CHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis

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1 ADDITIONAL MATHEMATICS MODULE 5 QUADRATIC FUNCTIONS

2 CHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercise Shape of Graphs of quadratic functions 5 Eercise The position of graphs of quadratic functions 6 Eercise Maimum and minimum value of a quadratic functions 9 Eercise Assessment 1 Answer 18 1

3 CHAPTER 3 : QUADRATIC FUNCTIONS Quadratic Functions f a b c f a p q a 0 a 0 p,q p,q f min q f ma q Minimum point f min q Corresponding value Coordinate minimum p p,q Symmetrical equation, p a 0 a 0 Maimum point f min q Corresponding value Coordinate maimum p p,q Symmetrical equation, =p

4 A: RECOGNISING THE QUADRATIC FUNCTIONS Eample 1: Epress the following quadratic functions in general form. a f 5 4 bg 7 3 Solution f 5 4 (a) 4 5 (b) g Eercise 1 1. Determine whether each of the following functions is a quadratic functions (a) f 3 1 (b) f 1 3 (c) f 5 (d) f 7 5. Epress the following quadratic functions in general form (a) f 3 (b) f 5 (c) f 4 (d) f

5 B. Graph of quadratic functions The graphs of quadratic functions, f a b c tabulated values of and f can be plotted based on the Eample 1 Plot the following quadratic function graphs. Then label the ais of symmetry, maimum point or minimum point for each graphs. (a) f for 3 3 (b) f 4 for 5 Solution (a) f f

6 (b) f f Eercise 1. Complete the table of values for each of the following quadratic functions. Plot the graph f(). Label the ais of symmetry, maimum point or minimum point of the graph. (a) f f

7 (b) f f 6

8 C. Shape of graphs of quadratic functions f a b c a 0 a 0 Eercise 1 1 Recognise the shapes of the graphs for the following quadratic functions. Quadratic functions a. f 4 Coefficient of Shape of the graph b. f 3 10 c. f 7 4 d. f 4 9 7

9 D. The position of the graphs of quadratic functions with types of roots for f 0 The roots of f a b c 0 are given by b b 4ac a b 4ac 0 b 4ac 0 b 4ac 0 Position of the graph a 0 a 0 Type of root a 0 a 0 Two different root 8

10 Eercise 4 1. Based on each of the following quadratic function graphs given, determine the type of the roots of f 0 (a) (b) (d) (c). Find the range of values of k if each of the following graphs of the quadratic functions meets the -ais at two points. (a) f k (b) f q 8 q 6 9

11 3. Find the values of p if each of the following graph of the quadratic functions touches the -ais at one point. (a) f p 6 9 (b) f 3 4 p 1 4. Find the range of values of n if each of the following graphs of the quadratic functions does not meet the -ais (a) f n 1 n 1 (b) f n (c) f 6n 3n 1 (d) f n 1 n 10

12 . Maimum and Minimum values of A Quadratic Functions f a f a p q b c a 0 a 0 f ma q f min q Eample 1 Find the minimum value of each of the following quadratic functions. Hence, find the corresponding value of. (a) f 4 3 (b) f 6 5 Solution (a) f The minimum value of f is -7 when 0 or 11

13 1 (b) 5 6 f The minimum value of f is 1 when 0 3 or 3 Eercise 5 1. Determine the maimum or minimum value of each of the following quadratic function and find the corresponding value of. (a) 7 8 f (b) 5 6 f

14 . Epress the following quadratic functions in the form of a p q find the maimum value or minimum value of f value of.. Hence, and state the corresponding (a) f 8 7 (b) f

15 Assessments 1. Given that the minimum value of the function f 4 m value of m. is 6, find the. Given the graph y p q has a maimum point of 3 4 value of p and q.,. Find the 3. The quadratic equation 1 p 4 value of p. has two distinct roots. Find the range of 14

16 4. Sketch the graph of 3 f for 3 and state the range of f. 15

17 y -5 (6,-5) 5. The diagram given shows the graph of the quadratic functions p 4 where p is a constant. Find (a) the value of p, (b) the ais symmetry, (c) the coordinates of the maimum point. y, 16

18 6. The minimum value of the quadratic function f 6h 10h 4 a 4h, where a and h are constants. is (a) Show that a h by using the method of completing the square. (b) Hence, find the values of h and a if the function has an ais of symmetry a 4 17

19 Answer Eercise 1 Eercise 1. (a)yes (b) No (c) Yes (d) No (a) Parabola with minimum point (b) Parabola with maimum point. (a) f 3 (b) f 5 (c) f 4 (d) f Eercise 3 Eercise 4 (a) 1, (b) -3, (c) -1, (d) 4, 1 (a) Has two equal roots (b) No real roots (c) Two different roots (d) Two different roots. (a) k (b) q 8 3. (a) p=1 (b) P=14 4. (a) n 1 (b) n (c) 1 n 6 (d) n

20 Eercise 5 1 (a) Minimum value = 9, (b) Maimum value, 8 4. (a) f 4 3, 5 min imum value, minimum value =3, 4 (b) f 3, 1. q 3. m=10 3. p 3 or p 5 4. f 9 Assessment 5. (a) p=3 (b) the ais symmetry 3 (c) maimum point 3,4 6. (b) a, a 5 19

21 ADDITIONAL MATHEMATICS MODULE 6 QUADRATIC FUNCTIONS 0

22 CHAPTER 3 : QUADRATIC FUNCTIONS MODULE Sketching graph of quadratic functions 3 Eercise Quadratic Inequalities 6-7 Eercise Assessment Answer

23 Chapter 3 : Quadratic Functions 3.1 Sketching Graph of quadratic functions Sketching Graph of quadratic functions shape of the graph a? position of the graph b -4ac ma, a(+p) +q (-p,q) min, a(+p) +q (-p,q) f()=0 =? f(0) mark all points and draw a smooth parabola Eample 1 Sketch the graph of the following quadratic function f

24 i) a 1 0 shape of the curve ii) b 4ac 41 0 = 4 0 iii) f = 1 1 = = 1 1 the minimum point is 1, 1 iv) f = 0 or v) f = 0 vi) Sketch the graph y f 11, 3

25 Eercise 1 1. Sketch the graph of the following quadratic functions. (a) f 4 6 (b) f 6 9 4

26 . Sketch the graph of f 3 for 3and state the range of f() 3. Given y h k p q. (a) Find the value of (i) p (ii) q In terms of and/ or k (b) If k=3, state the ais of symmetry of the curve. (c) It is given that the line y=4 touches the curve y h k (i) State h in term of k. (ii) Hence, sketch the graph of the curve 5

27 3. Quadratic Inequalities Quadratic Inequalities coefficient positive, a 0 f a b c Sketch the graph minimum/ maimum point, two intercepts, State the range 6

28 Eample 1 Find the range of values of for which 15 0 Solution 15 0 f Let 15 = 3 5 When f or 5 3 f 0 5 For or 3 Eercise 1. Find the range of value of for which 8 1. Find the range of values of for which

29 3. Given that the quadratic equation m 5m 1 has roots, find the range of value m. 4. (a) Find the range of values of q for which 6 36 q has roots (b) Show that 5 6 is always negative for all values of. 8

30 Assessment 1. Sketch the graph of the quadratic function 1 f.. Given the quadratic function f 5 3 f in the forms of m n p, (a) Epress Determine whether f (b) Sketch the graph of f. where m, n and p are constants. has maimum or minimum point and state the value. 9

31 3. Skecth the graph of y 4 3 and find the range of values of y for 0 4. Given that k k 1 has no roots, find the range of values of k. 30

32 5. (a) Given that 11 6, find the range of value of. (b) Given that the line y 3 4 and the curve 4y h 0 show that the line do not meet if h 1 31

33 Answer 3. y Eercise 1 1. (a) y 3, , 6 3 Assessment (b) y 1. 3 f y f. (a) f (b) a ,n, p 6 5,

34 3., k (a) 1 or 6 (b) n 1 1

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