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# MT T4 (Bab 3: Fungsi Kuadratik)

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Matematik Tmbahan Tingkatan 4 Bab 3 Fungsi Kuadratik.

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### MT T4 (Bab 3: Fungsi Kuadratik)

2. 2. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [2][2] General form of quadratic function:- f(x) = ax2 +bx + cf(x) = ax2 +bx + c a ≠ 0 b c constants Examples: f(x) = 3x2 + 5x + 2 f(x) = -2x2 - 5x + 4 a = 3 b = 5 c = 2 a = -2 b = -5 c = 4
3. 3. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [3][3]  Example 1: x -2 -1 0 1 2 3 4 f(x) -5 0 3 4 3 0 -5 By using a suitable scale, plot the tabulated graph below. f(x) x -1 ― -2 ― -3 ― -4 ― -5 ― -6 ― 1 ― 2 ― 3 ― 4 ― 5 ― I -1 I -2 I -3 I 1 I 2 I 3 I 4 I 5 xx xx xx xx xx xx xx Max. (1 , 4)Max. (1 , 4) Axis of symmetry, x=1Axis of symmetry, x=1
4. 4. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [4][4]  Exercise: Draw a table and find the value of f(x) for the following quadratic function with the given range of x. By using suitable scale, plot the graph. From the graph, state the axis of symmetry and a maximum or minimum point. f(x) = x2 – 3x -10 ; -3 ≤ x ≤ 5 x -3 -2 -1 0 1 2 3 4 5 f(x) 8 0 -6 -10 -12 -12 -10 -6 0 f(x) x -2 ― -4 ― -6 ― -8 ― -10 ― -12 ― 2 ― 4 ― 6 ― 8 ― 10 ― I -2 I -4 I 2 I 4 I 6 I 8 I 10 0 x x x x x x x x x Axis of symmetry, x=1.5Axis of symmetry, x=1.5 Min. (1.5 , -12.25)Min. (1.5 , -12.25)
5. 5. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [5][5] Shapes of graphs of quadratic function:- a > 0a > 0 (+ve)(+ve) a < 0a < 0 (-ve)(-ve) f(x) x 0 Minimum graph (Smile) Minimum point along the graph f(x) x 0 Maximum point along the graph Maximum graph (Sad)
6. 6. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [6][6] Shapes of graphs of quadratic function:- a > 0a > 0 (+ve)(+ve) a < 0a < 0 (-ve)(-ve) f(x) x 0 Minimum graph (Smile) Vertex f(x) x 0 Maximum graph (Sad) Vertex Axis of symmetry Axis of symmetry
7. 7. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [7][7] Identify the shape of each graph of the following.Identify the shape of each graph of the following.Exercise 3.1.3Exercise 3.1.3 a) f(x) = 2x2 + 3x - 1 a = 2  a > 0 b) f(x) = 5 + 3x - x2 = -x2 + 3x +5 a = -1  a < 0 Min. Max. c) f(x) = x(4 – 2x) = 4x – 2x2 = -2x2 +4x a = -2  a < 0 Max.
8. 8. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [8][8] f(x)=axf(x)=ax22 +bx+c+bx+c Value of discriminant bb22 -4ac-4ac Type of roots Shape of graphs Points of intersection with x-axis f(x)= x2 -6x+8 = 4 (> 0) Two distinct roots Two x-intercepts f(x)= x2 -2x+1 = 0 2 equal roots Vertex at x-axis f(x)= x2 +4x+6 = -8 (< 0) No real roots No x-intercept f(x)= -x2 +8x-15 = 4 (> 0) Two distinct roots Two x-intercepts f(x)= -x2 +4x-4 = 0 2 equal roots Vertex at x-axis f(x)= -x2 +2x-2 = -4 (< 0) No real roots No x-intercept
9. 9. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [9][9] Discriminant (bb22 – 4ac– 4ac) a > 0 (+ve)a > 0 (+ve) a < 0 (-ve)a < 0 (-ve) bb22 – 4ac > 0– 4ac > 0 Two distinct roots bb22 – 4ac = 0– 4ac = 0 Two equal roots bb22 – 4ac < 0– 4ac < 0 No real roots x x x y 0 x x y 0 x y 0 x x x y 0 x x y 0 x y 0 Min. Min. Min. Max. Max. Max.
10. 10. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [10][10]  Example: Find the values of p for which the x-axis is a tangent to the graph of f(x) = pxf(x) = px22 +8x + p - 6+8x + p - 6 Solution:Solution: Informations: •x-axis is a tangent to the graph •a = p, b = 8, c = (p – 6)a = p, b = 8, c = (p – 6)  bb22 – 4ac =– 4ac = 00  bb22 – 4ac =– 4ac = 00 x x y 0 Min. x x y 0 Max. OR tangent 82 – 4(p)(p – 6) = 0 64 – 4(p2 – 6p) = 0 64 – 4p2 + 24p = 0 -4p2 + 24p + 64 = 0 -4(p2 - 6p - 16) = 0 p2 - 6p - 16 = 0 Expand Rearranged  General form Factorised (p – 8)(p +2) = 0 p – 8 = 0 p = 8p = 8 or; p + 2 = 0 p = -2p = -2
11. 11. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS [11][11]  Example: Find the values of p for which the x-axis is a tangent to the graph of f(x) = pxf(x) = px22 +8x + p - 6+8x + p - 6 Solution:Solution: Informations: •x-axis is a tangent to the graph •a = p, b = 8, c = (p – 6)a = p, b = 8, c = (p – 6)  bb22 – 4ac =– 4ac = 00  bb22 – 4ac =– 4ac = 00 x x y 0 Min. x x y 0 Max. OR tangent Check:Check: SubtituteSubtitute p into f(xp into f(x),), When p = 8; f(x) = 8x2 +8x + (8-6) = 8x2 + 8x + 2 When p = -2; f(x) = -2x2 +8x + (-2-6) = -2x2 +8x - 8 SHOW GRAPHSHOW GRAPH SHOW GRAPHSHOW GRAPH
12. 12. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS 3.5 3 2.5 2 1.5 1 0.5 -0.5 -2 -1 1 2 3 4 xA = -0.5 f x( ) = 8⋅x2 +8⋅x+2 A BACKBACK
13. 13. SMT PONTIAN, JOHOR 3.1 QUADRATIC FUNCTIONS & THEIR3.1 QUADRATIC FUNCTIONS & THEIR GRAPHSGRAPHS 1 0.5 -0.5 -1 -1.5 -2 -2.5 1 2 3 4 f x( ) = -2⋅x2 +8⋅x( )-8 BACKBACK
14. 14. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [1][1]  In this subtopic, we will determine:- The vertex or the turning pointThe vertex or the turning pointThe vertex or the turning pointThe vertex or the turning point The axis of symmetryThe axis of symmetryThe axis of symmetryThe axis of symmetry
15. 15. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [2][2] Determine the vertex and axis of symmetry by using completing thecompleting the square.square. f(x) = a(x + p)2 +qf(x) = a(x + p)2 +q Still remember how to solve byStill remember how to solve by completing the square?????completing the square?????
16. 16. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [2][2] f(x) = ax2 + bx + c       ++= a c x a b xa 2       +      −      ++= a c a b a b x a b xa 22 2 22             −+      += 22 22 a b a c a b xa             −+      += 22 22 a b a c a a b xa a b p 2 =             −= 2 2a b a c aq f(x) = a(x + p)2 +qf(x) = a(x + p)2 +q x = -p y = q Vertex =Vertex = (-p , q)(-p , q)
17. 17. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [3][3] Example 1: State the maximum or minimum value of the following quadratic functions:- a) f(x) = 3(x + 2)2 - 6 a = 3 (+ve)  Minimum (smile) x = -2 y = -6 Why x = -2 ???? Why y = -6 ???? f(x) = 3(x + 2)2 - 6 When (x+2)2 = 0, then f(x) = -6 f(x) = 3[0] – 6 = -6 (x+2)2 = 0 x + 2 = 0 x = -2x = -2 Therefore; the vertex is (-2, -6)vertex is (-2, -6)
18. 18. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [4][4]  Example 2 Express f(x) = x2 – 10x + 9 in the form of f(x) =(x +a)2 +b , with a and b are constants. Hence, state the maximum or minimum value of f(x) and the corresponding value of x. Solution:-Solution:- a = 1, b = -10, c = 9 9 2 10 2 10 10)( 22 2 +      − −      − +−= xxxf ( ) ( ) 95510 222 +−−−+−= xx ( ) 9255 2 +−−= x ( ) 165 2 −−= x f(x) =(x +a)2 +b Therefore; a = -5, b = -16 a > 0 (+ve)  Minimum (smile) Min. value of f(x) = -16 Min. (5, -16)Min. (5, -16) x y 0 5 -16 - f(x) = x2 – 10x + 9 ( ) 5 05 05 2 = =− =− x x x
19. 19. SMT PONTIAN, JOHOR 3.2 MAX. & MIN. VALUES OF QUADRATIC3.2 MAX. & MIN. VALUES OF QUADRATIC FUNCTIONSFUNCTIONS [5][5]  Exercises: 1. Find the maximum value of the function f(x) = 5 + x – xf(x) = 5 + x – x22 . State the value of x, so that f(x) has a maximum value. 2. Express f(x) = 2xf(x) = 2x22 + 4x +7+ 4x +7 in the form of complete square. Hence, find the maximum or minimum value of function f(x). The maximum value of the function f(x) = ??f(x) = ?? x = ??x = ?? 4 21 )( =xf 2 1 =x f(x) = 2(x + 1)f(x) = 2(x + 1)22 + 5+ 5 The minimum value of the function f(x) = 5f(x) = 5
20. 20. SMT PONTIAN, JOHOR 3.3 SKETCH GRAPHS OF QUADRATIC3.3 SKETCH GRAPHS OF QUADRATIC FUNCTIONFUNCTION [1][1]  The steps involved in sketching graphs of quadratic functions f(x) = ax2 + bx +c are:- 11. Determine the shape of the graph: Value of a. 22. Find the maximum or minimum point by expressing in f(x) in the form of f(x) = a(x + p)2 +q. a > 0 : Min. (Smile) a < 0 : Max. (Sad) a > 0 : Min. (Smile) a < 0 : Max. (Sad) Completing the squareCompleting the square 33. Determine the point of intersection with x-axis, if its exists, by solving the equation f(x) = 0f(x) = 0. xx11 & x& x22xx11 & x& x22 44. Determine the point of intersection with the y-axis by finding f(x)f(x) when x = 0when x = 0 [value of f(0)f(0)]. f(x) = axf(x) = ax22 + bx +c+ bx +c = a(0)= a(0)22 + b(0) + c+ b(0) + c = c= c f(x) = axf(x) = ax22 + bx +c+ bx +c = a(0)= a(0)22 + b(0) + c+ b(0) + c = c= c 55. Mark the points and draw smooth parabola through all the points.
21. 21. SMT PONTIAN, JOHOR 3.3 SKETCH GRAPHS OF QUADRATIC3.3 SKETCH GRAPHS OF QUADRATIC FUNCTIONFUNCTION [2][2]  Example: Sketch the following quadratic function and state the max. or min. point. 1. a = -3 : Maximum (Sad) a = -3, b = -18, c = -22 2 .             − −      − −−= 3 22 3 18 3)( 2 xxxf ( )             − −−−−= 3 22 63 2 xx             ++−= 3 22 63 2 xx             +      −      ++−= 3 22 2 6 2 6 63 22 2 xx ( ) ( )             +−++−= 3 22 3363 222 xx ( )             +−+−= 3 22 933 2 x ( )     −+−= 3 5 33 2 x ( ) ( )3 3 5 33 2 −−+−= x ( ) 533 2 ++−= x When (x + 3)2 = 0, x = -3 Max. value of f(x) = y = 5 f(x) = -3x2 -18x - 22f(x) = -3x2 -18x - 22
22. 22. SMT PONTIAN, JOHOR 3.3 SKETCH GRAPHS OF QUADRATIC3.3 SKETCH GRAPHS OF QUADRATIC FUNCTIONFUNCTION [3][3]  Example: Sketch the following quadratic function and state the max. or min. point. a = -3, b = -18, c = -22 3. Find the value of x1 & x2 [f(x) = y = 0] ( ) 5330 2 ++−= x ( ) 533 2 =+x ( ) 3 5 3 ±=+x 3 5 3 ±−=x 291.4 709.1 2 1 −= −= x x 4. When x = 0; f(x) = -3(0)2 – 18(0) – 22 = -22 y x -22 ― | -3 ― 5 0 f(x) = -3x2 -18x - 22f(x) = -3x2 -18x - 22
23. 23. SMT PONTIAN, JOHOR 3.3 SKETCH GRAPHS OF QUADRATIC3.3 SKETCH GRAPHS OF QUADRATIC FUNCTIONFUNCTION [4][4] Exercises: 1. Sketch the graph of f(x) = 4 –(xf(x) = 4 –(x – 3)– 3)22 for the domain 0 ≤ x ≤6. 2. Find the max. or min. value of the function y = 3(2x – 1)(x + 1) – x(4xy = 3(2x – 1)(x + 1) – x(4x – 5) + 2.– 5) + 2. Hence, sketch the graph of function y. y x -5 ― | 1 (3, 4) 0 | 5 | 6 x y = 3(2x – 1)(x + 1) – x(4x – 5) + 2 y = 3(2x2 + x - 1) – 4x2 + 5x + 2 = 6x2 + 3x – 3 - 4x2 + 5x + 2 = 6x2 + 8x - 1 Minimum value = -9 y x -1 ― (-2, -9) | 0.1213 x | -4.121
24. 24. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [1][1] The range of quadratic inequalities can be determined from the shape of the graph. a x y 0 b f(x) < 0f(x) < 0 for a < x < b a x y 0 b f(x) > 0f(x) > 0 for a < x < b f(x) < 0f(x) < 0 for a < x or x > b Minimum Maximum f(x) > 0f(x) > 0 for a < x or x > b
25. 25. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [2][2]  Example 1: Find the range of values of x which satisfies the inequality 0 ≤ x2 – 4x ≤ 5 0 ≤ x2 – 4x ≤ 5 0 ≤ x2 – 4x x2 – 4x ≤ 5 a > 0 : Minimumy x 0 x2 – 4x – 5 ≤ 0 y x 0 a > 0 : Minimum
26. 26. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [2][2]  Example 1: Find the range of values of x which satisfies the inequality 0 ≤ x2 – 4x ≤ 5 0 ≤ x2 – 4x x2 – 4x = 0 x(x – 4) = 0 x = 0 or; (x – 4) = 0 x = 4 y x 0 4 0 ≤ f(x) x2 – 4x – 5 ≤ 0 0 ≥ f(x) x2 – 4x – 5 = 0 (x – 5)(x + 1) = 0 (x – 5) = 0 x = 5 (x + 1) = 0 x = -1 y x -1 5
27. 27. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [3][3]  Example 1: Find the range of values of x which satisfies the inequality 0 ≤ x2 – 4x ≤ 5 f(x) x0 | 4 | 5 | -1 f(x) = x2 – 4x – 5 f(x) = x2 – 4x The range of x; -1 ≤ x ≤ 0 or 4 ≤ x ≤ 5
28. 28. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [4][4]  Example 2: Given that f(x) = 2x2 + px + 30 and that f(x) < 0 only when 3 < x < k. Find the values of p and k. 3 < x < k a > 0 (Max.) f(x) x 0 | k | 3 30 - f(x) < 0 a =2, b = p, c = 30 ( ) 24 384 38424 024014424 24024144 24012 24012 )30)(2(4)4(3 )2(2 )30)(2(4 3 22 22 22 2 2 2 − = −= =+++− −=++ −=+ −±=+ −±−= −±− = p p ppp ppp pp pp pp pp p = -16 Use formula method;
29. 29. SMT PONTIAN, JOHOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [4][4]  Example 2: Given that f(x) = 2x2 + px + 30 and that f(x) < 0 only when 3 < x < k. Find the values of p and k. 3 < x < k a > 0 (Max.) f(x) x 0 | k | 3 30 - f(x) < 0 a =2, b = p, c = 30 Use formula method; 4 416 4 1616 4 24025616 )2(2 )30)(2(4)16()16( 2 ± = ± = −± = −−±−− =k Substitute p = -16 into formula; k1 = 5 or; k2 = 3 kk22 = 3= 3 is already exist in the range 3 < x < k. Therefore, k = 5k = 5
30. 30. SMK SULTAN SULAIMAN SHAH, SELANGOR 3.4 QUADRATIC INEQUALITIES3.4 QUADRATIC INEQUALITIES [6][6]  Exercises: 1. Given that f(x) = 5x2 - 4x – 1, find the range of values of x, so that f(x) is positive. 2. Given that f(x) = x2 + 4x – 1 and g(x) = 6x + 2, find the range of values of x if f(x) > g(x). 1, 5 1 >−< xx 3,1 >−< xx