Functions and Graphs

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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 log-shaped 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

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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 t-ais 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

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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

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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

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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

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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

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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

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45 PSf 67. [END F PAPER 1 SECTIN B] Questions marked c SQA hsn.uk.net Page 45

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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

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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

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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

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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

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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

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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

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