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Quadratic

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  • 1. QUADRATIC FUNCTIONS The word quadratic comes from the Latin word Quadratus which means square
  • 2. Chapter Objectives Understand the concept of quadratic functions and their graphs. Find maximum and minimum values of quadratic Functions.     Sketch graphs of quadratic functions Understand and use the concept of quadratic Inequalities.
  • 3. Recognising quadratic functions f : x  a x 2 + b x + c f (x) = a , b and c are constants a  0 The highest power of x is 2 x 2 x c + + a b
  • 4. Determine whether each of the following is a quadratic function.       f(x) = 2x 2  h(x) = 4x - 3x + 1 2  g(x) = x + 3x 2  k(x) = 5x + 7   g(x) = 3x + 1 2 x  h(x) = (x + 3) + 4 2
  • 5. a  0 f(x) = ax + bx + c 2 quadratic function 2 ax + bx + c quadratic expression 2 ax + bx + c = 3 quadratic equation
  • 6. Plotting the graphs of quadratic functions Based on given tabulated values   By constructing the table of values
  • 7. x f ( x ) -4 -3 0 -2 -1 1 2 3 4 -7 9 8 0 5 The table below shows some values of x and the corresponding values of f(x) of the function f(x) = 9 – x 2 8 5 0 -7 Plot the graph of the function Example   0 -2 -1 1 2 5 Select suitable scales on both axes and subsequently plot the Graph. Given the quadratic function f (x) = x 2 – 2x – 4. Plot the graph of the function for -3 ≤ x ≤ 5. We first construct the table of values of the function. 3 4 -5 -4 4 -1 -4 -1 4 11 11 -3 x f ( x )
  • 8. Shapes of graphs of quadratic functions f(x) = ax 2 + bx + c If a > 0 , then the graph of the function is a parabola with a min pt. If a < 0 , then the graph of the function is a parabola with a max pt. a > 0 a < 0 axes of symmetry Minimum point Maximum point
  • 9. Example Describe the shape of the graph of each of the following quadratic functions. Solution (a) f (x) = - 3x 2 – 4x + 5 (b) g (x) = 10x 2 + 6x + 3 (a) Since a = - 3 < 0 , the graph of the function is a parabola with a maximum pt. (b) Since a = 10 > 0 , the graph of the function is a parabola with a minimum pt.
  • 10. Relating the position of the graph of a quadratic function f(x) = ax 2 + bx + c with the for types of roots f(x) = 0 m n Referring to the graph, When f(x) = 0 , x = m and x = n m and n are the roots of the equation m and n are also the values of x where the graph intersects the x – axis. Therefore , the roots of f(x) = ax 2 + bx + c are the points where the graph of f(x) intersects the x – axis . Values of x when f(x) = 0
  • 11. In this respect , we have three cases : (I) If f (x) = ax 2 + bx + c has two distinct (different) roots , meaning b 2 – 4ac > 0 , then the graph of the function f (x) intersects at two distinct points. x x a > 0 a < 0
  • 12. Example 1 f (x) = 2x 2 –x -10 Points of intersection with the x – axis. f(x) = 0 When f(x) = 0 b 2 – 4ac = (-1) 2 – 4(2)(-10) = 81 , b 2 – 4ac > 0 a > 0 Hence 2x 2 –x -10 = 0 (2x – 5)( x + 2) = 0 X = 5/2 , -2
  • 13. Example 2 f (x) = -x 2 + 3x +10 Points of intersection with the x – axis. f(x) = 0 When f(x) = 0 b 2 – 4ac = 3 2 – 4(-1)(10) = 49 , b 2 – 4ac > 0 a < 0 Hence -x 2 + 3x +10 = 0 (5 - x)( x + 2) = 0 X = 5 , -2
  • 14. In this respect , we have three cases : (II) If f (x) = ax 2 + bx + c has two real and equal roots , meaning b 2 – 4ac = 0 , then the graph of the function f (x) intersects at only one point. x x a > 0 a < 0
  • 15. Example 3 f (x) = x 2 +6x + 9 When f(x) = 0 b 2 – 4ac = 6 2 – 4(1)(9) = 0 , b 2 – 4ac = 0 a > 0 Hence Point of intersection with the x – axis. f(x) = 0 x 2 + 6x + 9 = 0 (x + 3)( x + 3) = 0 x = -3
  • 16. In this respect , we have three cases : (III) If f (x) = ax 2 + bx + c does not have any real roots , meaning b 2 – 4ac < 0 , then the graph of the function f (x) does not intersect the x - axis. x x a > 0 a < 0
  • 17. Example 4 f (x) = 2x 2 + 5x + 7 There is NO point of intersection with the x – axis. When f(x) = 0 b 2 – 4ac = (5) 2 – 4(2)(7) = - 31 , b 2 – 4ac < 0 a > 0 Hence
  • 18. Summary (I) If b 2 – 4ac > 0 x x a < 0 (II) If b 2 – 4ac = 0 x x a < 0 a > 0 a > 0 (III) If b 2 – 4ac < 0 x x a < 0 a > 0
  • 19.  

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