = ax 2 +c. Quadratics y = a(x-b) 2. www.mathsrevision.com. Quadratics y = a(x-b) 2 + c. Factorised form y = (x-a)(x-b). Int 2. Starter. www.mathsrevision.com. Functions. Int 2. Learning Intention.

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Functions

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  1. Quadratic Functions Int 2 Functions Quadratic Functions y = ax2 Quadratics y = ax2 +c Quadratics y = a(x-b)2 www.mathsrevision.com Quadratics y = a(x-b)2 + c Factorised form y = (x-a)(x-b)

  2. Int 2 Starter www.mathsrevision.com

  3. Functions Int 2 Learning Intention Success Criteria • Understand the term function. • To explain the term function. • Work out values for a given function. www.mathsrevision.com

  4. Functions Int 2 A roll of carpet is 5m wide. It is solid in strips by the area. If the length of a strip is x m then the area. A square metres, is given by A = 5x. The value of A depends on the value of x. We say A is a function of x. We write : www.mathsrevision.com A(x) =5x Example A(1) = 5 x 1 =5 A(2) = 5 x 2 =10 A(t) = 5 x t = 5t

  5. Functions Int 2 Using the formula for the function we can make a table and draw a graph using A as the y coordinate. www.mathsrevision.com In the case The graph is a straight line We can this a Linear function.

  6. Functions Int 2 For the following functions write down the gradient and were the function crosses the y-axis f(x) = 2x - 1 f(x) = 0.5x + 7 f(x) = -3x Sketch the following functions. www.mathsrevision.com f(x) = x f(x) = 2x + 7 f(x) = x +1

  7. Functions Int 2 Now try MIA Ex 1 Ch14 (page 216) www.mathsrevision.com

  8. Int 2 Starter www.mathsrevision.com

  9. Quadratic Functions Int 2 Learning Intention Success Criteria • To know the properties of a quadratic function. • To explain the main properties of the basic quadratic function y = ax2 • using graphical methods. • Understand the links between graphs of the form y = x2 and y = ax2 www.mathsrevision.com

  10. Quadratic Functions Int 2 A function of the form f(x) = a x2 + b x + c is called a quadratic function www.mathsrevision.com The simplest quadratics have the form f(x) = a x2 Lets investigate

  11. Quadratic Functions Int 2 Now try MIA Ex 2 Q2 P 219 www.mathsrevision.com

  12. Quadratic of the form f(x) = ax2 Key Features Symmetry about x =0 Vertex at (0,0) The bigger the value of a the steeper the curve. -x2 flips the curve about x - axis

  13. Quadratic Functions Int 2 Example The parabola has the form y = ax2 graph opposite. The point (3,36) lies on the graph. Find the equation of the function. (3,36) Solution f(3) = 36 www.mathsrevision.com 36 = a x 9 a = 36 ÷ 9 a = 4 f(x) = 4x2

  14. Quadratic Functions Int 2 Now try MIA Ex 2 Q3 (page 219) www.mathsrevision.com

  15. Starter Int 2 Q1. Write down the equation of the quadratic. f(x) = ax2 (2,100) Solution f(2) = 100 100 = a x 4 www.mathsrevision.com a = 100 ÷ 4 a = 25 f(x) = 25x2 (x-4)(x-3)

  16. Quadratic Functions Int 2 Learning Intention Success Criteria • To know the properties of a quadratic function. • y = ax2+ c • To explain the main properties of the basic quadratic function • y = ax2+ c • using graphical methods. www.mathsrevision.com • Understand the links between graphs of the form y = x2 and y = ax2 + c

  17. Quadratic Functions Int 2 Now try MIA Ex 2 Q5 (page 220) www.mathsrevision.com Quadratic of the form f(x) = ax2 + c

  18. Quadratic of the form f(x) = ax2 + c Key Features Symmetry about x = 0 Vertex at (0,C) a > 0 the vertex (0,C) is a minimum turning point. a < 0 the vertex (0,C) is a maximum turning point.

  19. Quadratic Functions Int 2 Example The parabola has the form y = ax2 + c graph opposite. The vertex is the point (0,2) so c = 2. The point (3,38) lies on the graph. Find the equation of the function. (3,38) Solution f(x) = a x2 + c www.mathsrevision.com (0,2) f(3) = a x 32 + 2 38 = a x 9 +2 a = (38 -2) ÷ 9 f(x) = 4x2 + 2 a = 4

  20. Quadratic Functions Int 2 Now try MIA Ex 2 Q7 (page 221) www.mathsrevision.com

  21. Starter Int 2 Q1. Write down the equation of the quadratic. (9,81) Solution f(9) = 81 81 = a x 9 www.mathsrevision.com a = 81 ÷ 9 a = 9 f(x) = 9x2 (x-5)(x-6)

  22. Quadratic Functions Int 2 Learning Intention Success Criteria • To know the properties of a quadratic function. • y = a(x – b)2 • To explain the main properties of the basic quadratic function • y = a(x - b)2 • using graphical methods. www.mathsrevision.com • Understand the links between graphs of the form • y = x2 and y = a(x – b)2

  23. Quadratic Functions Int 2 Now try MIA Ex 3 Q2 (page 222) www.mathsrevision.com Quadratic of the form f(x) = a(x - b)2

  24. Quadratic of the form f(x) = a(x - b)2 Key Features Symmetry about x = b Vertex at (b,0) Cuts y - axis at x = 0 a > 0 the vertex (b,0) is a minimum turning point. a < 0 the vertex (b,0) is a maximum turning point.

  25. Quadratic Functions Int 2 Example The parabola has the form f(x) = a(x – b)2. The vertex is the point (2,0) so b = 2. The point (5,36) lies on the graph. Find the equation of the function. Solution f(x) = a (x - b)2 (5,36) www.mathsrevision.com f(5) = a ( 5 - 2)2 (2,0) 36 = a x 9 a = 36 ÷ 9 f(x) = 4(x-2)2 a = 4

  26. Quadratic Functions Int 2 Now try MIA Ex 3 Q4 and Q5 (page 222) www.mathsrevision.com

  27. Quadratic Functions Int 2 Homework MIA Ex 4 (page 222) www.mathsrevision.com

  28. Int 2 Starter f(x) www.mathsrevision.com (5,25) x

  29. Quadratic Functions Int 2 Learning Intention Success Criteria • To know the properties of a quadratic function. • To explain the main properties of the basic quadratic function • y = a(x-b)2 + c • using graphical methods. • Understand the links between the graph of the form • y = x2 • and • y = a(x-b)2 + c www.mathsrevision.com

  30. Quadratic Functions Int 2 Every quadratic function can be written in the form y = a(x - b)2+c The curve y= f(x) is a parabola axis of symmetry at x = b Y - intercept www.mathsrevision.com Vertex or turning point at (b,c) (b,c) Cuts y-axis when x = 0 y = a(x – b)2 + c x = b a > 0 minimum turning point a < 0 maximum turning point

  31. Quadratic Functions y = a(x-b)2+c Int 2 Example 1 Sketch the graph y = (x - 3)2 + 2 a = 1 b = 3 c = 2 = (3,2) Vertex / turning point is (b,c) y Axis of symmetry at b = 3 www.mathsrevision.com (0,11) y = (0 - 3)2 + 2 = 11 (3,2) x

  32. Quadratic Functions y = a(x-b)2+c Int 2 Example2 Sketch the graph y = -(x + 2)2 + 1 a = -1 b = -2 c = 1 = (-2,1) Vertex / turning point is (b,c) y Axis of symmetry at b = -2 www.mathsrevision.com (-2,1) y = -(0 + 2)2 + 1 = -3 x (0,-3)

  33. Quadratic Functions y = a(x-b)2+c Int 2 Example Write down equation of the curve Given a = 1 or a = -1 a < 0 maximum turning point a = -1 (-3,5) Vertex / turning point is (-3,5) www.mathsrevision.com b = -3 (0,-4) c = 5 y = -(x + 3)2 + 5

  34. Quadratic Functions Int 2 Now try MIA Ex 5 Q1 and Q2 (page 225) www.mathsrevision.com

  35. Quadratic of the form f(x) = a(x - b)2 + c Cuts y - axis when x=0 Symmetry about x =b Vertex / turning point at (b,c) a > 0 the vertex is a minimum. a < 0 the vertex is a maximum.

  36. Quadratic Functions Int 2 Now try MIA Ex6 (page 226) www.mathsrevision.com

  37. Int 2 Starter f(x) www.mathsrevision.com x (3,-6)

  38. Quadratic Functions Int 2 Learning Intention Success Criteria • To interpret the keyPoints of the factorised form of a quadratic function. • To show factorised form of a quadratic function. www.mathsrevision.com

  39. Quadratic Functions Int 2 Some quadratic functions can be written in the factorised form y = (x - a)(x - b) The zeros / roots of this function occur when y = 0 (x - a)(x - b) = 0 x = a and x = b www.mathsrevision.com Note: The a,b in this form are NOT the a,b in the form f(x) ax2 + bx + c

  40. Q. Find the zeros, axis of symmetry and turning point for f(x) = (x - 2)(x - 4) Zero’s at x = 2 and x = 4 Axis of symmetry ALWAYS halfway between x = 2 and x = 4 x =3 (3,-1) Y – coordinate - turning point y = (3 - 2)(3 - 4) = -1

  41. Quadratic Functions Int 2 Now try MIA Ex7 (page 227) www.mathsrevision.com

  42. Solving Quadratic Int 2 Finding roots graphically Finding roots by factorising www.mathsrevision.com Finding roots using formula created by Mr. Lafferty

  43. Starter Int 2 www.mathsrevision.com created by Mr. Lafferty

  44. Graphical Roots Int 2 Learning Intention Success Criteria • Use graphs to locate roots of a quadratic equation. • To sow how to solve quadratic equations graphically. • Solve quadratic equations by finding the roots. www.mathsrevision.com created by Mr. Lafferty

  45. f(x) f(x) x 1 3 x -2 -4 Graphical Roots Int 2 Finding the roots (solve) of a quadratic equation is simply finding where the graph crosses the x – axis ie. f(x) = 0 Example : Find the roots (solve) of the quadratic equations below. www.mathsrevision.com created by Mr. Lafferty

  46. Graphical Roots Int 2 www.mathsrevision.com created by Mr. Lafferty

  47. Graphical Roots Int 2 www.mathsrevision.com created by Mr. Lafferty

  48. Graphical Roots Int 2 www.mathsrevision.com created by Mr. Lafferty

  49. Graphical Roots Int 2 Now try MIA Ex 8 Ch14 (page 228) www.mathsrevision.com created by Mr. Lafferty

  50. Starter Int 2 Q. Write down the function represented by the graphs below given that they are of the form and a is either 1 or -1. y = a(x - b)2 + c f(x) f(x) www.mathsrevision.com (1,4) x (0,-2) x (-2,-4) (2,-2) created by Mr. Lafferty

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