1 / 42

Functions

Quadratic Functions. Int 2. Functions. Quadratic Functions y = ax 2. Quadratics y = ax 2 +c. Quadratics y = a(x-b) 2. Quadratics y = a(x-b) 2 + c. Factorised form y = (x-a)(x-b). Int 2. Starter. Functions. Int 2. Learning Intention. Success Criteria. Understand the term function.

rashad-fischer
Download Presentation

Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Int 2 Starter

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

  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 : 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. In the case The graph is a straight line We call this a Linear function.

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

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

  8. Int 2 Starter

  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

  10. Quadratic Functions Int 2 A function of the form f(x) = a x2 + b x + c is called a quadratic function 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

  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 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)

  15. Starter Int 2 Q1. Write down the equation of the quadratic. (2,100) Solution f(2) = 100 100 = a x 4 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. • 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) 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 (0,2) f(3) = a . 32 + 2 38 = a . 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)

  21. Starter Int 2 Q1. Write down the equation of the quadratic. (9,81) Solution f(9) = 81 81 = a x 9 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. • 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 2 Q5 (page 220) Quadratic of the form f(x) = ax2 + c

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

  25. 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.

  26. 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) f(5) = a ( 5 -2)2 (2,0) 36 = a × 9 a = 36 ÷ 9 f(x) = 4(x-2)2 a = 4

  27. Quadratic Functions Int 2 Now try MIA Ex 3 Q4 and Q5 (page 222)

  28. Quadratic Functions Int 2 Homework MIA Ex 4 (page 222)

  29. Int 2 Starter f(x) (5,25) x

  30. 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

  31. 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 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

  32. 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 (0,11) y = (0 - 3)2 + 2 = 11 (3,2) x

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

  34. 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) b = -3 (0,-4) c = 5 y = -(x + 3)2 + 5

  35. Quadratic Functions Int 2 Now try MIA Ex 5 Q1 and Q2 (page 225)

  36. 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.

  37. Quadratic Functions Int 2 Now try MIA Ex6 (page 226)

  38. Int 2 Starter f(x) x (3,-6)

  39. 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.

  40. 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 Note: The a,b in this form are NOT the a,b in the form f(x) ax2 + bx + c

  41. 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

  42. Quadratic Functions Int 2 Now try MIA Ex7 (page 227)

More Related