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Chapter 2 - LINEAR LAW
1.1 Draw the line of best fit 1
2
3
4
1.2 Write the equation for the line of best fit of the following graphs. 1
.
[ ]
2
[ ]
3
[y=2x2+6]
4
[y=-3x2+11]
5
[ ]
6
[ ]
Linear law 1
y
x
P(0,3)
Q(6,13)
.
.
XX
X
XX
X
XP(0,5)
Q(2,0)
y
x
.
.
X
X
XX
X
P(-1,4)
Q(5,16)
x2
y
.
.
xx
x
xx
A(3,0)
B(5,9)
x
x
x
xs
t
y2
x
y
x2
log 10 y
log 10 x
y
x
y
x2
P(1,8)
Q(3,2)
.
.
x
x xx
x
7
[ p = ]
8
[ ]
1.3 Determine the values of variables from lines of best fit 1
The diagram below shows a line of best fit. From the graph, find
i. the value of y when x = 0.5ii. the value of x when y = 7
iii.
[2.8, 3.3]
2
The diagram below shows a line of best fit. From the graph, find
i. the value of t when w = 38ii. the value of w when t = 1.6
[3.6,22]
3
The diagram below shows a line of best fit obtained by plotting the graph of d against t. The line intersects the vertical and the horizontal axes at points (0,2) and (6,0) respectively. Find
i. the equation of best fitii. the value of t when d=3
iii. the value of d when t=4
[ ]
4
Two variables, p and q are known to be linearly related as shown by the line of best fit in the diagram below. The line passes through points (1.6, 6) and (13.6 , 30). Determine
i. the equation of best fitii. the value of q when p= 15
iii. the value of p when q = 5
Linear law 2
B(5,3)
x
xp
q
A(2,)
d(0,2)
(6,0)
t
p(13, 30)
(1, 6)
q
x
x
x
x
x
x
x
1 2 3 4
2
4
6
8
x
y
x
x
x
x
100
20
30
1 2 3 4 t
W
40 x
[p=2q+4, 5.5, 14]2.1 Reduce non linear relations to linear form Reduce each of the equations to the form Y=m X+C where a and b are constants.
Non-linear equation Linear equation Y X m C1 y = a x2 + b x x
2 y = ax3 + bx2 x
3y =
y
4y =
xy x2
5xy =
y
6 x +by = axy
7 xy x2
8 xy
9 y = abx log10 y x
10 y = axb log10 y log10 x
11 y =a2xb log10 y log10 x
12 PV=a P
Linear law 3
2.2 Determine values of constants of non-linear relations given lines of best fit1 The diagram below shows the line of best fit for the
graph of y2 against x. Determine the non-linear equation connecting y and x.
[y2=-2x+4]
2 The diagram below shows the line of best fit for the
graph of y2 against . Determine the non-linear equation
connecting y and x.
[ ]
3 The diagram below shows the line of best fit for the
graph of against x. Determine the non-linear
equation connecting y and x.
[ ]
4 The diagram below shows the line of best fit for the
graph of against x. Determine the non-linear equation connecting y and x.
[ ]
Linear law 4
P(0,4)
Q(2,0)
y2
x
X
XP(0,2)
Q(2,12)y2
x
1
X
P(3,0)
Q(6,12)2x
y
xX
X
(2,4)
2x
y
(4, 5)
x0
5 The diagram below shows the line of best fit for the graph of log10 y against x. Determine the non-linear equation connecting y and x.
[log10 y = 3x ]
6 The diagram below shows the line of best fit for the graph of log10 y against log 10 x. Determine the non-linear equation connecting y and x.
[ log 10 y = 2 log10 x+2]7 The diagram below shows the line of best fit for the
graph of xy against x. Determine the relation between y and x.
[ ]
8 The diagram below shows the straight line graph of against x. Express y in terms of x.
[ ]
9 The diagram below shows the straight line graph of
against x. Express y in terms of x.
10 The diagram below shows the line of best fit for the graph of x2y against x. Determine the relation between y and x.
Linear law 5
(0,0)
(2,6)y10log
xX
X
(0,2)
(2,6)
y10log
x10log
X
X
(4,10)
x
(1, 1)
xy
P(2,4)
Q(4,12)
xX
Xxy
(4,10)
x
(1, 1)
x 2 y
P(2,4)
Q(4,12)
xX
Xx2 y
[ ] [ ]
11 The diagram below shows the line when against x
is drawn. Express y as a function of x.
[ ]
12The diagram below shows the line when against x is
drawn. Express y as a function of x.
[ ]
13
The diagram below shows the line of best fit for the
graph of against . Determine the relation
between y and x.
[ ]
14 The diagram below shows the line when against x is
drawn. Express y as a function of x.
[ ]
15The diagram below shows the line when
against x is drawn. Express y in terms of x.
16The diagram below shows the line when against x is
drawn. Determine the non-linear equation connecting y and x
Linear law 6
(12, -1)
5
x
y
x(12, -1)
5
x
y
x0
(0,8)
(-4,0)
x
y 1
x
(8,10)
(4,2)
y
1
x0
P(3,0)
Q(6,12)2x
y
xX
X
(2,4)
2x
y
(4, 5)
x0
3
(8, -3)
x
y
x
x
y
x0
[y=2x2+8x-1]
[ ]
17The diagram below shows the line when against x
is drawn. Determine the non-linear equation connecting y and x
[
]
18
The diagram below shows the line of best fit for the graph of log10 y against x. Determine the relation between y and x.
[ y = 10 3x ]
19 The diagram below shows part the graph of log10 y against x. Form the equation that connecting y and x.
[]
20
The diagram below shows the line of best fit for the graph of log10 y against log 10 x. Determine the relation between y and x.
[y= 100x2]
21 The diagram below shows part the graph of log10 y against log10 x. Form the equation that connecting y and x.
[ ]
22
The diagram below shows part the graph of log 2 y against log2 x. Determine the relation between y and x.
[ ]
Linear law 7
(3,6)
(0,3)
y
x
x0(0,0)
(2,6)y10log
xX
X
6
-3
log10 x
log10 y
2
(5,6)
log2 x
log2 y
0
log10 y
(3,4)
(4,0)
x (0,2)
(2,6)
y10log
x10log
X
X
2.3 Determine values of constants of non-linear relations when given lines of best fitExample :The variables x and y are related
by the equation , where a
and b are constants. The diagram below shows part of a line of best fit obtained by plotting a graph of xy against .
Find the values of a and b.
1 From the graph identify the representation of y-axis and x-axisY = , X =
2 Reduce the equation given to linear form, Y = mX + [ then ]
3 Compare with Y = mX + c Y = , m = , X = , c =
4 Find m from the graph :
5 Find c, substitute X = 1, Y = 35, m = 5 into the equation Y = mX +
c
6 Find the variables a and b :
Enrichment Exercise1.The variables x and y are related by the equation
. Find
(i) the value of k and c.(ii) the value ofd x when y =2
2. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.
3. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.
4. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.
Linear law 8
5. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.
6. The diagram below represents a linear graph of log y against x. It is known that the variables x and y are related by the equation y a where a and b are constants. Find the values of a and b.
7. The diagram below represents a line of against
It is known that the variables x and y are related by the
equation where p and q are constants. Find the
values of p and q.
8. Diagram shows a straight line graph of against
Given that x and y are related by the equation
where p and q are constants, find the values of p and q.
9. Diagram shows a straight line graph of against
Given that x and y are related by the equation
10. Diagram shows a straight line graph of against
Given that x and y are related by the equation
Linear law 9
O
(4,6)
(1,3) x²
(0,2)
10log x
10log y(4,10)
2x
y
x
(0,2)
(6,5)
where a and b are constants, find the values of a and b. where a and b are constants, find the values of a and b.
Further practices
The graph shows against . It is known that and are
related by . Find the value of and .
The diagram shows part of the graph against .
Given that , find the values of and .
•
1. Diagram 2 shows a straight line graph of .
Given that , calculate the value of and of . [3
The variables x and y are related by the equation, where k is a constant.
(a) Convert the equation to linear form.(b) Diagram 3 shows the straight line obtained by plotting against . Find the value of (i) (ii) . [4
Linear law 10
• (2, )
• (, 1)
O
x
(2, h)
• (, 3)
O
2x
y
(-1,p)(q,2)
x
x
y
(6 ,p)
(q ,2)
2x
SPM Past Year SeriesSPM 2003 P 1 Q10x and y are related by the equation , where p and
q are constants. A straight line is obtained by plotting
against x, as shown in the diagram below.
Calculate the values of p and q. [4 marks ]
[p= -2, q =13]
SPM 2004 P1Q13
Diagram below shows a straight line graph of against x
Given that y= 6x-x2, calculate the value of k and h [3 marks]
[h=3, k=4]
SPM 2005 P 1 Q13The variables x and y are related by the equation y=kx4, where k is a constant.
(a) Convert the equation y=kx4 to linear form.(b) Diagram below shows the straight line obtained by
plotting log10y against log10x
Find the value of(i) log10 k(ii) h [4 marks]
[3, 11]
SPM 2006P1 Q11
The first diagram shows the curve y = 3x2+5. The second graph shows the straight line graph obtained when y = -3 x2 +5 is expresses in the linear form Y= 5X+c. Express X and Y in terms of a and/or y [3 marks]
X=9/25 x2, Y=-3/5 y
SPM 2007 P1 Q12 SPM2008 P1 Q12
The variables x and y are related by the equation y = ,
where k is a constant. Diagram below shows the straight line graph obtained by plotting log10y against x.
Linear law 11
(2,9)
(6,1)
x0
(2, k)
(h, 3)
x0
(2, h )
(0, 3)
10log x0
The variables x and y are related by the equation y2=2x(10-x).
A straight line graph is obtained by plotting against x
Find the value of p and of q [3 marks]
[10,14]
(a) Express the equation y = ,in its linear form used to
obtain the straight line graph shown in the diagram above.
(b) Find the value of k [4 marks]
[log10 y= -xlog105 +log10k, 0.01]
2.3 Obtain information from (i) lines of best fit (ii) equations of lines of best fit.
A. Using a graph paper 1. Identify the graph to be drawn 2. Change the non-linear function with variables x and y to a linear form Y = mX + c 3. Construct a table for X and Y 4. Plot the linear graph using the scale given by the question & label both axes 5. Draw line of best fit 6. Determine : gradient , m and Y-intercept , c from the graph to find the values oj constants in the non-linear equation.
Example: 1. Use graph paper to answer this question.
The table below records the values of an experiment for two variables x and y which are related by where p
and q are constants.x 0.8 1 1.3 1.4 1.5 1.7y 108.75 79 45.38 36.5 26.67 8.19
(a) Plot xy against x3 using scale 2 cm represents 1 unit in x-axis and 2 cm represents 10 units for y-axis. Hence, draw the line of best fit
[5marks ] (b) From the graph, estimate the value of
(i) p and q
(ii) x when y= [5marks] Answer:p=-16.67, q=95, x=1.458
2. SPM 2004 2 Q 7
Linear law 12
Use graph paper to answer this question. Table below shows the values of two variables, x and y, obatained from an experiment. Variables x and y are related by the equation y = p k x , where p and k are constants.
x 2 4 6 8 10 12y 3.16 5.50 5.50 16.22 28.84 46.77
(a) Plot log10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit on the y-axis. Hence, draw the line of best fit [4 marks ](b) Use your graph from (a) to find the value of
(i) p(ii) k [ 6 marks]
Answer :p =1.820, k =1.309
3. SPM 2005 P 2 Q 7
Use graph paper to answer this question. Table below shows the values of two variables, x and y, obtained from experiment. The variables x and y are
related by the equation , where p and r are constants.
x 1.0 2.0 3.0 4.0 5.0 5.5y 5.5 4.7 5.0 6.5 7.7 8.4
(a) Plot xy against x2, by using a scale of 2 cm to 5 units on both axes. Hence,draw the line of best fit. [5 marks](b) Use the graph from (a) to find the value of
(i) p(ii) r [5 marks] Answer :[ p=1.37, r=5.48]
3. SPM 2006 P2 Q7 Use graph paper to answer this question. Table below shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation , where p and k are constants.
x 1 2 3 4 5 6y 4.0 5.7 8.7 13.2 20.0 28.8
(a) Plot log y against (x+1), using a scale of 2 cm to 1 unit on the (x+1) –axis and 2 cm to 0.2 unit on the log y-axis.Hence, draw the line of best fit. [5 marks]
(b) Use you graph from (a) to find the values of(i) p(ii) k [5 marks]Answers: 1.778, 1.483
4.SPM 2007 P2Q7 Use Graph paper to answer this question.Table below shows the values of two variables, x and y, obtained from an experiment.
Variables x and y are related by the equation , where p and k are constants.
x 2 3 4 5 6 7y 8 13.2 20 27.5 36.6 45.5
(a) Plot against x, using a scale of 2cm to 1 unit on both axes. Hence, draw the line of best fit.
[4marks](b) Use your graph in part (a) to find the value of
(i) p,(ii) k,(iii) y when x=1.2 [6marks]Answers : 0.754, 0.26, 4.2
5. SPM 2008 P2 Q8
Linear law 13
Table below shows the values of two variables, x and y , obtained from an experiment. Variables x and yare related by the equation y = hk2x, where h and k are constants.
x 1.5 3.0 4.5 6.0 7.5 9.0y 2.51 3.24 4.37 5.75 7.76 10.00
(a) based on the Table above, construct a table for the values of log10 y. [ 1 mark](b) Plot log10 y against x, using a scale of 2cm to 1 unit on the x-axis and 2 cm to 0.1 unit on the log10 y-axis Hence, draw the line of best fit. [4 marks](c) Use the graph in part (b) to find the value of (i) x when y = 4.8 (ii) h (iii) k [ 5 marks]Answers : 4.95, 1.905, 1.096
12 SPM 2009 P2 Q 8Use Graph paper to answer this question.Table below shows the values of two variables, x and y, obtained from an experiment.
The Variables x and y are related by the equation , where p and k are constants.
x 1.5 2.0 3.0 4.0 5.0 6.0y 2.502 0.770 0.465 0.385 0.351 0.328
a) Based on the Table above, construct a table for the values of and . [2 marks]
b) Plot against , using a scale of 2cm to 1 unit on both axes. Hence, draw the line of best fit. [5marks]
c) Use your graph in part (a) to find the value of(i) k,
(ii) p [5marks] Answers : { (c) (i) 0.256 (ii) – 1.331} Answer for 2.1
Non-linear equation Linear equation Y X m C1 y = a x2 + b x x a b
2 y = ax3 + bx2 x a b
3y =
y b a
4y =
xy x2 a b
5xy =
y b a
6 x +by = axy -b a
7 xy x2 -3 5
8 xy ab -a
9 y = abx log10 y x log 10 b log10 a
10 y = axb log10 y log10 x b log10 a
11 y =a2xb log10 y log10 x b 2log10 a
12 PV=a P a 0
Linear law 14