MASDHARU
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GRAPHS AND TRANSFORMATIONS
1. a . Raj measures the height,hcm,of 70 plants. The table shows the
information
Height (h cm)
Frequency
10
h ≤ 20
7
20
h ≤40
15
40
h ≤50
27
50
h ≤60
13
60
h ≤90
8
Calculate an estimate of the mean height of the plants.
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b
Find the four inequalities that define the region that is not shaded.
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2. The cumulative frequency diagram shows information about the time
taken t minutes by 60 students to complete a test
Cumulative frequency
Time taken - minutes
(a). Find
i.
The median
ii.
Inter quartile range
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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
0 - 40
Frequency
8
40 - 60
60 - 70
70 -80
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80 - 90
90 - 100
4
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(c). On the grid below complete the histogram to show the information in the
table in part (b)
Frequency
density
Time taken (minutes)
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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
-2
-1.5
-1.75 -1.06
-1 -0.75 -0.5
0.5
0.75
1
0
4.50
2.53
2
1.03
1.5
2
2.25
(a). Complete the table of values
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3
(b). On the grid draw the graph of
for
(c). Use the graph to solve the equation
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(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
-3
-2.5
y
-2
1.13
-2
-1.5
1.38
-1
-0.5
-1.38
0.5
1
-1.13
(a). Complete the table of values
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(b). On the grid draw the graph of
(c). By drawing a suitable line solve the equation
for -3
=
(d). By drawing a suitable tangent find an estimate of the gradient of the
curve at the point where x = -1.75
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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
*
+
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7. The table shows the height h cm of 40 children in a class
Height(h)
Frequency
120 - 130
130 - 140
140 - 144
144 - 150
150 - 170
3
14
4
6
13
(a). Write down the interval containing the median
(b). Calculate an estimate of the mean height
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(c). Complete the histogram
Frequency density
Height
8.
Write down the three inequalities which define the unshaded region
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9. The cumulative frequency diagram shows information about the distance
travelled in km by 60 people
Find
(a). the 80th percentile
(b). the interquartile range
(c). the number of people who travelled more than 60 km.
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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
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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
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11.
200 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
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12. The table shows some values for
x
y
-2
-1.5
4.25 2.58
-1 -0.5 -0.25 -0.2
2.06
2.54
0.2
0.25
-2.46 -1.94
0.5
1
1.5
1.92
(a). Complete the table of values
(b). On the grid draw the graph of
for -2
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2
(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
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13.
(a). Draw the image of
i.
Shape A after translation by ( )
ii.
Shape A after a rotation through 1800 about the point ( 0,0)
iii.
Shape A after the transformation represented by the matrix
(
)
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(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
0 - 10
10 - 24
24 - 30
30 – 40
40 - 60
60 - 70
Frequency
10
90
135
85
70
10
(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
Cumulative frequency
t
t
10
t
100
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t
t
t
70
400
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(ii). On the grid draw a cumulative frequency diagram to show this
information
Cumulative
frequency
(c). Use your graph to estimate
Time
i.
the median time
ii.
the inter quartile range
iii.
the 15 th percentile
iv.
the number of people who took more than 50 minutes
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15.
(a). Complete the table of values
x
-4
-3
-2
-1
y
21
11
-9
-23
-0.5
0.5
1
2
23.25
1
6
3
4
11
(b). On the grid draw the graph of
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(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
can be solved by drawing y = 3x
+ 1 on the grid. Find the value of a, b and c
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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
5
0
200
300
9
300
350
18
350
400
12
400
500
6
(i). Calculate an estimate of the mean
(ii). Complete the histogram to show thw information in the table
Frequency density
Mass
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(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
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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 1800 about (0,0)
(b). Describe fully the single transformation that maps
(i). triangle T onto triangle U
(ii). triangle T onto triangle V
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19. f(x) =
(a). Complete the table of values for f(x).
x
f(x)
-5
-4
-3
-2.2 -1.5 -0.6
-2
-1.5
1.5
2.8
4.3
2
2.5
3
3.5
2.5
2.4
2.4
4
5
2.8
(b). On the grid draw the graph of y = f(x)
(c). Solve f(X) = 0
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(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 = -3.
20. The table shows the times, t minutes, taken by 200 students to complete
an IGCSE paper
Time (t minutes) 40
Frequency
60
10
70
50
75
-9
-22.75
(a) By using mid-interval values, calculate an estimate of the mean time
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(b) On the grid, draw a histogram to show the information in the table.
Frequency density
Time
21.
.
Draw the image of the shape A after translation by the vector ( )
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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
0
1
2
3
4
5
6
Frequency
1
2
3
5
7
3
3
(a). Calculate the mean number of pets
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(b). Jennifer joins the group of 24 students. When the information for
Jennifer is added to the table the new mean is 3.44.
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 .
7
8
8
y
6
9
10
5
His mean mark is 7.5. Calculate the value of y.
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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
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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/s2
(c). Find the distance travelled by the car in 30 seconds in metres
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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
-3
y
0.125
-2
-1
0
0.5
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1
2
2
4
3
8
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(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)
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(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
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(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
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29. 120 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
72
21
9
11
5
2
Calculate an estimate of the mean time taken
(b) (i)
Using the table in part a complete this cumulative frequency
table
2
Time
Cumulative
frequency
72
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120
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(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
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(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
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30. The table shows aome values for
x
y
-2
-1.5
3.125
-1
-0.5
0
3.375
2
0.5
1
1.5
2
0
(a). Complete the table of values
(b). On thegrid draw the graph of
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(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 = - 1.5
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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
5
13
72
90
117
120
Rainfall (r mm)
Cumulative
frequency
(a). On the grid below, draw cumulative frequency diagram to show
these results
Cumulative
frequency
rainfall
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(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
5
59
3
(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
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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
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(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
〈
〉
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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
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(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
-3
-2
-1.5
f(x)
-3.1
-2.1
-1.7
-1
-0.5
-0.3
0.3
0.5
-2.5
-5.9
-5.3
-1.5
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1
1.5
2.
1.3
1.9
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(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
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(e). The equation
f(x) and a
can be solved using the graph of y =
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
50
30
40
40
(a). Calculate an estimate of the mean
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(b). On the grid draw the histogram to show the information above.
Frequency density
.
Mass
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36. Jenna draws a cumulative frequency diagram to show information about
the scores of 500 people in a quiz
mulative frequency
Score
Use the diagram to find
(i). the median
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(ii). inter quartile range
(iii). 40th percentile
(iv). The number of students who score d30 or less or more than 20
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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 (
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)
38. Ali leaves home at 10 00 to cycle to his grandmother’s house. He arrives
at 13 00. The distance time graph represents his journey.
Distance
from home
Time
(a). Calculate Ali’s speed between 10 00 and 11 30. Give your answer in
km/h.
(b). Show that Ali’s average speed for the whole journey to his
grandmother’s house is 12 km/h
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(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 16 42. Complete the distance time graph.
39.
(a). On the grid
(i). draw the image of shape A after translation by the vector 〈
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〉
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(ii). draw the image of shape A after rotation through 900 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
-2.5
-2
y
-7.125
-1
-1.5
-1
3
-0.5
0
0.5
1
1.5
2
2.5
1
-0.375
-1
-0.125
3
9.125
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(b). draw the graph of
(c). By drawing a suitable tangent estimate the gradient of the curve at the
point where x = 2
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(d). Use your graph to solve theequation
41.
Cumulative
frequency
Time
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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
12
21
33
14
(i). Calculate an estimate of the mean time
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(ii). Draw a histogram to represent the grouped frequency table
Frequency
density
Time
42. f(x) =
Complete the table of values
x
-3
-2.5
-2
-1.5
f(x)
6.1
5.2
4.3
3.4
-1
-0.5
0.4
5
5.5
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0.5
1
1.5
2
-2.6
-3.8
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(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
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(e). (i). Draw the tangent to the graph y = f(x) at the point where x = -1.5
(ii). Use the tangent to estimate the gradient of the graph y = f(x) where
x = -1.5
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43.
(a). Draw the reflection of the shape Q in the line x = -1
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(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 -2
44.
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
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(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
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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
-3
-2
y
-2.89
-1.75
-1
-0.5
0.5
3.5
1
2
2
2.25
3
4
4.06
(a). Complete the table of values
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(b). On the grid draw the graph of
(c). use your graph to solve the equation
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(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 √
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47.
By shading the unwanted regions of the grid find and label the region R
which satisfies the following four inequlities
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48. 72 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
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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
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(c).
Mass m grams
Frequency
16
74
104
6
The data in this frequency table is to be shown in a histogram.
Complete the frequency density table below.
Mass
Frequency density
4
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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
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(a). To begin with the car accelerated at
for 20 seconds to reach
a speed of v m/s.
(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
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