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Understanding Quadratic Functions in Mathematics

This comprehensive guide covers the fundamental aspects of quadratic functions, including their definitions, properties, and graphical representations. You'll learn about the standard form (y = ax² + c), factorized form (y = a(x - b)²), and how to determine characteristics such as the vertex and axis of symmetry. With examples and exercises, this resource helps you grasp the concepts of quadratic equations, their applications in real-world scenarios, and how to sketch their graphs effectively.

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Understanding Quadratic Functions in Mathematics

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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

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