Linear Law

“Transformation” of non-linear relationships into linear

relationships

How it works22 3y x Quadratic Curve: non-linear!

Transforming to linear relationship

22 3y x

cmXY

Linear ?

General equation of linear relationship:

22 3y x −3

y

x2

Plot y vs x2

yx

2

3 31

2

xy

3

y

1/x

baxy

2

1

baxy

bax

y

2

2

1

1

1

cmXY

Plot (1/y) vs x2 m = a, c = b

Example 1

cmXY

Plot xy vs x2 m = a, c = b

Example 2

x

baxy

baxxy 2

cmXY

m = b, c = a

Example 3

xbx

ay

xbxaxy

Plot xy vs xx

cmXY

Plot lg y vs x m = lg b, c = lg a

Q9

xaby

bxay

bay

abyx

x

lglglg

lglglg

)lg(lg

ya bx

x

bxaxy bx

ay

Plot xy vs x Plot y vs

Grad = b, xy-intercept = a Grad = a, y-intercept = b

x

1

Q16

y ab x 3

bbxay

bxay

bay

abyx

x

lg3lglglg

lg)3(lglg

lglglg

)lg(lg3

3

cmXY

Plot lg y vs x m = lg b, c = lg a + 3 lg b

Q17

y ab x 4 cmXY

bxay

aby

abyx

x

lglg)4lg(

)lg()4lg(

4

Plot lg (y – 4) vs x m = lg b, c = lg a

Q18

cmXY

Plot lg y vs lg x m = b, c = - lg a

Q19

bxay

axby

xbya

xay b

lglglg

lglglg

)lg()lg(

cmXY

m = p, c = - q

Q20

qxpxe y 2

qpxx

e y

Plot vs xx

e y

Express y in terms of x? y = ??

(0,1)

(4,9)

a) y

x2

cxmy )( 2

204

19

12

12

xx

yym

121

)0(21

)1,0(

)(2

2

2

xyc

c

At

cxy

12

)0(21

)(

)(

2

2

1

2

1

11

xy

xy

xxmyy

xxmyy

Express y in terms of x?

cx

my

11

2

1

40

02

12

12

xx

yym

22

11

21

2

112

)0(2

12

)2,0(

1

2

11

xy

xyc

c

At

cxy

22

11

)01

(2

12

1

)1

(1

)(

11

11

xy

xy

xx

myy

xxmyy

(4,0)

(0,2)y

1

x

1

(5, 9)

(2, 3)

x + 1

lg y

225

39

12

12

xx

yym

cxmy 1lg

12

10

11

11

10

12log

12lg

)1(23lg

)21(23lg

)1(2lg

)(

xy

xy

xy

xy

xy

xxyy

xxmyy

Q1

313

39

12

12

xx

yym

cxmy )]1[ln(ln

3

3

11

11

)1(

)1ln(ln

)1ln(3ln

3)1ln(33ln

]1)1[ln(33ln

])1[ln(3ln

)(

xy

xy

xy

xy

xy

xxyy

xxmyy

Q (3, 9)

P (1, 3)ln (x – 1)

ln y

Q2

The following table gives values of y corresponding to some value of x.

x 1 2 3 4 5

y 1 1.6 2 2.28 2.5

It is known that x and y are related by the equation 1a b

y x

.

(i)Explain how a straight-line graph of 1

y against

1

x

can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b.

(ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

.

(i)Explain how a straight-line graph of 1

y against

1

x

can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b.

(i) 1

1 1( ) ( ) 1

1 1 1( )

a b

y x

a by x

b

y a x a

In order to plot 1/y against 1/x, we need to arrange the equation into (1). b/a represents the gradient and 1/a represents the vertical intercept.

(1)

x 1 2 3 4 5

y 1 1.6 2 2.28 2.5

1/x 1 0.5 0.33 0.25 0.2

1/y 1 0.625 0.5 0.44 0.4

Choose appropriate scales1

y

1

x0.2 0.4 0.6 0.8 1.0

0.2

0.40.6

0.8

1.0

1

1 1( ) ( ) 1

1 1 1( )

a b

y x

a by x

b

y a x a

1, intercept 0.25

10.25

4

Fr graphy

aa

,

0.7 0.250.75

0.6 0

0.75

3

Fr graph

m

b

ab

1

y

1

x0.2 0.4 0.6 0.8 1.0

0.2

0.40.6

0.8

1.0

(0.6, 0.7)

(0,0.25)

1 1 1b

y a x a

(ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

1

plot vs with gradient and vertical intercept

a b

y x

y ya b y y b a

x xy

y b ax

Q1 The data for x and y given in the table below are related by a law of the form y px x q 2

, where p and q are constants.

x 1 2 3 4 5

y 41.5 38.0 31.5 22.0 9.5

By drawing a suitable straight line, find estimates for p and q.

qxpxy 2

qpxxy 2

Plot (y ─ x) against x2, p represents the gradient and q represents the (y-x) -intercept.

qpxxy 2 x 1 2 3 4 5

y 41.5 38.0 31.5 22.0 9.5

x2 1 4 9 16 25

y ─ x 40.5 36.0 28.5 18.0 4.5

42.5q

42.5 20

0 151.5

p

5 10 15 20 25

5

1015

20

25

2x

xy

303540

45 (0, 42.5)

)20,15(

qpxxy 2

Q2 The table shows the experimental values of two variables x and y which are known to be related by an equation of the form p(x + y – q) = qx 3, where p and q are constants.

x 0.5 1.0 1.5 2.0 2.5

y 1.06 1.00 1.69 3.50 6.81

Draw a suitable straight-line graph to represent the above data. Use your graph to estimate (i) the value of p and of q, (ii)the value of y when x = 2.2.

qxp

qyx

p

qxqyx

3

3

Plot (x + y) against x3, (q/p) represents the gradient and q represents the (x + y) - intercept.

qxp

qyx

3

x 0.5 1.0 1.5 2.0 2.5

y 1.06 1.00 1.69 3.50 6.81

x3 0.125 1 3.375 8 15.625

x+y 1.56 2 3.19 5.5 9.31

5 10 15 20

2

46

8

10

yx

3x

)10,20(

)6,10(

8.1q

5.44.0

8.1

4.08.1

4.01020

610

p

p

p

q

42.22.62.6,graphFr

648.102.2 3

yyx

xx

qxp

qyx

3

Q3 The table below shows experimental values of two variables, x and y. One value of y has been recorded incorrectly.

x 1 2 3 4 5

y 5.71 6.38 9.10 14.20 20.49

It is believed that x and y are related in the form y = x 2 – ax + b, where a and b are constants. Draw a suitable straight-line graph to represent the given data. Use your graph to estimate (i) the value of a and of b, (ii) a value of y to replace the incorrect value.

baxxy 2

Plot (y ─ x2) against x, ─ a represents the gradient and b represents the (y ─ x2) -intercept.

baxxy 2 x 1 2 3 4 5

y 5.71 6.38 9.10 14.20 20.49

x 1 2 3 4 5

y ─ x2 4.71 2.38 0.10 -1.80 -4.51

1 2 3 4 5

1

23

4

5

2xy

x

-5

-4-3

-2

-1

5.7b

5.2

5.230

05.7

a

a

)5.7,0(

)0,3(

8.1342.24

readingcorrect 2.2

readingincorrect 8.1

2

2

2

yx

xy

xy

baxxy 2

Identify the incorrect readings/ outliers!!1

y

1

xy x

2x

x 1 2 3 4 5

y 2.65 3.00 3.32 3.71 3.87

x+2 3 4 5 6 7

y2 7.02 9.00 11.02 13.76 14.98

2 ( 2)y m x c

2y

2x 1 2 3 4 5

2

4

6

8

10

6 7

12

14

16

? One of the values of y is subject to an abnormally large error

Identify the abnormal reading and estimate its correct value.

abnormal reading: y = 3.71Correct value should be

2 12.8

3.58

y

y

2 ( 2)y m x c

2y

2x 1 2 3 4 5

2

4

6

8

10

6 7

12

14

16

Estimate the value of x when y = 2

22 4y y

2When 4, 2 1.5y x

0.5x

2 1.99( 2) 1y x 22 1.99( 2) 1

3( 2)

1.990.492

x

x

x

2 ( 2)y m x c

2 ( 2)y m x c

Q4 The table below shows the experimental values of two variables x and y. It is known that one value of y has been recorded incorrectly x 0.5 1 1.5 2.0 2.5

y 1.20 1.00 0.86 0.70 0.66

It is known that x and y are related by an equation of the form a

yx b

, where a and b are constants. By plotting

1

y

against x, obtain a straight-line graph to represent the above data. Use your graph to estimate the value of a and of b.

(i) Use your graph to estimate a value of y to replace the incorrect value.(ii) Find the value of x when y = 10

9.

(iii) By inserting another straight line to your graph, find the value of x and of y which satisfy the simultaneous equations

ay

x b

and

10

15 12y

x

x 0.5 1 1.5 2.0 2.5

y 1.20 1.00 0.86 0.70 0.66

x 0.5 1 1.5 2.0 2.5

1/y 0.83 1 1.16 1.43 1.52

ay

x b

1

1 1

x b

y a

bx

y a a

1

y

x0.5 1.0 1.5 2.0 2.5

0.2

0.40.6

0.8

1.01.2

1.4

1.6

13.2

68.0

ba

b

)68.0,0(

)4.1,25.2(

13.3025.2

68.04.11

aa

1 1 bx

y a a

1

y

x0.5 1.0 1.5 2.0 2.5

0.2

0.40.6

0.8

1.01.2

1.4

1.6

)68.0,0(

)4.1,25.2(

abnormal reading: y = 0.70Correct value should be

x 0.5 1 1.5 2.0 2.5

y 1.20 1.00 0.86 0.70 0.66

x 0.5 1 1.5 2.0 2.5

1/y 0.83 1 1.16 1.43 1.52

11.35

0.741

y

y

1 1 bx

y a a

1

y

x0.5 1.0 1.5 2.0 2.5

0.2

0.40.6

0.8

1.01.2

1.4

1.6

Estimate the value of x when y = 10

910 1

0.99

yy

1When 0.9, 0.75x

y

10.319 0.68x

y

0.9 0.319 0.68

0.690

x

x

1 1 bx

y a a

ay

x b

and

10

15 12y

x

1 15 12

10

11.5 1.2

x

y

xy

Need to draw this and find the point of intersection of the 2 lines

Bear in mind: need to use the same axes as first line!

Vertical intercept (0, -1.2)

Horizontal intercept (0.8, 0)

1 11.5(0) 1.2 1.2

11.5 1.2

y y

xy

1.20 1.5 1.2 0.8

1.5x x

1

y

x0.5 1.0 1.5 2.0 2.5

0.2

0.40.6

0.8

1.01.2

1.4

1.6

Vertical intercept (0, -1.2)

Horizontal intercept (0.8, 0)

ay

x b

-0.2-0.4

-0.6

-0.8

-1.0

-1.2

10

15 12y

x

At point of intersection,

11.3 0.769

1.8

yy

x

Q5 The variables x and y are known to be connected by the equation

xCay

An experiment gave pairs of values of x and y as shown in the table.One of the values of y is subject to an abnormally large error.

x 1 2 3 4 5 6 7

y 56.20 29.90 25.10 8.91 6.31 3.35 1.78

Plot lg y against x and use the graph to

(i) identify the abnormal reading and estimate its correct value.(ii) estimate the value of C and of a.(iii) estimate the value of x when y = 1.

xCay

axCy

Cay x

lglglg

lglg

x 1 2 3 4 5 6 7

y 56.20 29.90 25.10 8.91 6.31 3.35 1.78

lg y 1.75 1.48 1.40 0.95 0.80 0.53 0.25

lg y

x1 2 3 4 5

0.2

0.40.6

0.8

1.0

1.2

1.4

1.6

6 7

1.8

2.0

(i) abnormal reading: y = 25.10 Correct value should be

lg 1.28

19.05

y

y

(ii) lg 2.0

100

C

C

2.0 0.4(ii) lg

0 6.51.76

a

a

)0.2,0(

)4.0,5.6(

lg lg lgy C x a

(iii) 1

lg 0

8.3

y

y

x

lg y

x1 2 3 4 5

0.2

0.40.6

0.8

1.0

1.2

1.4

1.6

6 7

1.8

2.0

8 9

estimate the value of x when y = 1.

lg lg lgy C x a