Functions CE/MAT/NOTES/FT

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1 數 Functions 數,! CE/MAT/NOTES/FT

2 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Contents 錄 : Functions 數 Change of subject in formula Notation and transformation on function 數 Knowledge and sketching of linear and quadratic function 數識 Eponential and logarithmic functions 數數數 Appendi I: Transformation on Functions 數 Appendi II: Solving Inequalities by Graphical Method 不

3 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: HKCEE Mathematics Syllabus Functions: 數 - 數

4 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: (A) Change of subject in formula e.g. Change the subject in the formula below to a. 3 a. e = c + a b b. d = a b + ef ac c. e = ab c + f a d e c = a e c = b 3 a = 3 3 a e c b 3 b = a e c b acd = a b + ef a b acd = ef a b acd + ef cd ± a = ( cd) b = 0 4bef Will not be eamined in HKCEE Maths (B) Notation and transformation on function 數 What is Function 數?. Function is a machine, which will have different output [ f () or y ] when having different input [ ] /. Function Transformation: y = ± c f ( a ± b) ± d / 拉 / 拉 / 3

5 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: (C) Knowledge and sketching of linear and quadratic function 數識 Linear Function: f ( ) m + c y O = or y = m + c, with slope m and y-intercept c. y = m + c A + By + C = 0 into y = m + c A C slope is and y-intercept is. B B 4

6 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Quadratic Function: y f ( ) = a + b + c y y = a + b + c y = a + b + c O O. Symmetry : A parabola is symmetrical about this vertical line.. Verte : i. The verte of a parabola is the point where the line of symmetry cuts the parabola. ii. At the verte, the parabola is either at its minimum point or maimum point. 3. Direction of opening : i. a > 0, opening upwards The verte is the minimum point of the parabola. iii. a < 0, opening downwards The verte is the minimum point of the parabola. 4. y-intercept y : i. The parabola cuts the y-ais at a point. ii. The value of y when = intercept : i. The parabola cuts the -ais at a point ii. May be distinct points, point or no point. iii. The value of when y=0 Questions: But How to find Verte?? Answers: Method of Completing Square.. 5

7 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Method of Completing Square Usage: To find the verte y = a + b + c => y = a( h) + k y = a + b + c b y = a + + c a y = a + b a b + a b + c a a y b b a + + c a a 4a = y = a + b a 4ac b + 4a The verte (h, k) of the quadratic function y = a + b + c are as follows: b h =, a b 4ac k = 4a e.g. find the verte of the following functions. y = 6 5 y = y = [ (3) + 3 ] 3 y = ( 3) 4 5 y = 3( 5) 8 y = 3( (.5) +.5 ) 8 + (3)(.5 ) Verte: (3, -4), Minimum Point y = 3(.5) 8 + (3)(.5 ) y = 3 (.5) Verte: (.5, 0.75), Maimum Point 6

8 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Summary of the Quadratic functions 數 7

9 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Solving Inequalities by Graphical Method 不 y O a y = k y = f ( ) i. f ( ) > k, < a ii. f ( ) < k, > a iii. f ( ) k, a iii. f ( ) k, a e.g. 6 5 > 0 e.g 6 5 < 0 e.g 6 5 < 4 e.g 6 5 > 4 8

10 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: (D) Eponential and logarithmic functions 數數數 Eponential Function: y = ka, a > 0, a, k 0. e.g. y 4( 5 ) =, 3 e.g. Sketch the curve of ( ) f = 9

11 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Properties of Eponential Functions: a. k > 0 and a > e.g. f ( ) = ( ) = f

12 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: b. k < 0 and > f = a e.g. ( ) ( ) = f ( ) = f 4 4

13 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: c. k > 0 and 0 < < a e.g. f ( ) = ( ) = f f ) ( ) = ( 4 4

14 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: d. k < 0 and 0 < < a e.g. f ( ) = ( ) = f f ) ( ) = (

15 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Application Eponential Problems: Depreciation e.g. The value of a car decreases % each year. Its value is $50000 this year. (a) What will be the value of the car after t years? Epress your answer in terms of t. (b) What will be the value of the car after 0 years? Correct your answer to the nearest dollar. (a) The value of the car after t years t = 50000( %) t = 50000(98%) (b) The value of the car after 0 years = 50000( %) 0 = 50000(98%) = $,

16 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Application Eponential Problems: Populations e.g. The population of a certain type of cell is doubled to the previous year. Assume the population of that cell is 0 today. (a) Complete the table. Days after today (D) Number of cells (n) (b) Epress n in terms of D. (c) Find n when D = 5. (d) Plot the graph n against D. (e) Estimate the minimum number of days required for 00 cells to be produced. 5

17 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: (a) Refer to the table above (b) n = 0( D ) (c) n = 0( 5 ) = = (d) 5 D 00 = 0( ) D 0 = ( ) D log 0 = log log 0 = D log log 0 D = = log 6

18 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Logarithmic Functions: e.g. y = log y = log, > 0.. Definition of Log 數 If 若 y = 0, Then = log y or = log y 0. log 0 =, log = 0, log 0 = undefined, 4. log( k ) = undefined, k is any positive number 數 =, M, N > 0 5. log ( MN ) log M + log N M 6. log = log M log N, M, N > 0 N 7. log M n = nlog M, M > 0 log 8. 0 a = a 7

19 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Application Logarithmic Problems: The Decibel Scale Decibel (db) is the unit for measuring the loudness (L) of sound, which is defined as L = 0log I I 0, where I is the intensity of sound and I 0 is the threshold of hearing. As scientists have found that I 0 = 0 W/m. e.g. Given that the intensity of a conversation is 0. 00W/m, find the loudness of the noise in decibels. L = 0log I I 0.00 = 0log 0 0 = 0log(0 = 0(9) = 90 db 9 ) e.g. The noise level of a Mong Kok MTR station is 50 db normally. When a MTR train enters the platform, the sound intensity record. is 00 times the noise level of the MTR station at normal. Find the loudness of sound in decibels of the sound intensity when the MTR train enters the platform. In Ln = 0log = 50, I 00In L = 0log I 0 In = 0(log00 + log ) 0 I 0 In = 0 log00 + 0log I = = 50 db 0 8

20 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Application Logarithmic Problems: Richter scale (M) Richter scale (M): A Scale for measuring Earth Quake log E = M e.g. The magnitude of Taiwan s 9 earthquake was 7.3 in the Richter scale. Find the energy released from the earthquake. log E = M log E = (7.3) log E = E =0 5 E = 5.60 J 9

21 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Eam Types Questions: e.g. f ( ) f = =, f =? = = = Remarks: Chang in variable in functions f = and + e.g. If ( ) ( ) g( ) f =? ( ) g() f = = = + 4 = => = Remarks: find variable in functions g( ) =, what is the value of such that 3 0

22 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. If f ( ) = + and g ( ) =, then [ g( ) ] = f [ g( ) ] = f [ ] = ( ) + = 4 Remarks: Composite functions f? e.g. If f ( ) = 4 +, then ( 3) = Let = 3, f 3 = 3 3 () 3 = 4( ) + ( ) = = f? = Remarks: Chang in variable in functions e.g. If 7 =, then =? 7 = log 7 = log log 7 = log log =.3. log7 Remarks: log a b = blog a!!!

23 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. If log < 0, then what is the range of? log < 0 log 0 0 < 0 < Remarks: Log Properties e.g. If 3 log = a, then log = 3 log = a 3 log = a a log = 3 a = 0 3 log = a 0 3 a = log0 6 log a = log0 6 log a = () 6 log a = 6 Remarks: Remarks: Log Properties

24 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. Given that log = a and log 3 = b, then log 5 = log5 = log(35) = log 3 + log5 0 = b + log = b + log0 = b + a log = b a + Remarks: Remarks: Log Properties e.g. What is the quadratic function represents the curve? Method (Slow Method, Little Knowledge): Let f ( ) = a + b + c put (-,0), (,0) and (0,-4) give 0 = a + b + c, 0 = 4a + b + c, 4 = c Solving the above 3 three equations give a =, b = - and c = -4 Method (Fast Method, More Knowledge): y = a( )( + ) {Think: [ a ( )( + ) = 0 ]} 4 = a (0 )(0 + ) a = y = ( )( + ) y = 4 3

25 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. What is the range for a, b and c and of the following quadratic function?. Opening Downwards: a < 0,. Symmetry on +ve -ais and and a < 0: b > ve y-intercept => c < 0 4. distinct real roots => > 0 e.g. Which of the following figures is the graph of A. B. y y =? y O O C. D. y y O O 黎 Sir = 0, y = 0 =, when increase, Answer: C : Answer: C y = increases. 4

26 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. The following figure shows the sketch of the graphs of y = logb and y = log c. y y = log a y = log a, y = logb O y = log c Which of the following is correct? A. c < a < b B. b < a < c C. a < b < c D. a < c < b 5

27 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. (a) Describe the way of translating the graph of f ( ) y = f ( + ) +. (b) If f ( ) =, sketch the graphs of the functions f ( ) y = f ( + ) + on the same figure. y = to the graph of y = and 6

28 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: e.g. (a) Sketch the graph C : y = f ( ) and C : y = f ( ) figure below for y = f ( ). (b) Find the -intercept and y-intercept of (i) C : y = f ( ) C : y = f (ii) ( ) 3 3 on the same 7

29 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Appendi I: Transformation on Functions 數 y = ± c f ( ± a ± b) ± d / -> ± d / -> ± b 拉 / -> 0 < + a <, + a > -> a 拉 / -> + c >, 0 < + c < -> c f ( ) = sin f ( ) = sin + f ( ) = sin f ( ) = sin f ( ) = sin( + ) f ( ) = sin( ) f ( ) = sin( ) f ( ) = sin f ( ) = sin 拉 f ( ) = sin 拉 f ( ) = sin 8

30 6/F, Hollywood Plaza, 60 Nathan Road, Mongkok, Kowloon, Hong Kong. Enrollment Hotline: Appendi II: Solving Inequalities by Graphical Method 不 e.g Solve for 3 + > > 0 ( )( ) > 0 < or > e.g. Solve for 3 + < < 0 ( )( ) < 0 < < e.g. Solve for 3 + < < 0.5 There is no solution in. e.g. Solve for 6 5 > > 0.5 All real values of ecept.5 數了.5 The End. 9

Math 103 Final Exam Review Problems Rockville Campus Fall 2006

Math 103 Final Exam Review Problems Rockville Campus Fall 2006 Math Final Eam Review Problems Rockville Campus Fall. Define a. relation b. function. For each graph below, eplain why it is or is not a function. a. b. c. d.. Given + y = a. Find the -intercept. b. Find