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2 Graphs & Transformations Mathematics - Revision Guide Page 1 of 75 MASDHARU
3 GRAPHS AND TRANSFORMATIONS 1. a. Raj measures the height,hcm,of 70 plants. The table shows the information Height (h cm) 10 h h h h h 90 Frequency Calculate an estimate of the mean height of the plants. Graphs & Transformations Mathematics - Revision Guide Page 1 of 75
4 b Find the four inequalities that define the region that is not shaded. Graphs & Transformations Mathematics - Revision Guide Page 2 of 75
5 2. The cumulative frequency diagram shows information about the time taken t minutes by 60 students to complete a test Cumulative frequency (a). Find Time taken - minutes i. The median ii. Inter quartile range Graphs & Transformations Mathematics - Revision Guide Page 3 of 75
6 iii. 40 th percentile iv. The number of students who took more than 80 minutes to complete the test (b). Use the cumulative frequency diagram to complete the frequency table below Time taken Frequency 8 4 Graphs & Transformations Mathematics - Revision Guide Page 4 of 75
7 (c). On the grid below complete the histogram to show the information in the table in part (b) Frequency density Time taken (minutes) Graphs & Transformations Mathematics - Revision Guide Page 5 of 75
8 3. (a). Describe fully the single transformation that maps (i). triangle x on to triangle y (ii). Triangle x on to triangle z (b). (i). Draw the image of triangle X after a translation by the vector. Label it as p (ii). Draw the reflection of triangle p in the line y = 3 (c). Draw the image of triangle x after the transformation represented by the matrix ( ) 4. The table shows some values of x y (a). Complete the table of values Graphs & Transformations Mathematics - Revision Guide Page 6 of 75
9 (b). On the grid draw the graph of for (c). Use the graph to solve the equation Graphs & Transformations Mathematics - Revision Guide Page 7 of 75
10 (d). The line y = ax + b can be drawn on the grid to solve the equation (i). Find the values of a and b (ii). Draw the line y = ax + b to solve the equation (e). By drawing a suitable tangent finds an estimate of the gradient of the curve at the point when x = 2 5. The table shows some values of X y (a). Complete the table of values Graphs & Transformations Mathematics - Revision Guide Page 8 of 75
11 (b). On the grid draw the graph of for -3 (c). By drawing a suitable line solve the equation = (d). By drawing a suitable tangent find an estimate of the gradient of the curve at the point where x = Graphs & Transformations Mathematics - Revision Guide Page 9 of 75
12 6. (a). Describe fully the single transformation that maps (i). flag A on to flag B (ii). flag a on to flag c (b). Draw the image of flag A after a translation by the vector ( ) (c). Draw the image of flag A after a reflection in the line x = 1 (d). Describe fully the single transformation represented by the matrix * + Graphs & Transformations Mathematics - Revision Guide Page 10 of 75
13 7. The table shows the height h cm of 40 children in a class Height(h) Frequency (a). Write down the interval containing the median (b). Calculate an estimate of the mean height Graphs & Transformations Mathematics - Revision Guide Page 11 of 75
14 (c). Complete the histogram Frequency density Height 8. Write down the three inequalities which define the unshaded region Graphs & Transformations Mathematics - Revision Guide Page 12 of 75
15 9. The cumulative frequency diagram shows information about the distance travelled in km by 60 people Find (a). the 80 th percentile (b). the interquartile range (c). the number of people who travelled more than 60 km. Graphs & Transformations Mathematics - Revision Guide Page 13 of 75
16 9. The diagram shows the speed time graph for 120seconds of a car journey. (a). Calculate the acceleration of the car during the first 20 seconds (b). Calculate the total distance travelled by the car during the 120 seconds (c). calculate the average speed for this 120 seconds journey Graphs & Transformations Mathematics - Revision Guide Page 14 of 75
17 10 Speed Time A car starts from rest and accelerates for u seconds until it reaches a speed of 10 m/s. The car then travels at 10 m/s for 2u seconds. The diagram shows the speed diagram journey. The distance travelled by the car in the first 3u seconds is 125 m. (a) Find the value of u (b) Find the acceleration of the car in the first u seconds Graphs & Transformations Mathematics - Revision Guide Page 15 of 75
18 students take a reaction time test. The cumulative frequency diagram shows the result Find (i). the median (ii). inter quartile range (iii). the number of students with a reaction time of more than 4 seconds Graphs & Transformations Mathematics - Revision Guide Page 16 of 75
19 12. The table shows some values for x y (a). Complete the table of values (b). On the grid draw the graph of for -2 Graphs & Transformations Mathematics - Revision Guide Page 17 of 75
20 (c). By drawing a sitable line use your graph to solve the equation. (d). Thegraph of the equation has only one solution. Write down th erange of values of k for which this is possible. (e). By drawing a suitable tangent find an estimate of thegradient of the curve at the point x = -1 Graphs & Transformations Mathematics - Revision Guide Page 18 of 75
21 13. (a). Draw the image of i. Shape A after translation by ( ) ii. Shape A after a rotation through about the point ( 0,0) iii. Shape A after the transformation represented by the matrix ( ) Graphs & Transformations Mathematics - Revision Guide Page 19 of 75
22 (b). Describe fully the single transformation that maps shape A onto shape B (c). Find the matrix which represents the transformation that maps the shape A on to shape C 14. The table shows the time t minutes that 400 people take to complete the test. Time taken Frequency (a). (i). Write down the modal time interval. (ii). Calculate an estimate of the mean time taken to complete the test (b). (i). Complete the table of cumulative frequencies Time taken t t t t t t 70 Cumulative frequency Graphs & Transformations Mathematics - Revision Guide Page 20 of 75
23 (ii). On the grid draw a cumulative frequency diagram to show this information Cumulative frequency (c). Use your graph to estimate i. the median time Time ii. the inter quartile range iii. the 15 th percentile iv. the number of people who took more than 50 minutes Graphs & Transformations Mathematics - Revision Guide Page 21 of 75
24 15. (a). Complete the table of values x y (b). On the grid draw the graph of Graphs & Transformations Mathematics - Revision Guide Page 22 of 75
25 (c). By drawing a suitable tangent find an estimate of the gradient of the graph at the point (1, 0.1) (d). The equation has exactly two distinct solutions. Use the graph to find (i). the value of k (ii). the solution of (e). The equation + 1 on the grid. Find the value of a, b and c can be solved by drawing y = 3x Graphs & Transformations Mathematics - Revision Guide Page 23 of 75
26 16. (a). a group of 50 students estimated the mass. M grams of sweets of jar. The resulats are shown in the table. Mass Number of students (i). Calculate an estimate of the mean (ii). Complete the histogram to show thw information in the table Frequency density Mass Graphs & Transformations Mathematics - Revision Guide Page 24 of 75
27 (b). A group of 50 adults also estimated the mass, of the sweets in the jar. The histogram below shows information about their estimates. Use the histogram to make two comparisons between the distributions of the estimates of the students and the adults 17. (i). On the grid draw the image of the triangle A after the transformation represented by the matrix ( ) (ii). Find the 2x2 matrix which represents the transformation that maps triangle A on to triangle B Graphs & Transformations Mathematics - Revision Guide Page 25 of 75
28 18. (a). On the grid, draw the image of (i). triangle t after reflection in the line x = -1 (ii). triangle T after a rotation through about (0,0) (b). Describe fully the single transformation that maps (i). triangle T onto triangle U (ii). triangle T onto triangle V Graphs & Transformations Mathematics - Revision Guide Page 26 of 75
29 19. f(x) = (a). Complete the table of values for f(x). x f(x) (b). On the grid draw the graph of y = f(x) (c). Solve f(x) = 0 Graphs & Transformations Mathematics - Revision Guide Page 27 of 75
30 (d). By drawing a suitable line on the grid, solve the equation f(x) = 1 x (e). By drawing a tangent at the point (-3, -0.6), estimate the gradient of the graph of y = f(x) when x = The table shows the times, t minutes, taken by 200 students to complete an IGCSE paper Time (t minutes) Frequency (a) By using mid-interval values, calculate an estimate of the mean time Graphs & Transformations Mathematics - Revision Guide Page 28 of 75
31 (b) On the grid, draw a histogram to show the information in the table. Frequency density 21. Time. Draw the image of the shape A after translation by the vector ( ) Graphs & Transformations Mathematics - Revision Guide Page 29 of 75
32 22. (a). Describe fully the single transformation that maps triangle S on to T (b). Find the matrix which represents thetransformation that maps triangle S onto triangle T 23. The table shows information about the numbers of pets owned by 24 students No of pets Frequency (a). Calculate the mean number of pets Graphs & Transformations Mathematics - Revision Guide Page 30 of 75
33 (b). Jennifer joins the group of 24 students. When the information for Jennifer is added to the table the new mean is Calculatethe number of pets that Jennifer has. 24. (a). Draw the image of A after translation by the vector ( ) (b). Jim scores the following marks in 8 tests y His mean mark is 7.5. Calculate the value of y. Graphs & Transformations Mathematics - Revision Guide Page 31 of 75
34 25. A car passes through a check point at time t = 0 seconds, travelling at 8m/s. it travels at this speed for 10 seconds. The car then decelarates at a constant rate until it stops when t = 55 seconds. (a). On the grid draw the speed - time graph Speed Time seconds (b). Calculate the total distance travelled by the car after passing through the check point Graphs & Transformations Mathematics - Revision Guide Page 32 of 75
35 26. Speed km/h Time seconds The diagram shows the speed time graph of a car. The car travels at 45 km/h for 20 seconds. The car then decelarates for 10 seconds until it stops. (a). Change 45 km/h into m/s (b). Find the decelaration of the car in m/s 2 (c). Find the distance travelled by the car in 30 seconds in metres Graphs & Transformations Mathematics - Revision Guide Page 33 of 75
36 27. (a). Calculate the (b). Find the value of x in each of the following (i). (ii). (iii). (iv). (c). (i). Complete the table of values for x y Graphs & Transformations Mathematics - Revision Guide Page 34 of 75
37 (ii). on the grid draw the graph of for (iii). Use your graph to solve (iv). Find the equation of the line joining the points (1, 2) and (3, 8) Graphs & Transformations Mathematics - Revision Guide Page 35 of 75
38 (v) By drawing a suitable line on the graph solve the equation 28. Ai buys x rose bushes and y lavender bushes. He buys at least 5 rose bushes at most 8 lavender bushes at most 15 bushes in total More lavender bushes than rose bushes (a) Write down the four inequalities in terms of x and x or y to show this information Graphs & Transformations Mathematics - Revision Guide Page 36 of 75
39 (b). On the grid show the information in part (a)i by drawing four straight lines. Label the region R where the four iequlities are true c. Rose bushes $6 each and lavender bushes cost $4.50 each. What is the greatest amount of money Ali could spend Graphs & Transformations Mathematics - Revision Guide Page 37 of 75
40 students take a mathematics examination. (a). The time taken in m minutes, for each student to answer question 1 is shown in the table below Time 9 (m minutes) Frequency Calculate an estimate of the mean time taken (b) (i) table Using the table in part a complete this cumulative frequency Time 2 Cumulative frequency Graphs & Transformations Mathematics - Revision Guide Page 38 of 75
41 (ii). draw the cumulative frequency diagram Cumulative frequency Time seconds (iii). Use your cumulative frequency diagram to find (a). the median (b). the inter quartile range Graphs & Transformations Mathematics - Revision Guide Page 39 of 75
42 (c). the 35 th percentile (c). A new frequency table is formed from the table showed in part (a) Time (m minutes) Frequency 72 (i). Complete the table above (ii). A histogram was drawn and the height of the first block representing the time was 3.6 cm. Calculate the heights of the other two blocks Graphs & Transformations Mathematics - Revision Guide Page 40 of 75
43 30. The table shows aome values for x y (a). Complete the table of values (b). On thegrid draw the graph of Graphs & Transformations Mathematics - Revision Guide Page 41 of 75
44 (c). By drawing suitable line solve the equation (d). By drawing a suitable tangent find an estimate of the gradient of the curve at the point where x = Graphs & Transformations Mathematics - Revision Guide Page 42 of 75
45 31. Leo measured the rainfall each day, in millimetres, for 120 days. The cumulative frequency table shows the result r r r r r r Rainfall (r mm) Cumulative frequency (a). On the grid below, draw cumulative frequency diagram to show these results Cumulative frequency rainfall Graphs & Transformations Mathematics - Revision Guide Page 43 of 75
46 (b). (i). Find the median (ii). Use your diagram to find out the number of days when the rainfall was more than 50 mm (c). Use the information in the cumulative frequency table complete the frequency table below Rainfall(r mm) Frequency (d). Use your frequency table to calculate an estimate of the mean (e). In a histogram drawn to show the information in the table in part (c), the frequency density for the interval is 5.9. Calculate the frequency density for the intervals and Graphs & Transformations Mathematics - Revision Guide Page 44 of 75
47 32. (a). Describe fully the single transformation that maps (i). Shape A on to shape B (ii). Shape A on to shape C (iii). Shape A on to shape D Graphs & Transformations Mathematics - Revision Guide Page 45 of 75
48 (b). Find the 2 x 2 matrix that represents the transformation in part a(iii). (c). on the grid, draw the image of the shape a after translation by the vector ( ) (d). describe fully the single transformation represented by the matrix Graphs & Transformations Mathematics - Revision Guide Page 46 of 75
49 33. (a). On the grid draw the image of (i). triangle T after translation by the vector (ii). triangle T after reflection in the line y = -1 Graphs & Transformations Mathematics - Revision Guide Page 47 of 75
50 (b). Describe fully the single transformation that maps triangle T onto triangle U (c). (i) Describe fully the single transformation that maps triangle T onto triangle W (ii). Find the 2 x 2 matrix that represents the transformation in part c.(i) 34. (a) Complete the table of values X f(x) Graphs & Transformations Mathematics - Revision Guide Page 48 of 75
51 (b) On the grid draw the graph of y = f(x) (c). Use your graph to solve the equation f(x) = -1 (d). There is only negative integer value, k, for which f(x) = k has only one solution for all real x. write down this value of k Graphs & Transformations Mathematics - Revision Guide Page 49 of 75
52 (e). The equation can be solved using the graph of y = f(x) and a straight line graph. (i). Find the equation of the straight line (ii). On the grid draw this straight line and solve the equation 35. The table shows information about masses, m grams, of 160 apples Mass (m grams) Frequency (a). Calculate an estimate of the mean Graphs & Transformations Mathematics - Revision Guide Page 50 of 75
53 (b). On the grid draw the histogram to show the information above. Frequency density. Mass Graphs & Transformations Mathematics - Revision Guide Page 51 of 75
54 36. Jenna draws a cumulative frequency diagram to show information about the scores of 500 people in a quiz ulative frequency Use the diagram to find (i). the median Score Graphs & Transformations Mathematics - Revision Guide Page 52 of 75
55 (ii). inter quartile range (iii). 40 th percentile (iv). The number of students who score d30 or less or more than 20 Graphs & Transformations Mathematics - Revision Guide Page 53 of 75
56 37. (a). Draw the image of triangle A after translation by the vector (b). Describe fully the single transformation which maps triangle A onto triangle B (c). Draw the image of triangle a after transformation by the matrix ( ) Graphs & Transformations Mathematics - Revision Guide Page 54 of 75
57 38. Ali leaves home at to cycle to his grandmother s house. He arrives at The distance time graph represents his journey. Distance from home (a). Calculate Ali s speed between and Give your answer in km/h. Time (b). Show that Ali s average speed for the whole journey to his grandmother s house is 12 km/h Graphs & Transformations Mathematics - Revision Guide Page 55 of 75
58 (c). Change 12 km/h into m/minute (d). Ali stays for 45 minutes at his grand mother s house and return home. He arrives at Complete the distance time graph. 39. (a). On the grid (i). draw the image of shape A after translation by the vector Graphs & Transformations Mathematics - Revision Guide Page 56 of 75
59 (ii). draw the image of shape A after rotation through 90 0 clockwise about the origin (b). (i). On the grid draw the image of shape A after transformation represented by the matrix ( ) ( ) (ii). Describe fully the single transformation represented by the matrix 40. (a). Complete the table of values for X y Graphs & Transformations Mathematics - Revision Guide Page 57 of 75
60 (b). draw the graph of (c). By drawing a suitable tangent estimate the gradient of the curve at the point where x = 2 Graphs & Transformations Mathematics - Revision Guide Page 58 of 75
61 (d). Use your graph to solve theequation 41. Cumulative frequency Time Graphs & Transformations Mathematics - Revision Guide Page 59 of 75
62 The time taken by 80 people to complete a charity swim were recorded. The results are shown in the cumulative frequency diagram above. (a). Find (i). the median (ii). inter quartile range (iii). 70 th percentile (b). The time taken by the 80 people are shown in this frequency table Time frequency (i). Calculate an estimate of the mean time Graphs & Transformations Mathematics - Revision Guide Page 60 of 75
63 (ii). Draw a histogram to represent the grouped frequency table Frequency density Time 42. f(x) = Complete the table of values x f(x) Graphs & Transformations Mathematics - Revision Guide Page 61 of 75
64 (a). On the grid draw the graph of y = f(x) (c). Solve the equation f(x) = 2 (d). solve the equation f(x) = 2x + 3 Graphs & Transformations Mathematics - Revision Guide Page 62 of 75
65 (e). (i). Draw the tangent to the graph y = f(x) at the point where x = -1.5 x = -1.5 (ii). Use the tangent to estimate the gradient of the graph y = f(x) where Graphs & Transformations Mathematics - Revision Guide Page 63 of 75
66 43. (a). Draw the reflection of the shape Q in the line x = -1 Graphs & Transformations Mathematics - Revision Guide Page 64 of 75
67 (b).(i). Draw the enlargement of the shape Q, centre (0, 0),scale factor 2 (ii). Find the 2 x 2 matrix that represents an enlargement, center (0, 0), scale factor Frequency density Mass The histogram shows some information about the mass of 39 apples. (a) (i). Show that there are 12 apples in the interval Graphs & Transformations Mathematics - Revision Guide Page 65 of 75
68 (ii). Calculate an estimate of the mean mass of the 39 apples. (b). The mean amss of 20 oranges is 70 g. One orange is eaten. The mean mas of the remainining orange is 70.5 g. Find the mass of the orange that was eaten. 45. Frequency density Amount Graphs & Transformations Mathematics - Revision Guide Page 66 of 75
69 A survey asked 90 people how much money they gave to charity in one month. The histogram shows the results of the survey (a). Complete the frequency table for the six columns in the histogram Amount frequency 4 (b). Use your frequency table to calculate an estimate of the mean amount these 90 people gave to charity 46. The table shows some values for the function X y (a). Complete the table of values Graphs & Transformations Mathematics - Revision Guide Page 67 of 75
70 (b). On the grid draw the graph of (c). use your graph to solve the equation Graphs & Transformations Mathematics - Revision Guide Page 68 of 75
71 (d). Use your graph to solve the equation (e). By drawing a suitable tangent, find an estimate of the gradient of the curve at the point where x = 2 (f). Using algebra show that you can use the graph at y = 0 to find Graphs & Transformations Mathematics - Revision Guide Page 69 of 75
72 47. By shading the unwanted regions of the grid find and label the region R which satisfies the following four inequlities Graphs & Transformations Mathematics - Revision Guide Page 70 of 75
73 students are given homework one evening. They are told to spend no more than 100 minutes completing their homework. The cumulativ frequency diagram shows the number of minutes they spend. Cumulative frquency Minutes (a). How many students spend more than 48 minutes to completing their homework. (b). Find (i). the median (ii). inter quartile range Graphs & Transformations Mathematics - Revision Guide Page 71 of 75
74 49. The mass m grams of corn flakes in each of 200 boxes is recorded. The cumulative frequency diagram shows the result C. F Mass (a). Use the diagram to estimate the inter quartile range (b). Find the probability that a box chosen at random has amass of 500 grams or less Graphs & Transformations Mathematics - Revision Guide Page 72 of 75
75 (c). Mass m grams Frequency The data in this frequency table is to be shown in a histogram. Complete the frequency density table below. Mass Frequency density 4 Graphs & Transformations Mathematics - Revision Guide Page 73 of 75
76 50. (a). N = ( ) Describe fully the single transformation represented by N (b). Find the matrix which represents the single transformation that maps triangle A onto triangle B 51. The diagram shows the speed time graph for a car travelling along a road for T seconds Speed m/s Time seconds Graphs & Transformations Mathematics - Revision Guide Page 74 of 75
77 (a). To begin with the car accelerated at a speed of v m/s. for 20 seconds to reach (i). Show that the speed, v, of the car is 15m/s. (ii). The total distance travelled is 1.8 kilometres Calculate the total time T of the journey. (b). Asma runs 22 km, correct to th enearest kilometre. She takes hours, correct to the nearest half hour. Calcuate the upper bound of Asma s speed Graphs & Transformations Mathematics - Revision Guide Page 75 of 75
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