Functions and Graphs


 Abner Gibson
 2 years ago
 Views:
Transcription
1 PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form hsn.uk.net Page 1 Questions marked c SQA
2 PSf PSf 3. The diagram shows a sketch of part of the graph = log of = log 2 (). 2 () (8, b) (a) State the values of a and b. 1 (b) Sketch the graph of = log 2 ( + 1) 3. (a, 0) 3 Part Marks Level Calc. Content Answer U1 C2 (a) 1 A/B CN A7 a = 1, b = P1 Q10 (b) 3 A/B CN A3 sketch 1 pd: use log p q = 0 q = 1 and evaluate log p p k 2 ss: use a translation 3 ic: identif one point 4 ic: identif a second point 1 a = 1 and b = 3 a logshaped graph of the same orientation 3 sketch passes through (0, 3) (labelled) 4 sketch passes through (7, 0) (labelled) 2 4. hsn.uk.net Page 2 Questions marked c SQA
3 PSf hsn.uk.net Page 3 Questions marked c SQA
4 PSf 7. hsn.uk.net Page 4 Questions marked c SQA
5 PSf 8. hsn.uk.net Page 5 Questions marked c SQA
6 PSf hsn.uk.net Page 6 Questions marked c SQA
7 PSf n a suitable set of real numbers, functions f and g are defined b f () = and g() = 1 2. Find f ( g() ) in its simplest form. 3 hsn.uk.net Page 7 Questions marked c SQA
8 PSf 13. f () = 2 1, g() = 3 2 and h() = 4 1 (5 ). (a) Find a formula for k() where k() = f ( g() ). 2 (b) Find a formula for h ( k() ). 2 (c) What is the connection between the functions h and k? A function f is defined on the set of real numbers b f () =, = 1. 1 Find, in its simplest form, an epression for f ( f () ). 3 hsn.uk.net Page 8 Questions marked c SQA
9 PSf 15. The functions f and g, defined on suitable domains, are given b f () = and g() = (a) Find an epression for h() where h() = g ( f () ). Give our answer as a single fraction. 3 (b) State a suitable domain for h Functions f and g, defined on suitable domains, are given b f () = 2 and g() = sin + cos. Find f ( g() ) and g ( f () ). 4 hsn.uk.net Page 9 Questions marked c SQA
10 PSf 17. Given f () = , epress f () in the form ( + a) 2 b. 2 Part Marks Level Calc. Content Answer U1 C2 2 C NC A5 ( + 1) P1 Q4 1 ss: e.g. start to complete square 2 pd: complete process 1 ( + 1) ( + 1) 2 9 or 1 a = 1 2 b = 9 or a + a 2 b 2 a = 1 and b = (a) Epress in the form a ( + b) 2 and write down the values of a and b. 2 (b) State the maimum value of and justif our answer Epress (2 1)(2 + 5) in the form a( + b) 2 + c. 3 hsn.uk.net Page 10 Questions marked c SQA
11 PSf 20. Epress in the form ( + a) 2 + b and hence state the maimum value 1 of Show that can be written in the form ( + a) 2 + b. Hence or otherwise find the coordinates of the turning point of the curve with equation = (a) Show that f () = can be written in the form f () = a( + b) 2 + c. 3 (b) Hence write down the coordinates of the stationar point of = f () and state its nature. 2 hsn.uk.net Page 11 Questions marked c SQA
12 PSf (a) Show that the function f () = can be written in the form f () = a( + b) 2 + c where a, b and c are constants. 3 (b) Hence, or otherwise, find the coordinates of the turning point of the function f. 1 hsn.uk.net Page 12 Questions marked c SQA
13 PSf 25. The Water Board of a local authorit discovered it was able to represent the approimate amount of water W(t), in millions of gallons, stored in a reservoir t months after the 1st Ma 1988 b the formula W(t) = 1 1 sin πt 6. The board then predicted that under normal conditions this formula would appl for three ears. (a) Draw and label sketches of the graphs of = sin πt 6 and = sin πt 6, for 0 t 36, on the same diagram. 4 (b) n a separate diagram and using the same scale on the tais as ou used in part (a), draw a sketch of the graph of W(t) = 1 1 sin πt 6. 3 (c) n the 1st April 1990 a serious fire required an etra 1 4 million gallons of water from the reservoir to bring the fire under control. Assuming that the previous trend continues from the new lower level, when will the reservoir run dr if water rationing is not imposed? 3 hsn.uk.net Page 13 Questions marked c SQA
14 PSf 26. (a) Epress f () = in the form f () = ( a) 2 + b. 2 (b) n the same diagram sketch: (i) the graph of = f (); (ii) the graph of = 10 f (). 4 (c) Find the range of values of for which 10 f () is positive. 1 Part Marks Level Calc. Content Answer U1 C2 (a) 2 C NC A5 a = 2, b = P1 Q7 (b) 4 C NC A3 sketch (c) 1 C NC A16, A6 1 < < 5 1 pd: process, e.g. completing the square 2 pd: process, e.g. completing the square 3 ic: interpret minimum 4 ic: interpret intercept 5 ss: reflect in ais 6 ss: translate parallel to ais 7 ic: interpret graph 1 a = 2 2 b = 1 an two from: parabola; min. t.p. (2, 1); (0, 5) 4 the remaining one from above list 5 reflecting in ais 6 translating +10 units, parallel to ais 3 7 ( 1, 5) i.e. 1 < < 5 hsn.uk.net Page 14 Questions marked c SQA
15 PSf 27. A sketch of the graph of = f () where f () = is shown below. The graph has a maimum at A and a minimum at B(3, 0). PSf A = f () B(3, 0) (a) Find the coordinates of the turning point at A. 4 (b) Hence sketch the graph of = g() where g() = f ( + 2) + 4. Indicate the coordinates of the turning points. There is no need to calculate the coordinates of the points of intersection with the aes. 2 (c) Write down the range of values of k for which g() = k has 3 real roots. 1 Part Marks Level Calc. Content Answer U1 C3 (a) 4 C NC C8 A(1, 4) 2000 P1 Q2 (b) 2 C NC A3 sketch (translate 4 up, 2 left) (c) 1 A/B NC A2 4 < k < 8 1 ss: know to differentiate 2 pd: differentiate correctl 3 ss: know gradient = 0 4 pd: process 5 ic: interpret transformation 6 ic: interpret transformation 7 ic: interpret sketch 1 d d =... 2 d d = = 0 4 A = (1, 4) translate f () 4 units up, 2 units left 5 sketch with coord. of A ( 1, 8) 6 sketch with coord. of B (1, 4) 7 4 < k < 8 (accept 4 k 8) hsn.uk.net Page 15 Questions marked c SQA
16 PSf PSf 28. The diagram shows the graphs of two quadratic functions = f () and = g(). Both graphs = f () have a minimum turning point at (3, 2). = g() Sketch the graph of = f () and on the same diagram sketch the graph of = g (). (3, 2) 2 Part Marks Level Calc. Content Answer U1 C3 2 C CN A3 sketch 2001 P1 Q9 1 ss: use d d (quadratic) = linear 2 ic: interpret stationar point 1 st. line for f though (3, 0), m f > 0 2 st. line for g through (3, 0), m f > m g > hsn.uk.net Page 16 Questions marked c SQA
17 PSf PSf 30. The graph of a function f intersects the ais at ( a, 0) and (e, 0) as shown. There is a point of infleion at (0, b) and a maimum turning point at (c, d). Sketch the graph of the derived function f. ( a, 0) 3 (e, 0) = f () (0, b) (c, d) Part Marks Level Calc. Content Answer U1 C3 3 C CN A3, C11 sketch 2002 P1 Q6 1 ic: interpret stationar points 2 ic: interpret main bod of f 3 ic: interpret tails of f roots at 0 and c (accept a statement to this effect) min. at LH root, ma. between roots both tails correct 31. The point P( 2, b) lies on the graph of the function f () = (a) Find the value of b. 1 (b) Prove that this function is increasing at P. 3 hsn.uk.net Page 17 Questions marked c SQA
18 PSf 32. A ball is thrown verticall upwards. The height h metres of the ball t seconds after it is thrown, is given b the formula h = 20t 5t 2. (a) Find the speed of the ball when it is thrown (i.e. the rate of change of height with respect to time of the ball when it is thrown). 3 (b) Find the speed of the ball after 2 seconds. Eplain our answer in terms of the movement of the ball. 2 hsn.uk.net Page 18 Questions marked c SQA
19 PSf 33. A function f is defined b the formula f () = ( 1) 2 ( + 2) where R. (a) Find the coordinates of the points where the curve with equation = f () crosses the  and aes. 3 (b) Find the stationar points of this curve = f () and determine their nature. 7 (c) Sketch the curve = f (). 2 hsn.uk.net Page 19 Questions marked c SQA
20 PSf 34. hsn.uk.net Page 20 Questions marked c SQA
21 PSf 35. If = 2, show that d d = If f () = k and f (1) = 14, find the value of k. 3 hsn.uk.net Page 21 Questions marked c SQA
22 PSf 37. hsn.uk.net Page 22 Questions marked c SQA
23 PSf 38. (a) The function f is defined b f () = The function g is defined b g() = 1. Show that f ( g() ) = (b) Factorise full f ( g() ). 3 (c) The function k is such that k() = 1 f ( g() ). For what values of is the function k not defined? 3 hsn.uk.net Page 23 Questions marked c SQA
24 PSf 39. hsn.uk.net Page 24 Questions marked c SQA
25 PSf 40. hsn.uk.net Page 25 Questions marked c SQA
26 PSf 41. hsn.uk.net Page 26 Questions marked c SQA
27 PSf 42. hsn.uk.net Page 27 Questions marked c SQA
28 PSf 43. hsn.uk.net Page 28 Questions marked c SQA
29 PSf 44. Functions f and g are defined on the set of real numbers b f () = 1 and g() = 2. (a) Find formulae for (i) f ( g() ) (ii) g ( f () ). 4 (b) The function h is defined b h() = f ( g() ) + g ( f () ). Show that h() = and sketch the graph of h. 3 (c) Find the area enclosed between this graph and the ais. 4 hsn.uk.net Page 29 Questions marked c SQA
30 PSf 45. A function f is defined b the formula f () = 4 2 ( 3) where R. (a) Write down the coordinates of the points where the curve with equation = f () meets the  and aes. 2 (b) Find the stationar points of = f () and determine the nature of each. 6 (c) Sketch the curve = f (). 2 (d) Find the area completel enclosed b the curve = f () and the ais. 4 hsn.uk.net Page 30 Questions marked c SQA
31 PSf 46. hsn.uk.net Page 31 Questions marked c SQA
32 PSf 47. hsn.uk.net Page 32 Questions marked c SQA
33 PSf 48. hsn.uk.net Page 33 Questions marked c SQA
34 PSf Functions f () = sin, g() = cos and h() = + π 4 set of real numbers. (a) Find epressions for: (i) f (h()); are defined on a suitable (ii) g(h()). 2 (b) (i) Show that f (h()) = 1 2 sin cos. (ii) Find a similar epression for g(h()) and hence solve the equation f (h()) g(h()) = 1 for 0 2π. 5 Part Marks Level Calc. Content Answer U2 C3 (a) 2 C NC A4 (i) sin( + π 4 ), (ii) 2001 P1 Q7 cos( + π 4 ) (b) 5 C NC T8, T7 (i) proof, (ii) = π 4, 3π 4 1 ic: interpret composite functions 2 ic: interpret composite functions 3 ss: epand sin( + π 4 ) 4 ic: interpret 5 ic: substitute 6 pd: start solving process pd: process 7 hsn.uk.net Page 34 1 sin( + π 4 ) 2 cos( + π 4 ) 3 sin cos π 4 + cos sin π 4 and complete 4 g(h()) = 1 2 cos 1 2 sin 5 ( 1 2 sin cos ) ( 1 2 cos 1 2 sin ) sin 7 = π 4, 3π 4 accept onl radians Questions marked c SQA
35 PSf 51. Functions f and g are defined on suitable domains b f () = sin( ) and g() = 2. (a) Find epressions for: (i) f (g()); (ii) g( f ()). 2 (b) Solve 2 f (g()) = g( f ()) for Part Marks Level Calc. Content Answer U2 C3 (a) 2 C CN A4 (i) sin(2 ), (ii) 2 sin( ) 2002 P1 Q3 (b) 5 C CN T10 0, 60, 180, 300, ic: interpret f (g()) 2 ic: interpret g( f ()) 3 ss: equate for intersection 4 ss: substitute for sin 2 5 pd: etract a common factor 6 pd: solve a common factor equation 7 pd: solve a linear equation or 1 sin(2 ) 2 2 sin( ) 3 2 sin(2 ) = 2 sin( ) 4 appearance of 2 sin( ) cos( ) 5 2 sin( ) (2 cos( ) 1) 6 sin( ) = 0 and 0, 180, cos( ) = 1 2 and 60, sin( ) = 0 and cos( ) = , 60, 180, 300, hsn.uk.net Page 35 Questions marked c SQA
36 PSf 53. (a) Solve the equation sinpsfrag 2 cos = 0 in the interval = sin 2 (b) The diagram shows parts of two trigonometric graphs, = sin 2 and = cos. 180 Use our solutions in (a) to write 90 down the coordinates of the point P. 1 P = cos Part Marks Level Calc. Content Answer U2 C3 (a) 4 C NC T10 30, 90, P1 Q5 (b) 1 C NC T3 (150, 3 2 ) 1 ss: use double angle formula 2 pd: factorise 3 pd: process 4 pd: process 1 2 sin cos 2 cos (2 sin 1) 3 cos = 0, sin = , 30, ic: interpret graph or PSf 3 sin = 1 2 and = 30, cos = 0 and = 90 (150, 5 ) The diagram shows the graph of a cosine function from 0 to π. (a) State the equation of the graph. 1 2 (b) The line with equation = 3 π π intersects this graph at point A 2 A B and B. = 3 Find the coordinates of B. 2 3 Part Marks Level Calc. Content Answer U2 C3 (a) 1 C NC T4 = 2 cos P1 Q8 (b) 3 C NC T7 B( 7π 12, 3) 1 ic: interpret graph 1 2 cos 2 2 ss: equate equal parts 3 pd: solve linear trig equation in radians 4 ic: interpret result hsn.uk.net Page cos 2 = = 5π 6, 7π 6 3 = 7π 12 Questions marked c SQA
37 PSf 55. Solve 2 sin 3 1 = 0 for Solve the equation 2 cos 2 = 1 2, for 0 π Find the eact solutions of the equation 4 sin 2 = 1, 0 < 2π. 4 hsn.uk.net Page 37 Questions marked c SQA
38 PSf ( ) Solve the equation 2 sin 2 π 6 = 1, 0 < 2π. 4 hsn.uk.net Page 38 Questions marked c SQA
39 PSf 60. hsn.uk.net Page 39 Questions marked c SQA
40 PSf 61. hsn.uk.net Page 40 Questions marked c SQA
41 PSf 62. hsn.uk.net Page 41 Questions marked c SQA
42 PSf 63. hsn.uk.net Page 42 Questions marked c SQA
43 PSf 64. (a) Evaluate π 2 0 cos 2 d. 3 (b) Draw a sketch and eplain our answer Given f () = (sin + 1) 2, find the eact value of f ( π 6 ). 3 hsn.uk.net Page 43 Questions marked c SQA
44 PSf 66. hsn.uk.net Page 44 Questions marked c SQA
45 PSf 67. [END F PAPER 1 SECTIN B] Questions marked c SQA hsn.uk.net Page 45
46 PSf Paper hsn.uk.net Page 46 Questions marked c SQA
47 PSf hsn.uk.net Page 47 Questions marked c SQA
48 PSf 5. f () = 3 and g() = 3, = 0. (a) Find p() where p() = f (g()). 2 (b) If q() = 3, = 3, find p(q()) in its simplest form. 3 3 Part Marks Level Calc. Content Answer U1 C2 (a) 2 C CN A P2 Q3 (b) 2 C CN A4 (b) 1 A/B CN A4 1 ic: interpret composite func. 2 pd: process 3 ic: interpret composite func. 4 pd: process 5 pd: process 1 f ( ) 3 stated or implied b ) stated or implied b 4 3 p ( Functions f and g are defined b f () = and g() = = ±5. The function h is given b the formula h() = g ( f () ). where R, For which real values of is the function h undefined? 4 hsn.uk.net Page 48 Questions marked c SQA
49 PSf The functions f and g are defined on a suitable domain b f () = 2 1 and g() = (a) Find an epression for f ( g() ). 2 (b) Factorise f ( g() ). 2 hsn.uk.net Page 49 Questions marked c SQA
50 PSf 9. hsn.uk.net Page 50 Questions marked c SQA
51 PSf 10. hsn.uk.net Page 51 Questions marked c SQA
52 PSf hsn.uk.net Page 52 Questions marked c SQA
53 PSf 13. hsn.uk.net Page 53 Questions marked c SQA
54 PSf 14. hsn.uk.net Page 54 Questions marked c SQA
55 PSf 15. (a) n the same diagram, sketch the graphs of = log 10 and = 2 where 0 < < 5. Write down an approimation for the coordinate of the point of intersection. 3 (b) Find the value of this coordinate, correct to 2 decimal places. 3 hsn.uk.net Page 55 Questions marked c SQA
56 PSf 16. hsn.uk.net Page 56 Questions marked c SQA
57 PSf 17. The diagram shows part of the graph of the curve with equation = = f () (a) Find the coordinate of PSfrag the maimum turning point. 5 (b) Factorise (c) State the coordinates of the point A and A hence find the values of for which (2, 0) < 0. 2 Part Marks Level Calc. Content Answer U2 C1 (a) 5 C NC C8 = P2 Q3 (b) 3 C NC A21 ( 2)(2 + 1)( 2) (c) 2 C NC A6 A( 1 2, 0), < ss: know to differentiate 2 pd: differentiate 3 ss: know to set derivative to zero 4 pd: start solving process of equation 5 pd: complete solving process 6 ss: strateg for cubic, e.g. snth. division 7 ic: etract quadratic factor 8 pd: complete the cubic factorisation 9 ic: interpret the factors 10 ic: interpret the diagram 1 f () = = 0 4 (3 1)( 2) 5 = ( 2)(2 + 1)( 2) 9 A( 1 2, 0) 10 < 1 2 hsn.uk.net Page 57 Questions marked c SQA
58 PSf 18. hsn.uk.net Page 58 Questions marked c SQA
59 PSf 19. hsn.uk.net Page 59 Questions marked c SQA
60 PSf 20. (a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that, for cos θ, it has equal roots. 3 (b) Show that there are no real roots for θ. 1 hsn.uk.net Page 60 Questions marked c SQA
61 PSf 21. hsn.uk.net Page 61 Questions marked c SQA
62 PSf 22. hsn.uk.net Page 62 Questions marked c SQA
63 PSf 23. hsn.uk.net Page 63 Questions marked c SQA
64 PSf 24. hsn.uk.net Page 64 Questions marked c SQA
65 PSf hsn.uk.net Page 65 Questions marked c SQA
66 PSf 27. hsn.uk.net Page 66 Questions marked c SQA
67 PSf hsn.uk.net Page 67 Questions marked c SQA
68 PSf hsn.uk.net Page 68 Questions marked c SQA
69 PSf 32. hsn.uk.net Page 69 Questions marked c SQA
70 PSf 33. hsn.uk.net Page 70 Questions marked c SQA
71 PSf 34. The displacement, d units, of a wave after t seconds, is given b the formula d = cos 20t + 3 sin 20t. (a) Epress d in the form k cos(20t α ), where k > 0 and 0 α (b) Sketch the graph of d for 0 t (c) Find, correct to one decimal place, the values of t, 0 t 18, for which the displacement is 1 5 units. 3 hsn.uk.net Page 71 Questions marked c SQA
72 PSf 35. hsn.uk.net Page 72 Questions marked c SQA
73 PSf 36. (a) Show that 2 cos( + 30 ) sin can be written as 3 cos 2 sin. 3 (b) Epress 3 cos 2 sin in the form k cos( + α ) where k > 0 and 0 α 360 and find the values of k and α. 4 (c) Hence, or otherwise, solve the equation 2 cos( + 30 ) = sin + 1, [END F PAPER 2] hsn.uk.net Page 73 Questions marked c SQA
Polynomials Past Papers Unit 2 Outcome 1
PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor
More informationIntegration. Each correct answer in this section is worth two marks.
PSf Integration Paper 1 Section A 1. Evaluate A. 2 B. 7 16 4 1 Each correct answer in this section is worth two marks. 1/2 d. C. 1 2 D. 2 Ke utcome Grade Facilit Disc. Calculator Content Source D 2.2 C
More informationFurther Calculus Past Papers Unit 3 Outcome 2
PSf Further Calculus Past Papers Unit 3 utcome 2 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Differentiate 3 cos ( 2 π ) 6 with respect to. A. 3 sin(2) B. 3 sin(2
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationy hsn.uk.net Circle Paper 1 Section A O xy
PSf Circle Paper 1 Section A Each correct answer in this section is worth two marks. 1. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0).
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationStraight Line Past Papers Unit 1 Outcome 1
PSf Straight Line Past Papers Unit 1 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation
More informationSL Calculus Practice Problems
Alei  Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationMathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section
04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Boardapproved calculators ma be
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationMathematics Paper 1 (NonCalculator)
H National Qualifications CFE Higher Mathematics  Specimen Paper A Duration hour and 0 minutes Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More informationINTEGRATION FINDING AREAS
INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper 1 Practice Paper A Time allowed 1 hour 0 minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions 1 0 (40 marks) Instructions
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationCALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES
6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationPractice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.
Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to
More informationQuadratic Functions. MathsStart. Topic 3
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information(d) The line y = 2x 6 is a tangent to the graph of h at the point P. Find the xcoordinate of P. (5) (Total 12 marks)
Math SL Test 1 of Year 2 Questions 1. Let f() = 2 + 4 and g() = 1. Find (f g)(). The vector 3 translates the graph of (f g) to the graph of h. 1 Find the coordinates of the verte of the graph of h. (c)
More informationx 2 would be a solution to d y
CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y
More informationTime: 1 hour 10 minutes
[C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Noncalculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may
More informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
More informationC100/SQP321. Course Assessment Specification 2. Specimen Question Paper 1 5. Specimen Question Paper Specimen Marking Instructions Paper 1 23
00/SQP Maths Higher NTIONL QULIFITIONS ontents Page ourse ssessment Specification Specimen Question Paper 5 Specimen Question Paper 7 Specimen Marking Instructions Paper Specimen Marking Instructions Paper
More informationMATHEMATICAL METHODS (CAS)
Victorian Certificate of Education 05 SUPERVISR T ATTACH PRCESSING LABEL HERE Letter STUDENT NUMBER MATHEMATICAL METHDS (CAS) Section Written eamination Thursda 5 November 05 Reading time: 3.00 pm to 3.5
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More information(12) and explain your answer in practical terms (say something about apartments, income, and rent!). Solution.
Math 131 Fall 01 CHAPTER 1 EXAM (PRACTICE PROBLEMS  SOLUTIONS) 1 Problem 1. Many apartment complees check your income and credit history before letting you rent an apartment. Let I = f ( r) be the minimum
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More informationFinal Exam PracticeProblems
Name: Class: Date: ID: A Final Exam PracticeProblems Problem 1. Consider the function f(x) = 2( x 1) 2 3. a) Determine the equation of each function. i) f(x) ii) f( x) iii) f( x) b) Graph all four functions
More informationGraphing and transforming functions
Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationMATHEMATICS. Prelim Revision. (with answers)
MATHEMATICS Prelim Revision (with answers) FRMULAE LIST The roots of b b 4ac a b c 0 are a Sine Rule: a sina b sin c sinc Cosine Rule: a b c b c a bccos A or cos A bc Area of a triangle: 1 A absinc Volume
More informationSTRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula
UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationCourse 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook
Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationInvestigation. So far you have worked with quadratic equations in vertex form and general. Getting to the Root of the Matter. LESSON 9.
DA2SE_73_09.qd 0/8/0 :3 Page Factored Form LESSON 9.4 So far you have worked with quadratic equations in verte form and general form. This lesson will introduce you to another form of quadratic equation,
More information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More information1.6. Determine a Quadratic Equation Given Its Roots. Investigate
1.6 Determine a Quadratic Equation Given Its Roots Bridges like the one shown often have supports in the shape of parabolas. If the anchors at either side of the bridge are 4 m apart and the maximum height
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Noncalculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM
X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Noncalculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion
More informationExponential Functions
CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about
More informationPractice for Final Disclaimer: The actual exam is not a mirror of this. These questions are merely an aid to help you practice 1 2)
Practice for Final Disclaimer: The actual eam is not a mirror of this. These questions are merel an aid to help ou practice Solve the problem. 1) If m varies directl as p, and m = 7 when p = 9, find m
More informationPreCalculus Review Lesson 1 Polynomials and Rational Functions
If a and b are real numbers and a < b, then PreCalculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b
More informationMidterm 1. Solutions
Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationContents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...
Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationMark Scheme (Results) January 2008
Mark (Results) Januar 8 GCE GCE Mathematics (6666/) Edecel Limited. Registered in England and Wales No. 449675 Registered Office: One9 High Holborn, London WCV 7BH . (a) Januar 8 6666 Core Mathematics
More informationAlgebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete
More informationAQA Level 2 Certificate FURTHER MATHEMATICS
AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More informationLesson 8.3 Exercises, pages
Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5  b) = 5 c) =  d) =. Sketch broken lines to represent the vertical and horizontal
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X
Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general seconddegree
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More informationPrecalculus Workshop  Functions
Introduction to Functions A function f : D C is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set C. D is called the domain of f. C is called the codomain of f.
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationx 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac
Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More informationInvestigating Horizontal Stretches, Compressions, and Reflections
.7 YOU WILL NEED graph paper (optional) graphing calculator Investigating Horizontal Stretches, Compressions, and Reflections GOAL Investigate and appl horizontal stretches, compressions, and reflections
More informationQuadratic Equations in One Unknown
1 Quadratic Equations in One Unknown 1A 1. Solving Quadratic Equations Using the Factor Method Name : Date : Mark : Ke Concepts and Formulae 1. An equation in the form a + b + c, where a, b and c are real
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationDIFFERENTIATION OPTIMIZATION PROBLEMS
DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More information3.1 Quadratic Functions
33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56
More informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the collegelevel mathematics placement test An answer ke is provided
More information1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.
1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form a + by + c = 0, where a, b and c are integers. (b) Find the length of AB, leaving your answer in surd form.
More informationCHAPTER 54 SOME APPLICATIONS OF DIFFERENTIATION
CHAPTER 5 SOME APPLICATIONS OF DIFFERENTIATION EXERCISE 0 Page 65. An alternating current, i amperes, is given b i = 0 sin πft, where f is the frequenc in hertz and t the time in seconds. Determine the
More information4.3 Connecting f ' and f '' with the graph of f Calculus
4.3 CONNECTING f ' AND f '' WITH THE GRAPH OF f First Derivative Test for Etrema We have alread determined that relative etrema occur at critical points. The behavior of the first derivative before and
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationUnit 1 Quadratic Functions & Equations
1 Unit 1 Quadratic Functions & Equations Graphing Quadratics Part I: What is a Quadratic? A quadratic is an epression of degree. E) Graph of y : The most basic quadratic function is given by y. Create
More information