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8/20/2019 Form 5 _ Maths - Chapter 2
http://slidepdf.com/reader/full/form-5-maths-chapter-2 1/38
Graphs of Functions 1
CHAPTER 11 : GRAPHS OF FUNCTIONS II
Steps involved in solving an equation graphically:
1. Find the y value by substituting x value into the given function.
2. Draw the graph by using the given scale for x-axis and y- axis.
3. Find the value of y or x when given value of x or y from the graph.
4. On the same axes, draw a suitable straight line which satisfies the equation.
5. Determine the solutions by reading off the x-coordinates of the point of
intersection of the two graphs.
8/20/2019 Form 5 _ Maths - Chapter 2
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8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 3
Example 2
Complete the following table for theequation 32 += x y
-1 0 3 y 1 3 9
By using scientific calculator, substituteeach of the x value in equation above.
3)(2 += x y -1 2(-1) + 3 = 1
0 2(0) + 3 = 33 2(3) + 3 = 9
Example 3
Complete the following table for the
equation 12
−= x
y
2 4 8 y 0 1 3
By using scientific calculator, substituteeach of the x value in equation above.
12
−= x
y
201
22
=−
411
2
4=−
831
28
=−
Exercise 2
Complete the following table for theequation 13 += x y
-1 0 3
By using scientific calculator, substituteeach of the x value in equation above.
1)(3 += x y -1
03
Exercise 3
Complete the following table for the
equation 23
−= x
y
2 4 8
By using scientific calculator, substituteeach of the x value in equation above.
x
2
4
8
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 4
Example 4
Complete the following table for theequation x y 23 −=
-1 0 1 y 5 3 1
By using scientific calculator, substituteeach of the x value in equation above.
)(23 x y −= -1 5)1(23 =−−
0 3)0(23 =− 1 1)1(23 =−
Exercise 4
Complete the following table for theequation x y 35 −=
-1 0 1
By using scientific calculator, substituteeach of the x value in equation above.
)(35 x y −=
-1
01
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 5
11.2: Quadratic FunctionComplete the following table below
Example:1. Complete the following table for
the equation y = 2x 2
x 1 2 3y 2 8 18
By using scientific calculator, substituteeach of the x value in equation above.
x y1 2(1) 2 = 22 2(2) 2 = 83 2(3) 2 = 18
Exercise:Complete the following table for theequations:
1. y = - 2x 2 x -1 0 1y
By using scientific calculator, substitute eachof the x value in equation above.
x y=- 2(x) 2 -101
2. y =21
x 2
x -1 0 1y
By using scientific calculator, substitute eachof the x value in equation above.
xy =
21
(x) 2
-101
3. y = 3x 2
x -1 0 1y
By using scientific calculator, substitute eachof the x value in equation above.
x y = 3(x) 2 -101
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 6
4. y = -25
x 2
x -1 0 1y
By using scientific calculator, substitute eachof the x value in equation above.
xy = -
25
(x) 2
-10
1
5. y = 5x 2 x -1 0 1y
By using scientific calculator, substitute eachof the x value in equation above.
x y = 5(x) 2 -101
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 7
11.3: Cubic Function
Complete the following table below
Example 1
Complete the following table for theequation 33 x y =
x 1 2 3 y 3 24 81
By using scientific calculator, substitute
each of the x value in equation above.
x 3)(3 x y = 1 3)1(3 3 = 2 24)2(3 3 = 3 81)3(3 3 =
Exercise 1
Complete the following table for theequations:
i. 35 x y =
x 1 2 3 y
By using scientific calculator, substituteeach of the x value in equation above.
3)(5 x y = 123
ii. 32 x y −=
x 1 2 3 y
By using scientific calculator, substituteeach of the x value in equation above.
3)(2 x y −= 123
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 8
Example 2
Complete the following table for theequation 52 3 += x y
x -2 0 2 y -11 5 21
By using scientific calculator, substituteeach of the x value in equation above.
x 5)(2 3 += x y -2 115)2(2 3 −=+− 0
55)0(23 =+
2 215)2(2 3 =+
Example 3
Complete the following table for theequation 43 +−= x y
x -1 -2 3 y 5 12 -23
By using scientific calculator, substituteeach of the x value in equation above.
x 4)( 3 +−= x y -1 54)1( 3 =+−− -2 124)2( 3 =+−− 3 234)3( 3 −=+−
Exercise 2
Complete the following table for theequation 53 += x y
-2 0 2
By using scientific calculator, substituteeach of the x value in equation above.
x 5)( 3 += x y
-20
2
Exercise 3
Complete the following table for theequation
i. 33 +−= x y x -1 -2 3
By using scientific calculator, substituteeach of the x value in equation above.
3)( 3 +−= x y
-1-23
ii. 12 3 −= x y x -1 -2 3
By using scientific calculator, substituteeach of the x value in equation above.
1)(2 3 −= x y -1-23
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 9
Example 4
Complete the following table for theequation 943 −−= x x y
-3 0 2-24 -9 -9
By using scientific calculator, substituteeach of the x value in equation above.
x 9)(4)( 3 −−= x x y -3 249)3(4)3( 3 −=−−−− 0 99)0(4)0( 3 −=−− 2 99)2(4)2( 3 −=−−
Exercise 4
Complete the following table for theequations:
i. 733 +−= x x y
x -3 0 2 y
By using scientific calculator, substituteeach of the x value in equation above.
7)(3)( 3 +−= x x y
-30
2
ii. 3612 x x y −−=
x 1 0 2 y
By using scientific calculator, substituteeach of the x value in equation above.
x 3)()(612 x x y −−= 10
2
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 10
11.4: Reciprocal Function
Complete the following table below
Example 1:
Complete the following table for the
equation y = x1
x -2 -1 1 2y -1/2 -1 1 1/2
By using scientific calculator, substitute eachof the x value in equation above.
x y-2 - ½-1 - 1/1 = -11 1/1 = 12 ½
Exercise:
Complete the following table for theequations:
1. y = - x2
x -2 -1 1 2
y
By using scientific calculator, substituteeach of the x value in equation above.
xy = -
x2
-2-112
2. y =3
x -2 -1 1 2y
By using scientific calculator, substituteeach of the x value in equation above.
xy =
3
-2-112
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 11
3. y = -
x2
1
x -2 -1 1 2y
By using scientific calculator, substituteeach of the x value in equation above.
xy = -
x21
-2-112
4. y =16
x -2 -1 1 2y
By using scientific calculator, substituteeach of the x value in equation above.
xy =
x16
-2-112
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 12
5. y = - x5
x -2 -1 1 2y
By using scientific calculator, substituteeach of the x value in equation above.
xy = -
x5
-2-112
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 13
11.21 : Quadratic Graph
Example :
a) Complete the following table for the equation 542 2 −−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 542 2 −−= x x y for 52 ≤≤− x
c) From your graph, find
i) the value of y, when x = -1.5ii) the values of x, when y= 0
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 0382 2 =+− x x for 52 ≤≤− x . State the values of x.
Answer For Example :
a) x = -1, y = 1x = 2, y = -5x = 4, y = 11
b) Refer graph on the next page
c) i) y = 5.5ii) x = -0.9, 2.9
d)
Therefore, the appropriate graph that should be drawn is
x 0 1y − 8 − 4
From the graph, the solutions of the equation are
x = 0.4, 3.65
x -2 -1 0 1 2 3 4 5 y 11 -5 -7 1 25
0382 2 =+− x x
382 2 −= x x
5438542 2 −−−=−− x x x x
84 −= x y
Rearrange the equationso that one side of theequation is 2 x2 − 4 x − 5
84 −= x y
0382 2 =+− x x
8/20/2019 Form 5 _ Maths - Chapter 2
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8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 15
Graph for Example Quadratic Graph
-2 -1 0 1 2 3 4 5
-5
--10
5
10
15
20
25
y
x
x
x
x
x
x
x
x
x
x
x
y = 4x - 8
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 16
Exercise 1 :
a) Complete the following table for the equation 592 2 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 592 2 +−= x x y for 70 ≤≤ x
c) From your graph, findi) the value of y, when x = 1.7ii) the value of x, when y = 15
d) Draw a suitable straight line on your graph to find the values of x which satisfy the
equation 041022
=+− x x for 70 ≤≤ x . State the values of x.
Exercise 2 :
a) Complete the following table for the equation 752 2 −−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw the
graph of 7522
−−= x x y for 53 ≤≤− x c) From your graph, find
i) the value of y, when x = -2.5ii) the value of x, when y = 15
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 0472 2 =+− x x for 53 ≤≤− x . State the values of x.
Exercise 3:
a) Complete the following table for the equation ( ) 952 −−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of ( ) 952 −−= x x y for 53 ≤≤− x
c) Draw a suitable straight line on your graph to find the values of x which satisfy theequation ( ) x x x 24952 −=−− for 53 ≤≤− x . State the values of x.
x 0 1 2 3 4 5 6 75 -2 -5 1 10 40
x -3 -2 -1 0 1 2 3 4 526 0 -7 -10 -9 5 18
x -3 -2 -1 0 1 2 3 4 524 -2 -9 -12 -11 3 6
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 17
Exercise 4 :
a) Complete the following table for the equation 523 2 ++−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 523 2 ++−= x x y for 43 ≤≤− x
c) From your graph, findi) the value of y, when x = -0.5ii) the value of x, that satisfy the equation of 523 2 =− x x
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 02523 2 =−+ x x for 43 ≤≤− x . State the values of x.
Exercise 5 :
a) Complete the following table for the equation ( ) x x y −= 821
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 1 unit on the y-axis, draw the
graph of ( ) x x y −= 821
for 70 ≤≤ x
c) Draw a suitable straight line on your graph to find the values of x which satisfy theequation ( ) 108 =− x x for 70 ≤≤ x . State the values of x.
Exercise 6 :
a) Complete the following table for the equation 2238 x x y −+=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 2238 x x y −+= for 43 ≤≤− x
c) From your graph, findi) the value of y, when x = 1.35ii) the values of x, when y = -10
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 02632 2 =+−− x x for 43 ≤≤− x . State the values of x.
x -3 -2 -1 0 1 2 3 4-11 0 4 -16 -35
x 0 0.5 1 2 3 4 5 6 70 1.88 6 7.5 7.5 3.5
x -3 -2 -1 0 1 2 3 4-19 3 8 9 6 -12
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 18
Exercise 7 :
a) Complete the following table for the equation 2275 x x y −+=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 2275 x x y −+= for 53 ≤≤− x
c) From your graph, findi) the value of y, when x = 3.2ii) the values of x, when y = -20
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 01132 2 =−− x x for 53 ≤≤− x . State the values of x.
Exercise 8 :
a) Complete the following table for the equation 2273 x x y −+=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 2273 x x y −+= for 53 ≤≤− x
c) From your graph, findi) the value of y, when x = 1.3ii) the values of x, when y = -25
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 01332 2 =−− x x for 53 ≤≤− x . State the values of x.
x -3 -2 -1 0 1 2 3 4 5 y -34 -4 5 11 8 -10
x -3 -2 -1 0 1 2 3 4 5-36 -6 3 9 6 -12
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Graphs of Functions 19
11.31 : Cubic Graph
Example 1:
a) Complete the following table for the equation 583 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 583 +−= x x y for 43 ≤≤− x
c) From your graph, findi) the value of y, when x = -1.4
ii) the value of x, when y= 25d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 01123 =−− x x for 43 ≤≤− x . State the values of x.
Answer For Example :
a) x = -2, y = 13x = 1, y = -2x = 3, y = 8
b) Refer graph on the next page
c) i) y = 13. 5ii) x = 3.7
d)
Therefore, the appropriate graph that should be drawn is
x 0 1y 6 10
From the graph, the solutions of the equation are
x = − 0.1, 3.5
x -3 -2 -1 0 1 2 3 3.5 42 12 5 -3 19.9 37
01123 =−− x x
1123 += x x
58112583 +−+=+− x x x x64 += x y
Rearrange the equationso that one side of theequation is x3 − 8 x + 5
64 += x y
01123 =−− x x
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 20
Graph for Example Cubic Graph
-3 -2 -1 0 1 2 3 4
-5
5
10
15
20
25
30
35
40
y
x
x
x
x
xx
x
x
x
x
x
= y 64 + x
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 21
Exercise 1:
a) Complete the following table for the equation 5103 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 5103 +−= x x y for 5.35.3 ≤≤− x
c) From your graph, find the value of y, when x = -2.5
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 6103 =− x x for 5.35.3 ≤≤− x . State the values of x.
Exercise 2:
a) Complete the following table for the equation 18103 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 18103 +−= x x y for 43 ≤≤− x
c) From your graph, find the value of y, when x = -0.5
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 010103 =−− x x for 4.3 ≤≤− x . State the values of x.
Exercise 3:
a) Complete the following table for the equation 7123 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 7123 +−= x x y for 44 ≤≤− x
c) From your graph, findi) the value of y, when x = 2.5ii) the values of negative x that satisfy the equation 7123 −= x x
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 02153 =+− x x for 44 ≤≤− x . State the values of x.
x -3.5 -3 -2 -1 0 1 2 3 3.5-2.9 8 14 5 -7 12.9
x -3 -2 -1 0 1 2 3 3.5 421 27 18 6 25.88 42
x -4 -3 -2 -1 0 1 2 3 4-9 16 18 7 -4 -9 23
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 22
Exercise 4:
a) Complete the following table for the equation 6123 +−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 10 unit on the y-axis, drawthe graph of 6123 +−= x x y for 45 ≤≤− x
c) From your graph, findi) the value of y, when x = -2.5ii) the values of positive x, when y = 0
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 12123 =− x x for 44 ≤≤− x . State the values of x.
Exercise 5:
a) Complete the following table for the equation 1253 −−= x x y
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 1253 −−= x x y for 5.33 ≤≤− x
c) From your graph, findi) the value of y, when x = 0.8ii) the value of x which satisfies the equation 1253 =− x x
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 493 =− x x for 5.33 ≤≤− x . State the values of x.
x -5 -4 -3 -2 -1 0 1 2 3 4 y -59 -10 22 17 6 -5 -3 22
x -3 -2 -1.5 -1 0 1 2 3 3.5-10 -7.9 -8 -12 -16 0 13.4
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Graphs of Functions 23
11.41 : Reciprocal Graph
Example 1:
a) Complete the following table for the equation x
y 8=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 2 unit on the y-axis, draw the
graph of y8
= for 55 ≤≤− x
c) From your graph, findi) the value of y, when x = -3.2
ii) the value of x, when y = 4.4d) From the graph, state the values of x when x and y have a same value.
Answer For Example :
a) x = -5, y = -1.6x = -2.5, y = -3.2x = 1.6, y = 5
b) Refer graph on the next page
c) i) y = -2.6ii) x = 1.8
d) x = -2.7, 2.73
x -5 -4 -2.5 -1.6 -1 1 1.6 2.5 4 5-2 -5 -8 8 3.2 2 1.6
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 24
x
x
55
Graph for Example Reciprocal Graph
-2
-4
-6
-8
0
2
4
6
8
10
y
-1-2-3-4 1 2 3 4
x
x
x
x
x
x
x
x
x
x
x
y = x
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 25
Exercise 1:
a) Complete the following table for the equation x
y1
=
b) By using a scale of 2 cm to 1 unit on the x-axis and 4cm to 1 unit on the y-axis, draw the
graph of y1
= for 44 ≤≤− x
c) From your graph, find the value of y, when x = -2.4d) Draw a suitable straight line on your graph to find the values of x which satisfy the
equation ( )0,11
≠=− x x for 44 ≤≤− x . State the values of x.
Exercise 2:
a) Complete the following table for the equation y6
−=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 2 unit on the y-axis, draw the
graph of y6
−= for 44 ≤≤− x
c) From your graph,i) find the value of y, when x = 1.3ii) find the value of x, when y = 3.5
d) Draw a suitable straight line on your graph to find the values of x which satisfy the
equation x=+ 13
for 44 ≤≤− x . State the values of x.
Exercise 3:
a) Complete the following table for the equation x
y2
=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 1 unit on the y-axis, draw the
graph of y 2= for 44 ≤≤− x
c) From your graph,i) find the value of y, when x = 1.5ii) find the value of x, when y = -1.8
d) Draw a suitable straight line on your graph to find the values of x which satisfy the
equation x
x4
12 =+ for 44 ≤≤− x . State the values of x.
x -4 -3 -2 -1 -0.5 0.5 1 2 3 4-0.25 -0.33 -1 -2 1 0.5 0.25
x -4 -2.5 -1 -0.6 0.6 1 2 3 41.5 6 10 -10 -6 -3 -1.5
x -4 -3 -2 -1 -0.5 0.5 1 2 3 4-0.5 -0.7 -1 -2 4 1 0.7 0.5
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 26
Exercise 4:
a) Complete the following table for the equation x
y4
=
b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 2unit on the y-axis, draw the
graph of y4
= for 55 ≤≤− x
c) From your graph,i) find the value of y, when x = -1.4ii) find the value of x, when y = 4.4
d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 0
23
=− x x
for 45.0 ≤≤ x . State the values of x.
Exercise 5:
a) Complete the following table for the equation 123
+= y
b) By using a scale of 2 cm to 0.5 unit on the x-axis and 2cm to 0.5unit on the y-axis, draw
the graph of 123 += x
y for 45.0 ≤≤ x
c) From your graph,i) find the value of y, when x = 1.3ii) find the value of x, when y = 2.8
d) Draw a suitable straight line on your graph to find the values of x which satisfy the
equation 023
=− x x
for 45.0 ≤≤ x . State the values of x.
x -5 -4 -2 -1 -0.5 0.5 0.8 1.5 2.5 5-1 -4 -8 8 5 1.6
x 0.5 1 1.5 2 2.5 3 3.5 44 2.5 1.75 1.6 1.43 1.38
8/20/2019 Form 5 _ Maths - Chapter 2
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Graphs of Functions 27
PAST YEAR EXAMINATION QUESTIONS
1. Nov 2003, Q12
(a) Complete the following table for the equation y 4−= .
(b)
x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4y 1 1.6 8 -8 -4 -1.25 -1
[2 marks ]
For this part of the question, use the graph paper. You may use a flexible curve.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis,
draw the graph of x
y 4−= for -4 ≤ x ≤ 4.
[5 marks ](c) From your graph, find
(i) the value of y when x = 1.8,(ii) the value of x when y = 3.4.
[2 marks ](d)
(e) Draw a suitable straight line on your graph to find all the values of x which
satisfy the equation 324
+= x x
for -4 ≤ x ≤ 4.
State these values of x.[3 marks ]
2. July 2004, Q12
(a) Table 2 shows values of x and y which satisfy the equation 1053 +−= x x y .
x -3.4 -3 -2 -1 0 1 2 3 3.4y -12.3 k 12 14 10 6 m 22 32.3
TABLE 2 Calculate the value of k and of m. [2 marks ]
(b) For this part of the question, use the graph paper on page 33. You may use a flexible curve rule. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,draw the graph of 1053 +−= x x y for -3.4 ≤ x ≤ 3.4.
[4 marks ](c) From your graph, find
(i) the value of y when x = 0.7 ,(ii) the value of x when 01053 =+− x x .
[2 marks ]
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(d) Draw a suitable straight line on your graph to find all the values of x whichsatisfy the equation 0263 =−− x x for -3.4 ≤ x ≤ 3.4.State these values of x.
[4 marks ]
Answer :
(a) k = ………………..
m = ……………….
(b) Refer graph on page 33
(c) (i) y = ………………..
(ii) x = ……………….
(d) x = ………………………………………………………………………………..
3. Nov 2004, Q12
(a) Table 1 shows values of x and y which satisfy the equation 342 2 −−= x x y .
x -2 -1 0 1 2 3 4 4.5 5y k 3 -3 -5 -3 m 13 19.5 27
Table 1Calculate the value of k and of m. [2 marks ]
(b) For this part of the question, use the graph paper. You may use a flexible curve. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,draw the graph of 342 2 −−= x x y for -2 ≤ x ≤ 5.
[5 marks ](c) From your graph, find
i. the value of y when x = -1.5,ii. the value of x when y = 0.
[2 marks ](d) Draw a suitable straight line on your graph to find a value of x which satisfies the
equation 0232 2 =−+ x x for -2 ≤ x ≤ 5.State this value of x.
[3 marks ]
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4. Nov 2005, Q12
(a) Complete Table 2 for equation 32 2 −−= x x y .
x -2 -1 0.5 1 2 3 4 4.5 5y 7 -2 -2 3 12 33 42
Table 2[2 marks ]
(b) For this part of the question, use the graph paper. You may use a flexible curve. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of 32 2 −−= x x y for -2 ≤ x ≤ 5.
[4 marks ](c) From your graph, find
i. the value of y when x = 3.6,ii. the value of x when y = 37.
[2 marks ](d) Draw a suitable straight line on your graph to find all value of x which satisfy the
equation 1032 2 =− x x for -2 ≤ x ≤ 5.State these values of x.
5. July 2005, Q12
(a) Complete Table 1 in the answer space for the equation 223 x x y −−= .
[2 mark ] Answer :
(a)x -3 -2 -1 0 1 2 3 3.5 4y -12 -3 3 0 -7 -25 -33
TABLE 1
(b) For this part of the question, use the graph paper provided on page 33.
You may use a flexible curve rule.By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,draw the graph of 223 x x y −−= for -3 ≤ x ≤ 4.
[4 marks ] Answer :
(b) Refer graph on page 33 .
(c) From your graph, find(i) the value of y when x = 1.7 ,
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Graphs of Functions 30
(ii) the value of x when y = − 26 .
Answer :
(c) (i) y = ……………………….
(ii) x = ………………………
(d) Draw a suitable straight line on your graph to find all the values of x which
satisfy the equation 0221
18 2 =−+ x x for -3 ≤ x ≤ 4.
State these values of x.[4 marks ]
Answer :
(d) x = ………………………… , ……………………………
6. July 2006, Q12
(a) Complete Table 2 in the answer space for the equation 383 +−= x x y by writing down the values of y when x = -2 and 2 .
[2 marks ]
(b) For this part of the question, use the graph paper provided on page 35.
You may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,draw the graph of 383 +−= x x y for -3.5 ≤ x ≤ 3.5 .
[4 marks ](c) From your graph, find
(i) the value of y when x = − 0.7 ,(ii) the value of x when y = − 9 .
[2 marks ]
(d) Draw a suitable straight line on your graph to find all the values of x which
satisfy the equation 8133 −=
x x for -3.5 ≤ x ≤ 3.5 .State these values of x.[4 marks ]
Answer :
(a)x -3 -2 -1 0 1 2 3 3.5 4y -12 -3 3 0 -7 -25 -33
TABLE 1
(b) Refer graph on page 33 .
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(c) (i) y = ……………………….
(ii) x = ………………………
(d) x = ………………………… , ……………………………
7. Nov 2006, Q13
(a) Complete Table 1 in the answer space for the equation y = x24
by writing down
the value of y when x = -3 and x = 1.5. [2 marks ]
(b) For this part of the question, use the graph paper provided on page 35. You mayuse a flexible curve rule .
By using a scale of 2cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,
draw the graph of y =24
for 44 ≤≤− x . [5 marks ]
(c) From your graph, findi. the value of y when x = 2.9
ii. the value of x when y = -13.
[ 2 marks ](d) Draw a suitable line on your graph to find a value of x which satisfies the
equation 2x 2 + 5x = 24 for 44 ≤≤− x .State this value of x.
[ 3 marks ]Answer :
(a)
x -4 -3 -2 -1 1 1.5 2 3 4y -6 -12 -24 24 12 8 6
TABLE 1
(b) Refer graph on page 35 (c) (i) y = ………………………………………..
(ii) x = ………………………………………..
(d) x = ……………………………..
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8 Jun 2007 Q 16
(a) Complete Table 3 in the answer space for the equation 132 2 −+= x x y by writing down the values of y when x = –3 and x = 1 .
(b) For this part of the question, use the graph paper provided on page 37.You may use a flexible curve rule .
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 unitson the y-axis, draw the graph of 132 2 −+= x x y for 5.34 ≤≤− x .
(c) From your graph, find
(i) the value of y when x = – 3.6 ,
(ii) the value of x when y = 24 .
(d) Draw a suitable straight line on your graph to find the values of x which satisfiy the equation 072 2 =−+ x x for 5.34 ≤≤− x .State these values of x .
Answer :
(a)
x – 4 –3 –2 –1 0 1 2 2.6 3.5 y 19 1 –2 –1 13 20.3 34
(b) Refer graph on page 37 .
(c) (i) y = …………………………………….
(ii) x = …………………………………….
(d) x = …………………………. , …………………………….
[2 marks ]
[4 marks ]
[2 marks ]
[4 marks ]
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9 November 2007 Q 12
(a) Complete Table 1 in the answer space for the equation 36 x y −= by writing down the values of y when x = –1 and x = 2 .
(b) For this part of the question, use the graph paper provided on page 23.You may use a flexible curve rule .
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 unitson the y-axis, draw the graph of 36 x y −= for 5.23 ≤≤− x .
(c) From your graph, find
(i) the value of y when x = 1.5 ,
(ii) the value of x when y = 10 .
(d) Draw a suitable straight line on your graph to find the values of x which satisfiy the equation 36 x y −= for 5.23 ≤≤− x .State these values of x .
Answer :
(a)
x – 3 –2.5 –2 –1 0 1 2 2.5 y 33 21.63 14 6 5
(b) Refer graph on page 23 .
(c) (i) y = …………………………………….
(ii) x = …………………………………….
(d) x = …………………………. , …………………………….
[2 marks ]
[4 marks ]
[2 marks ]
[4 marks ]
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10 Jun 2008 Q 12
(a) Complete Table 12 in the answer space for the quation 343 +−= x x y by writing down the values of y when x = –1 and x = 2 .
(b) For this part of the question, use the graph paper provided on page 25You may use a flexible curve rule .
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 unitson the y-axis, draw the graph of 343 +−= x x y for 45.3 ≤≤− x .
(c) From your graph in 12( b), find
(i) the value of y when x = – 0.7 ,
(ii) the value of x when y = 25 ,
(d) Draw a suitable straight line on the graph in 12( b) to find the values of x which satisfiy the equation 017163 =−− x x for .45.3 ≤≤− x State these values of x .
Answer :
(a) x –3.5 –3 –2 –1 0 1 2 3 4
y –25.9 –12 3 3 0 18 51
(b) Refer graph on page 25 .
(c) (i) y = …………………………………….
(ii) x = …………………………………….
(d) x = …………………………. , …………………………….
[2 marks ]
[4 marks ]
[2 marks ]
[4 marks ]
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11 Nov 2008 Q 12
(a) Complete Table 12 in the answer space for the quation y36
= by writing down the values of y when x = 6 and x = 10 .
(b) For this part of the question, use the graph paper provided on page 23 . You may use a flexible curve rule .
Using a scale of 1 cm to 1 unit on the x-axis and 1 cm to 1 unit
on the y-axis, draw the graph of y36
= for 142 ≤≤ x .
(c) From your graph in 12( b), find
(i) the value of y when x = 2.8 ,
(ii) the value of x when y = 5 ,
(d) Draw a suitable straight line on the graph in 12( b) to find the values of x
which satisfiy the equation 01436
=−+ x x
for .142 ≤≤ x
State these values of x .
Answer :
(a) x 2 2.4 3 4 6 8 10 12 14
y 18 15 12 9 4.5 3 2.6
(b) Refer graph on page 23 .
(c) (i) y = …………………………………….
(ii) x = …………………………………….
(d) x = …………………………. , …………………………….
[2 marks ]
[4 marks ]
[2 marks ]
[4 marks ]
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Graphs of Functions 36
ANSWERS
1 SPM 2003(N)
(a) 4 , – 2
(b) Graph
(c) (i) y = – 2.2 (ii) x = – 1.2
(d) 32 −−= x y , x = – 2.3 , 0.85
2 SPM 2004(J)
a) k = – 2 , m = 8
(b) Graph
(c) (i) y = 6.8 (ii) x = – 2.9
(d) 122 += x y , x = – 2.25 , – 0.32 , 2.6
3 SPM 2004(N)
(a) k = 13 , m = 3
(b) Graph
(c) (i) y = 7 (ii) x = 2.6 , – 0.5
(d) 205 +−= x y , 3.1 ≤ x ≤ 3.2
4 SPM 2005(J)
(a) 2 , – 18
(b) Graph(c) (i) y = – 4.7 (ii) x = 3.55
(d) 723
−−= x y , x = – 2.85 , 3.1
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5 SPM 2005(N)
(a) 0 , 25
(b) Graph
(c) (i) y = 19 (ii) x = 4.7
(d) 72 += x y , x = – 1.6 , 3.1
6 SPM 2006(J)
(a) y = 11 , y = – 5
(b) Graph
(c) (i) y = 8 (ii) x = – 3.4
(d) 55 −= x y , x = 0.64 , x = 3.24 ,
7 SPM 2006(N)
(a) y = – 8 , y = 16
(b) Graph
(c) (i) y = 8 (ii) x = – 1.85
(d) 52 += x y , x = 2.45
8 SPM 2007(J)
(a) y = 8 , y = 4
(b) Graph
(c) (i) y = 14 (ii) x = 2.85
(d) 62 += x y , x = – 2.14 , x = 1.64 ,
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9 SPM 2007(N)
(a) y = 7 , y = – 2
(b) Graph
(c) (i) y = 2.5 (ii) x = – 1.65
(d) x y 8−= , x = – 0.8 , x = – 2.4 ,
10 SPM 2008(J)
(a) y = 7 , y = – 2
(b) Graph
(c) (i) y = 2.5 (ii) x = – 1.65
(d) x y 8−= , x = – 0.8 , x = – 2.4 ,
11 SPM 2008(N)
(a) y = 6 , y = 3
(b) Graph
(c) (i) y = 5.5 (ii) x = 3.75
(d) 2012 += x y , x = – 3.3