Edexcel (I)GCSE
Mathematics
Higher Tier
Sequences functions and graphs - Graphs
Total Marks: 104
You must have:
Ruler
Calculator
Instructions:
Use black ink or ball-point pen.
Answer All questions.
Answer the questions in the spaces provided there may be more space than you need
Show all the steps in any calculations and state the units.
Information:
The total mark for this paper is 104
The marks for each question are shown in brackets use this as a guide as to how much time to spend
on each question.
Advice:
Read each question carefully before you start to answer it.
Keep an eye on the time.
Write your answers neatly and in good English.
Try to answer every question.
Check your answers if you have time at the end.
51. On the grid, draw the graph of y xíIURPx íWRx y
20
15
10
5
–2
–1
O
1
2
3
4
x
–5
–10
(Total for Question 15 is 4 marks)
Do NOT write in this space.
2. (a) Find the gradient of the line with equation 3xy 13
...........................................
(3)
(b) Find the coordinates of the point of intersection of the line with equation 3xy and the line with equation 5xíy Show your working clearly.
(. .. . .. . .. .. . .. . .. .. . .. . .. . .. .. . . , . . .. .. . . . .. . . . . .. . . . .. . . . . .. . . . )
(5)
2 is 8 marks)
(Total for Question 13
14
3. (a) The equation of a line L is 2xíy = 6
Find the gradient of L.
. .. . .. . .. . .. .. . .. . .. .. . .. . .. . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . .
(3)
(b) Find the equation of the line which is parallel to L and passes through
the point (6, 9).
. .. . .. . .. .. . .. . .. . .. .. . .. . .. . . .. . . . .. . . . . .. . . . .. . . . .. . . . . .. . . .
(2)
3 is 5 marks)
(Total for Question 14
Do NOT write in this space.
104. The points (0, –1) and (4, 5) lie on the straight line L.
y
6
L
5
4
3
2
1
–4
–3
–2
–1 O
1
2
3
4
5
6
x
–1
–2
–3
(a) Work out the gradient of L.
. . .. .. . .. . .. . .. .. . .. . .. .. . .. . . .. . . . . .. . . . .. . . . .. . . . . .. . . . .. . . .
(2)
(b) Write down an equation of L.
. .. . .. . .. .. . .. . .. . .. .. . .. . .. . . .. . . . .. . . . . .. . . . .. . . . .. . . . . .. . . .
(1)
(c) Find an equation of the line which is parallel to L and passes through the point (– 2, 0)
.. .. . .. . .. . .. .. . .. . .. .. . .. . .. . ... . . . .. . . . . .. . . . .. . . . . .. . . . .. . .
(2)
(Total for Question 10
4 is 5 marks)
135. The points with coordinates (0, 3) and (8, 5) lie on the straight line L.
y
6
L
5
4
3
2
1
–4
–2
O
2
4
6
8
x
–1
–2
–3
(a) Work out the gradient of L.
... ... .... ... .... .... . ....... ....... ........ ....... ...
(2)
(b) Write down an equation of L.
. ... .... ... .... ... .... ......... ....... ....... ....... ..
(1)
(c) Find an equation of the line which is parallel to L and which passes through
the point (–4, –2)
.. ... .... ... .... ... ... ...... ....... ....... ....... .....
(2)
(Total for Question 13
5 is 5 marks)
46. The diagram shows three points, A, B and P, on a centimetre grid.
y
8
7
P
6
A
5
4
3
2
1
–2
–1 O
–1
1
2
3
4
5
6
7
8
9
x
B
–2
The point A has coordinates (4, 5) and the point B has coordinates (2, –1).
(a) Find the coordinates of the midpoint of AB.
( .. .. . .. . . .. . . . . . ,
(2)
AB is a diameter of a circle.
P is the point (7, 6)
C is the point on the circle such that PA = PC.
(b) On the diagram, mark with a cross (×) the point C.
Label your point C.
(2)
(Total for Question 46 is 4 marks)
Do NOT write in this space.
. . . .. . . . . .. . . . ..
)
207. (a) Complete the table of values for y = x 2 +
2
x
x
0.1
0.2
1
1.5
2
3
y
20.01
10.04
3
3.58
5
9.67
0.5
4
(1)
(b) On the grid, draw the graph of y = x 2 +
2
for 0.1 - x - 4
x
y
20
15
10
5
O
1
2
3
4
x
(2)
(c) Use your graph to find estimates for the solutions of x 2 +
2
= 14
x
in the interval 0.1 - x - 4
Give your estimates correct to 1 decimal place.
.. . .. . .. . .. .. . .. . .. .. . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . . .. .
(2)
(d) x = 1 is one solution of the equation x 2 +
2
= mx
x
(i) Find the value of m.
m = . . . . .. . . . . .. . . . .. . . . .. . . . . .. . .
(ii) Draw a suitable straight line on your graph to find an estimate for the second
2
positive solution of the equation x 2 + = mx for the value of m found in
x
part (d)(i).
Give your estimate correct to 1 decimal place.
x = .. . . . . .. . . . .. . . . .. . . . . .. . . . .. .
(3)
(Total for Question 20
7 is 8 marks)
Do NOT write in this space.
6 8. Mansi left her home at 09 00 to walk to the shops.
She stopped at the newspaper shop and then carried on to the fish shop.
Here is the distance-time graph for Mansi’s journey from her home to the fish shop.
Distance
from home
(km)
Fish
shop
Newspaper
shop
2
1.5
1
0.5
Home
0
09 00
09 30
10 00
10 30
Time
(a) How many minutes did it take Mansi to walk from the newspaper shop to the
fish shop?
.. .... ... .... .... .. ... ....... ..
minutes
(1)
(b) Work out the average speed, in kilometres per hour, for Mansi’s journey from her
home to the newspaper shop.
.... ... .... .. ....... ....... ....
(2)
Mansi stopped for 10 minutes in the fish shop.
She then walked home at a constant speed of 3 km/h.
(c) Show this information on the graph.
(2)
(Total for Question 86 is 5 marks)
km/h
9. (a) The straight line L passes through the points (0, 12) and (10, 4).
13
Find an equation for L.
.. .... .... ... .... ... .... ... .... ..... ....... ....... ....... .....
(3)
(b) Find an equation of the straight line which is parallel to L and passes through the
SRLQWí
.... ... .... .... ... .... ... .... . .... ....... ....... ....... .......
(2)
(Total for Question 13
9 is 5 marks)
Do NOT write in this space.
810. (a) Complete the table of values for 2x + y = 4
x
–1
2
4
y
(2)
(b) On the grid, draw the graph of 2x + y = 4 for values of x from –1 to 4
(2)
y
8
6
4
2
–2
–1
O
1
2
3
4
5
x
–2
–4
–6
(c) Show, by shading on the grid, the region which satisfies all three of the inequalities
x . –1, y . 2 and 2x + y - 4
Label the region R.
(2)
(Total for Question 10
8 is 6 marks)
11.
8 (a) On the grid, draw the line with equation x + 2y = 8 for values of x from 0 to 9
y
4
2
O
2
4
6
8
10
x
–2
–4
(2)
(b) Show, by shading on the grid, the region defined by all three inequalities
x + 2y - 8
x.2
y.1
Label your region R.
(3)
(Total for Question11
8 is 5 marks)
12.
8
(a) Complete the table of values for y = x2 – 5x + 4
0
x
1
2
3
4
5
–2
y
4
(2)
(b) On the grid, draw the graph of y = x2 – 5x + 4 for all values of x from x = 0 to x = 5
(2)
y
5
4
3
2
1
–0.5 O
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5 x
–1
–2
–3
(Total for Question 12
8 is 4 marks)
Do NOT write in this space.
13. (a) Complete the table of values for y = x2 + 2xí
11
x
í
y
í
í
í
0
í
í
0
1
2
5
(2)
(b) On the grid, draw the graph of y = x2 + 2xíIRUYDOXHVRIxIURPíWR
y
6
5
4
3
2
1
–5
–4
–3
–2
–1 O
1
2
3
x
–1
–2
–3
–4
–5
(2)
(Total for Question 13
11 is 4 marks)
Do NOT write in this space.
14
13
The line LSDVVHVWKURXJKWKHSRLQWVíDQG
(a) Find an equation of the line L.
....... ....... ....... .......
(3)
(b) Find an equation of the line that is parallel to L and which passes through
WKHSRLQWí
.... ....... ....... ....... ...
(2)
(Total for Question 14
13 is 5 marks)
15 (a) Find the gradient of the line with equation 3y – 2x = 6
10
.... ... .... ... ...... ....... ....... ....... .....
(2)
(b) Find an equation of the line with gradient –3 that passes through the point (2, 5).
.. .... ... .... .... ..... ....... ....... ....... ...
(2)
(Total for Question 15
10 is 4 marks)
16.
18 y = x3 – 4x2 + 4x + 3
(a) Find
dy
dx
.... ... .... ... ..... ....... ........ ....... .....
(2)
y
P
O
x
The diagram shows a sketch of the curve with equation y = x3 – 4x2 + 4x + 3
The point P is a turning point on the curve.
(b) Work out the coordinates of P.
Show clear algebraic working.
(. . .. . .. . .. .. . .. . .. .. . .. . .. . .. .. . , .. . . .. . . . .. . . . . .. . . . .. . . . . .. . . . . )
(4)
(c) Write down the range of values of x for which the curve has a negative gradient.
. .... ... .... .... ..... ....... ....... ....... ....
(2)
16 is 8 marks)
(Total for Question 18
517. On the grid, draw the graph of y
3x + 2 for values of x from í2 to 4
y
20
18
16
14
12
10
8
6
4
2
–2
–1
O
1
2
3
4
x
–2
–4
–6
(Total for Question 17
5 is 4 marks)
6
A is the point with coordinates (4, 1)
B is the point with coordinates (1, 9)
Find the coordinates of the midpoint of AB.
(.... ... .... .... ... .... . , ..... ....... ....... ..... )
18.
(Total for Question 6 is 2 marks)
7
y
11
10
9
8
B
7
6
5
4
3
2
A
1
O
1
2
3
4
5
6
7
8
x
Describe fully the single transformation that maps shape A onto shape B.
.. ... .... .... .... .... .. .. . . . . . . . . . . . . . . . . . . . . . . . .. ... .. .. . .. .. .. . .. .. .. . .. .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. . .. ... .... ... .... .... ... ... .... .... ... .... ... .... ... .... ... .... ... .... ... .... ... .... .... ... .... ... .... ... .. .... ....... ....... ....... ......
.. ... .... .... .... .... .. .. . . . . . . . . . . . . . . . . . . . . . . . .. ... .. .. . .. .. .. . .. .. .. . .. .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. . .. ... .... ... .... .... ... ... .... .... ... .... ... .... ... .... ... .... ... .... ... .... ... .... .... ... .... ... .... ... .. .... ....... ....... ....... ......
(Total for Question 18
7 is 2 marks)
419. On the grid, draw the graph of y = 3x – 4 for values of x from –2 to 3
y
6
5
4
3
2
1
–2
–1
O
1
2
3
x
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
–12
(Total for Question19
4 is 4 marks)
20. Here is the graph of y = x2 – 2x – 1
15
y
5
4
3
2
1
–2
–1
O
1
2
3
x
4
–1
–2
–3
(a) Use the graph to solve the equation x2 – 2x – 1 = 2
.. .... ... .... .... ... .... ..... ....... ....... ....... ......
(2)
The equation x2 + 5x – 7 = 0 can be solved by finding the points of intersection of the
line y = ax + b with the graph of y = x2 – 2x – 1
(b) Find the value of a and the value of b.
a = . ....... ....... ....... ......
b = ...... ........ ....... .......
(2)
(Total for Question 20
15 is 4 marks)
19
21.
y
y = kx
A(p,q )
y=
N
x
x
O
The diagram shows the straight line with equation y = kx intersecting
N
the curve with equation y =
at the point A(p, q).
x
(a) Find p and find q.
Give each answer in its simplest form, in terms of k and N.
p = .. ..... ....... ....... ....... ...
q = ...... ....... ....... ....... ....
(3)
Given that p = 2q
(b) find the value of k.
k = ......... ....... ....... ....... .
(2)
(Total for Question 21
19 is 5 marks)