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Graphs function
1. On the diagrams, sketch the graphs of the following functions.
(a) y = x + 1 (b) y = 2 - x²
y y
x
0
x
0
2. On the diagrams, sketch the graphs of the following functions.
2
(a) y = x³ + 1 (b) y = -
x
y y
x
x 0
0
3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3)
y
P
x
0 Q
Find the coordinates of
(i) P (ii) Q
Answer :
1
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(b) On the Cartesian plane below, sketch the graph of function y = x(x + 5)
y
x
0
4. (a) The diagram shows the graph of function y = xn – 8 .
y
x
0 a
-8
State the value of
(i) n (ii) a
Answer :
8
(b) On the diagram, sketch the graph of the function y = for x ≥ 0 .
x
y
x
0
5.Write the inequalities which define the shaded region in the graph.
y x=5 Answer :
y=x-5
x
0
y = -x
2
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6. State the three inequalities which define the shaded region in the graph.
y
x=6
y = 12 Answer :
x
0 y + 2x = 12
7. State the three inequalities which define the shaded region in the graph.
y
y = x-3 Answer :
2
0
2y+x = 0
8. On the graph provided, shade the region which satisfies the two inequalities
x – y ≥ 4 and x < 4.
y
x-y = 4
x
0 4
-4
3
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9. On the graph provided, shade the region which satisfies the three inequalities
y ≤ 2x+6, y ≥ x and y < 6.
y
y=x
x
0
y = 2x+ 6
10. 10.On the graph provided, shade the region which satisfies the three inequalities
x + 2y ≥ 6 , x ≤ 4 and y ≤ x .
y
y=x
x
0
y=4 x + 2y = 6
4
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EXERCISE 2
1.(a)
x -4 - 2.5 -1 0 1 2 2.5 3 4
y p 6 -3 -4 -1 q 11 17 32
Table 1
Table 1 shows the values of x and y which satisfy the equation y = 2x2 + x – 4.
Calculate the value of p and of q. [2 marks]
(b) Use graph paper to answer this part of the question. You may use a flexible curve
ruler.
Using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the
graph of y = 2x2 + x – 4 for – 4 ≤ x ≤ 4.
[4 marks]
(c) From your graph in (b), find
(i) the value of y when x = - 3,
(ii) the values of x when y = 5. [2 marks]
(d) Draw a suitable straight line on your graph to find all the values of x which satisfy the
equation y = 2x2 + 5x – 12 for – 4 ≤ x ≤ 4.
[4 marks]
2.(a)
x -3 -2 -1 - 0.5 0.5 1 1.5 2 3
y -1 -3 6 3 1.5 1
Table 2
3
Copy and complete Table 2 for the equation y = . [3 marks]
x
(b) For this part of the question, use a graph paper. You may also use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph
3
of y = for – 3 ≤ x ≤ 3. [3 marks]
x
(d) From your graph, find
(i) the value of y when x = - 2.5,
(ii) the values of x when y =1.8. [3 marks]
(d) Draw a suitable straight line on your graph to find all the values of x which satisfy the
3
equation - = - 2x + 3 for – 3 ≤ x ≤ 3. State these values of x. [3 marks]
x
5
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3.(a) Copy and complete the table below for the equation y = x2 - 3x + 5.
x -2 - 1.5 -1 - 0.5 1 2 3.5 4 5
y 1.5 11.75 6.75 3 6.75 9 15
Table 3 [2 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph
of y = x2 - 3x + 5 for – 2 ≤ x ≤ 5. [4 marks]
(e) From the graph, find
(i) the value of y when x = 1.5,
(ii) the value of x when y = 8. [2 marks]
(d) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation x2 + 5x – 2=0 for – 2 ≤ x ≤ 5. State these values of x. [4 marks]
4.(a)
x -4 -3 - 2.5 -2 -1 0 1 2 3
y -60 -28 -15.75 -6 p 8 0 q -46
Table 4
Table 4 shows the values of x and y which satisfy the equation y = 8 – 3x – 5x2.
Calculate the values of p and q. [2 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the
graph of y = 8 – 3x – 5x2 for – 4 ≤ x ≤ 3. [4 marks]
(f) From the graph, find
(i) the value of y when x = 1.5,
(ii) the value of x when y = -20. [3 marks]
(d) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation 10 – 4x – 5x2 =0 for – 4 ≤ x ≤ 3.
[3 marks]
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5.(a) Complete Table 5 for the equation y = 12 + 5x – 2x2
x -3 -2 -1 0 1 2.5 3 4 5
y -21 5 12 15 12 9 -13
Table 5 [2 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the
graph of y = 12 + 5x – 2x2 for – 3 ≤ x ≤ 5.
[4 marks]
(c) From the graph, find
(iii) the value of y when x = -2.5,
(iv) the value of x when y = 14. [2 marks]
(d) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation 15 + 3x – 2x2 = 0 for – 3 ≤ x ≤ 5.
[4 marks]
6.(a) Complete Table 6 for the equation y = x3 – 3x + 6.
x -3 -2 -1 0 1 1.5 2 2.5 3
y -12 8 6 4 14.1 24
Table 6 [3 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 50 units on y-axis, draw the
graph of y = x3 – 3x + 6 for – 3 ≤ x ≤ 3. [4 marks]
( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation x3 – 6x + 4 = 0 for – 3 ≤ x ≤ 3. [3 marks]
(d) Shade the region which represents the inequalities x ≥ 0, y ≥ - 8 and y ≤ - 4x.
[2 marks]
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7.(a) Complete Table 7 for the equation y = 3x – x2.
x 0 1 2 3 4 5 6 7 8
y 0 2 0 -4 - 18 - 28
Table 7 [3 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph
of y = 3x – x2 for 0 ≤ x ≤ 8. [4 marks]
( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation 5x – x2 = 2 for 0 ≤ x ≤ 8.
[3 marks]
(d) Shade the region which represents the two inequalities and y ≤ 3x – x2 and y ≥ - 10.
[2 marks]
6
8.(a) Complete Table 8 for the equation y = – .
x
x -4 - 2.5 -1 - 0.6 0.6 1 2 3 4
y 1.5 6 10 - 10 -6 -3 - 1.5
Table 2
[3 marks]
(b) For this part of the question, use a graph paper. You may also use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 2 units on y-axis, draw the graph
6
of y = – for – 4 ≤ x ≤ 4. [3 marks]
x
(c) From your graph, find
(v) the value of y when x = 1.3,
(vi) the values of x when y =3.5. [3 marks]
(d) Draw a suitable straight line on your graph to find all the values of x which satisfy the
3
equation + 1 = x for – 4 ≤ x ≤ 4.
x
[3 marks]
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9.(a) Complete Table 9 for the equation y = x3 – 4x – 6.
x -4 -3 -2 -1 0 1 2 3 3.5
y - 54 - 21 -3 -6 -6 22.88
Table 9 [3 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 10 units on y-axis, draw the
graph of y = x3 – 4x – 6 for – 4 ≤ x ≤ 3.5. [4 marks]
( c) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation x3 – 4x + 2 = 0 for – 4 ≤ x ≤ 3.5.
[3 marks]
(d) Shade the region which represents the three inequalities y ≤ x3 – 4x – 6, y ≥ 5x - 10
and ≤ 2.5.
[2 marks]
10.(a) Complete Table 10 for the equation y = x3 – 10x + 5.
x - 3.5 -3 -2 -1 0 1 2 3 3.5
y - 2.9 8 14 5 -7 12.88
Table 10 [3 marks]
(b) For this part of the question, use a graph paper. You may use a flexible curve
ruler.
By using a scale of 2cm to 1 unit on x-axis and 2cm to 5 units on y-axis, draw the graph
of y = x3 – 10x + 5 for – 3.5 ≤ x ≤ 3.5. [4 marks]
(c) From your graph, find the value of y when x = - 2.5. [2 marks]
(d) Draw a suitable straight line on the graph to find all the values of x which satisfy the
equation x3 – 6x + 4 = 0 for – 3 ≤ x ≤ 3. [3 marks]
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ANSWERS : EXERCISE 1
1. On the diagrams, sketch the graphs of the following functions.
(a) y = x + 1 (b) y = 2 - x²
y y
2
1
x
0
x
-1 0
2. On the diagrams, sketch the graphs of the following functions.
2
(a) y = x3 + 1 (b) y = -
x
y y
1
x 1 x
0 0
-2
3. (a) The diagram shows the graph of the function y = (x – 1)(x – 3)
y
P
x
1 Q
Find the coordinates of
(i) P (ii) Q
Answer :
(i) When x = 0, y = 4 (ii) When y = 0, x = 1 or x = 3
P ( 0, 4 ) Q ( 3, 0 )
10
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(b) On the Cartesian plane below, sketch the graph of function y = x2 - 2
y
0 x
-2
4. (a) The diagram shows the graph of function y = xn – 8 .
y
x
1 a
-8
State the value of
(i) n (ii) a
Answer :
(i) n = 3 (ii) When y = 0, x3 – 8 = 0
x3 = 8
x =38
x=a=2
8
(b) On the diagram, sketch the graph of the function y = for x ≥ 0 .
x
y
8
8
y=
x x
0 1
5.Write the inequalities which define the shaded region in the graph.
y x=5 Answer :
y=x–5 x≤5
y≤x–5
x
0 y > -x
y = -x
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6. State the three inequalities which define the shaded region in the graph.
y
x=6
y = 12 Answer :
x<6
y ≤ 12
y + 2x ≥ 12
x
1 y + 2x = 12
7. State the three inequalities which define the shaded region in the graph.
y
y = x-3 Answer :
2 y≥x–3
y≤2
2y + x ≥ 0
0
2y+x = 0
8. On the graph provided, shade the region which satisfies the two inequalities
x – y ≥ 4 and x < 4.
y
x-y = 4
x
1 4
x=4
-4
12
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9. On the graph provided, shade the region which satisfies the three inequalities
y ≤ 2x+6, y ≥ x and y < 6.
y
y=x
y=6
0 x
y = 2x+ 6
10.On the graph provided, shade the region which satisfies the three inequalities
x + 2y ≥ 6 , x ≤ 4 and y ≤ x .
y
y=x
x
0
y=4 x + 2y = 6
13
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ANSWERS : EXERCISE 2
1. (a) p = 24, q = 6
(b)
y
y = - 4x + 8
x
0
(c) (i) 11
(ii) - 2.4, 1.9
(d) - 4, 1.5
2. (a) x = - 2, y = - 1.5
x = - 0.5, y = - 6
x = 1.5, y = 2
(b)
y
y=2x - 3
x
0
(c) (i) x = - 2.5, y = - 1.2
(ii) y = 1.8, x = 1.7
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(d) y = 2x – 3
x = 2.19, - 0.69
3. (a) y = 9, y = 3
(b)
y
x
0
y = - 8x + 7
(c) (i) y = 2.75
(ii) x = - 0.8, 3.8
(d) y = - 8x + 7, x = 0.38
4. (a) p = 6, q = - 18
(b)
y
x
0
y=x-2
(c) (i) y = - 7.75
(ii) x = 2.09, x = - 2.69
(d) x = - 1.83, x = 1.1, y = x - 2
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5. (a) x = - 2, y = - 6
x = 4, y = 0
(b)
y
y = 2x - 3
x
0
(c) (i) y = - 13
(ii) x = 0.5 , x = 2
(d) x = - 2.1, x = 3.6, y = 2x - 3
6. (a) x = - 2, y = 4
x = 1.5, y = 4.75
x = 2, y = 8
(b) , (d)
y
x
0
y = -8
y = - 4x
(c) y = 3x + 2
x = - 2.8, 0.85, 2
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7. (a) x = 2, y = 2
x = 5, y = - 10
x = 8, y = - 40
(b), (d)
y
x
0
y = - 10
(c) (i) y = 2 – 2x
(ii) x = 0.45, x = 4.55
8. (a) x = - 2.5, y = 2.4
x = 3, y = - 2
(b)
y
x
0
y=- 2x + 2
(c) (i) y = - 4.6
(ii) x = - 1.7
(d) y = - 2x + 2, x = - 1.3, 2.3
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9. (a) x = - 2, y = - 6
x = 1, y = - 9
x = 3, y = 9
(b) , (d)
x = -2.5 y
y = 5x - 10
x
0
(c) y =-8
x = - 2.2, 0.5, 1.5
10. (a) x = - 2, y = 17
x = 1, y = - 4
x = 3, y = 2
(b)
y
5
x
0
(c) y = 13.5
(d) y = 11, x = - 2.75, - 0.65, 3.45
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DIAGNOSTIC TEST
1. Complete the following table for the equation y = 5 + 7x – 2x2.
x -3 -2 -1 0 1 2 3 4 5
y - 34 -4 5 11 8 - 10
6
2. Complete the following table for the equation y = – .
x
x -4 -2.5 -1 -0.6 0.6 1 2 3 4
y 1.5 6 10 -10 -6 -3 - 1.5
3. Use the graph paper to answer this question.
Table 11 shows the values of two variables, x and y, of a function.
x -4 -3 -2 -1 0 1 2 3
y 9 0 -5 -6 -3 4 15 30
Table 11
The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 5 units.
By using the scale given, draw the graph of the function.
4. Use the graph paper to answer this question.
Table 12 shows the values of two variables, x and y, of a function.
x -4 -3.5 -3 -2.5 -2 -1 0 1 2
y -56 -35 -19 -8 0 7 8 9 16
Table 12
The scale on the x-axis is 2cm to 1 unit. The scale on the y-axis is 2cm to 10 units.
By using the scale given, draw the graph of the function.
5. Use the graph paper to answer this question.
Table 13 shows some values of x and y which satisfy the equation y = 2x2 – 4x – 5.
x -2 -1 0 1 2 3 4 5
y 11 m -5 -7 n 1 11 25
Table 13
(a) Calculate the values of m and n.
(b) Using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis,
draw the graph of y = 2x2 – 4x – 5 for – 2 ≤ x ≤ 5.
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ANSWERS – Diagnostic Test
1. x = - 2, y = - 17; x = 1, y = 10; x = 4, y = 1
2. x = - 2.5, y = 2.4; x = 3, y = - 2
3.
y
x
0
4
y
x
0
20
