Linear law
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Transcript of Linear law
additionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsadditionalmathematicsa
LINEAR LAW
Name
........................................................................................
LINEAR LAW
2.1 Understand and use the concept of lines of best fit
2.1.1 Draw lines of best fit by inspection of given data
Exercise 1. Draw the line of best fit.
1
2
3
4
2.1.2 Write equation for lines of best fit
y = mx + c is the linear equation of a straight line
Criteria of the best fit line :
1. points lie as close as possible to the line
2. line pass through as many points as possible
3. points that do not fit onto the line should be more or
less the same on both sides of the line
m = gradient
c = y-intercept
2 1 1 2
2 1 1 2
y y y yor
x x x x
c
y
x
Exercise 2. Write the equation of the line of best fit for each of the following graphs
(i)
Example:
(ii) Example:
Find m:
6 2
4 0
1
m
Find m:
6 1
1 5
5
4
m
Find c: c = y-intercept
c = 2
Find c: V = mt + c
The line passes through (1, 6)
t , V
56 1
4
29
4
c
c
Substitute into y = mx + c
The equation of the line : y = x + 2
Substitute into V= mt + c
The equation of the line : 5 29
4 4y x
a
y = x + 9
b
2 17
3 3y x
c
y = 2x 8
d
y = 3x 3
2
y
x O
(4, 6)
x (1, 6)
V
t O
(5, 1)
x
x
9
y
x O
(8, 1) x
(3, 6)
S
t O 1
x
(1, 5)
V
t O
(7, 1)
x
x
F
v O
(6, 4) x
(8, 8) x
3.1.3 Determine the values of variables
a) from lines of best fit
Exercise 3
1
(2.8, 3.3)
2
(3.6, 22)
b) from equations of lines of best fit
1
2
2.2 Applications of Linear Law to Non-linear Relations
2.2.1 Reduce non-linear relations to the linear form
LINEAR FORM OF EQUATION
Y = mX + c or Y = mX , c = 0
Variables for the y-axis Constant (no variable)
coefficient = 1 m = gradient y-intercept
Variable for the x-axis
Example : Identify Y, X, m and c for the following linear form
(i) y = 2x2 3
Y = y, m = 2, X = x2 , c = 3
(ii) y
x= 3x
2 5
Y = y
x, m = 3, X = x
2 , c = 5
Exercise 4. Reduce the following non-linear equation to linear equation in the form of Y = mX + c.
Hence, identify Y, X, m and c.
Example:
3y = 5x2 + 7x
Example:
y = px2
Create the y-intercept
23 5 7
35 7
y x xx :
x x x
yx
x
Create the y-intercept
210
210 10 10
10 10 102
takelog : log y log px
log y log p log x
log y log p log x
Create coefficient of y = 1 &
arrange in the form Y = mX + c
3 5 73
3 3 3
5 7
3 3
y x:
x
y x
x
Create coefficient of y = 1 &
arrange in the form Y = mX + c
10 10 102log y log x log p
Compare to Y = mX + c
5 7
3 3
y x
x
Y = y
x, m =
5
3, X = x, c =
7
3
Compare to Y = mX + c
10 10 102log y log x log p
Y =log10 y, m = 2, X = log10 x, c = log10 p
Equation Y = mX + c Y X m c
a) y2 = ax + b
b) y = ax2 + bx
c) y2 = 5x
2 + 4x
d) y =
5c
x
e) xy = a + bx
f) axy bx
x
g) y = a ( x+ b)2
h) y = ba x
x
i) 1a b
y x
j) y = abx
2.2.2a Determine values of constants of non-linear relations when given lines of best fit
Example :
The variables x and y are related by the equation a
by xx
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 x2.
Find the values of a and b.
From the graph identify the representation of y-axis and x-axis
Y = xy, X = x2
Reduce the equation given to linear form, Y = mX + c
21
by x ab :
b b b x
x ay
b bx
x x a xx : x y
b bx
axy x
b b
Compare with Y = mX + c
Y = xy , m = 1
b, X = x
2 , c =
a
b
Find m from the graph : 50 35
54 1
m
Find c, substitute X = 1, Y = 35, m = 5 into the equation Y = mX + c
35 = 5 (1) + c
c = 30
Find the variables a and b :
15
1
5
mb
b
30
301
56
a
ba
a
Exercise 5.
1
[y = 2x + 4]
2
[y2 = 5(1/x) + 2]
(1, 35)
xy
x2
O
(4, 50)
7 x
y
x
The variables x and y are related by the equation
y = px2 + qx.
Find the values of p and q.
p = 2, q = 13
8 y2
x
1
The variables x and y are related by the equation
cx
ky 2 . Find
(i) the value of k and c
(ii) the value of x when y = 2
(i) k = 2, c = 2 (ii) x = 1
9 y
x
1
The above figure shows part of a straight line
graph drawn to represent the equation of
xy = a + bx.
Find the value of a and b.
a = 3, b = 3
10 log y
log x
The above figure shows part of a straight line
graph drawn to represent the equation of
y = axb.
Find the value of a and b.
a = 11, b = 8
(2, 9)
(6, 1)
(4, 10)
(2, 6)
(3, 6)
1
(1, 9) 1
1
2.2.2b Determine values of constants of non-linear relations when given data
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. Choose a suitable scale & label both axes
5. Draw line of best fit
6. Determine : gradient , m and Y-intercept , c from the graph to find the values of
constants in the non-linear equation.
Example:
The table shows the corresponding values of two variables, x and y, obtained from an experiment.
The variables x and y are related by the equation k
y hxhx
, where h and k are constants
x 1.0 2.0 3.0 4.0 5.0 6.0
y 6.0 4.7 5.2 6.2 7.1 7.9
a) Using a scale of 2 cm to 10 units on both axes, plot a graph of xy against x2.
Hence, draw a line of best fit.
b) Use your graph from (a) to find the value of
(i) h,
(ii) k.
Solution :
1. Drawn graph : Y = xy and X = x2.
2. Change non-linear to linear form.
2
2
ky hx
hxk x
x: x y hx xhx
kxy hx
hk
Y xy, X x , m h, ch
3.
x 1.0 2.0 3.0 4.0 5.0 6.0
y 6.0 4.7 5.2 6.2 7.1 7.9
x2
1.0 4.0 9.0 16.0 25.0 36.0
xy 6.0 9.4 15.6 24.8 35.5 47.4
6. From the graph, gradient , 37
1 23330
m .
Hence, h = 1.233
From the graph, Y-intercept , c = 4.5
Hence,
4 5
4 51 233
5 549
h.
k
k.
.
k .
0 5 10
0
15
0
20
0
25 30
0
5
10
0
15
0
20
0
25
30
0
35
40
0
35
0
40
0
45
50
0
xy
x2
x
x
x
x
x
x
Exercise 6.
1. The table below shows some experimental data of two related variable x and y. It is known that
x and y are related by an equation in the form y = ax + bx2, where a and b are constants.
x 1 2 3 4 5 6 7
y 7 16 24 24 16 0 24
a) Draw the straight line graph of y
x against x.
b) Hence, use the graph to find the values of a and b. a = 1, b = 10
2. The table below shows some experimental data of two related variable x and y.
x 0 2 4 6 8 10
y 1.67 1.9 2.21 2.41 2.65 2.79
It is known that x and y are related by an equation in the form
ax by
y y , where a and b are constants.
a) Draw the straight line graph of y2 against x.
b) Hence, use the graph to find the values of a and b. a = 0.5, b = 2.8
3. The table below shows two variable x and y, which are obtained from an experiment. The
variables are related by the equation r
y pxpx
, where p and r are constants.
x 0 2 4 6 8 10
y 1.67 1.9 2.21 2.41 2.65 2.79
a) Plot xy against x2. Hence draw the line of best fit.
b) Based on the graph, find the values of p and r. p =1.38, r = 5.52
