AM_F5_C2(T)

WAJA 2009

ADDITIONAL MATHEMATICS

FORM FIVE

( Teacher’s Copy )

Name: ___________________________

Class : ___________________________

WAJA 2009 – ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law

Learning Objective:2.Understand and use the concept of lines of best fit

Learning Outcome:2.1.1 Draw lines of best fit by inspection of given data

1) Which one between the pair is the best fit line? Tick the appropriate cup. Then write down the reason for your choice in the rectangular box.

Example:Draw a line of best fit by inspecting the given data on the graph

( i )

2

×

×

×

××

×

×

×

×

×

××

×

×

√

xx

xx

xxx

y

x

Criteria the line of best fit:1.points lie as close as possible to the line2.line pass through as many points as possible 3.number of points above and below should b e the same distance of the line

x

√

Line pass through as many points as possible and number of points above and below should be the same distance of the line


( ii )

( iii )

3

×

×

××

×

×

×

×

×

××

×

×

×

×

×

×

××

×

×

×

×

×

××

×

×

√

√

Number of points lie as close as possible to the line and line pass through as many points as possible of the line

Line pass through as many points as possible and number of points above and below should be the same distance of the line


2) Draw a line of best fit by inspecting the given data on the graph for each of the following graph

(a) (b) x x x x x x x x

0 0

c)

Learning Outcome:

2.1.2 Write equation for lines of best fit

4

0x

y2

xx

xx

0x

y

xx

xx

xx

xx

y

x

y

x

x

x

x

(d)

y=mx+c is the linear equation of a straight line

21

21

12

12

xx

yyor

xx

yy

c

x

m=gradientc=y-intercept

is a value of y where the graph cuts the y-axis


1. Match the correct answer

2. Write the equation of the line of best fit for each of the following graphs Example:

i) ii)

Find m:

Find m:

Find c: y-intercept, c = 2

Find c:

Substitute into The equation of the line is

Substitute into

The equation of the line :

5

2

0x

y

(4, 6)

x

x

x

0

(1,6)

(( 8,1)xx

x

x

t

P

a) find y-intercept given y=2x+8

b) find the gradient given y=-5x-8

c) find y-intercept from the graph below

2

d) find the gradient from the graph below

P(4,2)

Q(8,14)

e) find the value of y-intercept from the graph below

(5,0)

(0,-3)

-3

3

8

2

-5


a)b)

Find m:

=

Find m:

=

Find c: P=mV+c

5=

c=

Find c : F=mX+c

2=

c=

P= F=

3. Determine the horizontal and vertical axes of the following graph

Example:

Plot graph y2 against x

6

x

y2

horizontal axes

vertical axes

0

F

X0

(2,2)

(8,10)x

x

x

x0 V

(1,5))

x

(6,1)x

x

x

P


a) Plot graph y against x2 b) Plot graph P against V

4. Based on the table given, plot the points and draw a line of best fit. Hence write the equation for the line of best fit.

Example:

i) The values of variables G and H in an experiment are given in the table below

G 0.3 0.6 0.8 0.9 1.0H 0.35 0.50 0.59 0.65 0.7

a) Plot G against H and draw a line of best fit b) Write an equation for the line of best fit

7

c) Plot graph xy against d) Plot graph against

y

x2

P

V

xy

x

1

a

b

1


Solution:

Plot the graph:

Find m from the graph:

Find c from the graph: c = 0.21

Substitute into

a) The table below represents the experimental values of two variables L and W.

W 0.1 0.2 0.3 0.4 0.5L 10.8 11.3 12.1 12.4 12.7

i) Plot L against W and draw the line of best fitii) Write an equation for the line of best fit

8

[Ans : m = 5.3, c = 10.2, L = 5.3W +10.2]

×

×

×

××

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.00.90.80.70.60.50.40.30.20.1

G

H

(0,0.22)

(1.0,0.7))


5

10

15

0.1 0.2 0.3 0.4 0.5


= 5.3Find c from the graph: c = 10.2

Substitute into y=mx+c

L=5.3W+10.2

b) The table below represents the experimental values of two variables p and q.

p 1.0 2.8 5.6 8.4 11.5 14.2q 2.2 3.4 5.7 7.4 9.7 11.2

i) Plot q against p and draw the line of best fit

ii) Write an equation for the line of best fit

9

[Ans : m = 0.7, c = 1.4, q = 0.7p + 1.4]

L

W

××

× ×

×


5

5

10

15

0 10 15


= 0.7

Find c from the graph: c = 1.4


q=0.7p+1.4

10

q

p

×

×

×

××

×


Learning Outcomes:

2.1.3 Determine values of variables

2.1.3 ( a ) Determine values of variables from lines of best fit

1. Determine the values of the variables from the given lines of best fit.

Example:

Two variables, x and y, are related by the line of best fit as shown in the graph above. From the graph, determine the value of

( i ) y when x = 1.4

( ii ) x when y = 4.4

11

×

×

××

××

y

x

0 0

5

4

3

1

2

3 1 2


Solution

From the graph,

( i ) when x = 1.4 , y = 2.9

( ii ) when y = 4.4 , x = 2.2

12

00

3

1

2

31 2

×

××

××

y

x

y=2.2

y=1.4

x=2.6x=0.8 x=2.2

x=1.2

y=2.99

y=4.4

×

×

××

××

y

x

0 0

5

4

3

1

2

3 1 2

x=1.4 x=2.2


a) Two variables , x and y, are related by the line of best fit as shown in the graph above. From the graph , determine the value of( i ) y when x = 0.8 ( ii ) y when x = 2.2( iii ) x when y = 1.2

b) The table below shows the corresponding values of the variables T and V obtained from an experiment.

T 20 40 60 80 100 120V 122 130 136 148 154 163

i) Plot a graph of V against T and draw a line of best fit. ii) Use the graph obtained in ( a ) to determine the value of

a) V when T = 72 b) T when V = 150

100

110

120

130

140

150

160

170

0 20 40 60 80 100 120

13

[ Answer: a) i) 2.2 ii) 1.4 iii) 2.6 b) ii) a) 143 b) 88 ]

V

T

×

×

×

×

×

×

V=143333

T=72

V=150

T=88


2.1.3 ( b ) Determine values of variables from the equations of lines of best fit

Example :

y P(0.2,16.5)

Q(0.8,9.0) x 0

The diagram above shows a line of best fit obtained by plotting a graph of y against x. The line passes through points P(0.2,16.5) and Q(0.8,9.0)

i) Find the equation of the line of best fitii) Determine the value of

( a) y when x = 0.7( b ) x when y = 22

Solution i)

Find m:

Find c: At point P(0.2,16.5),

Substitute into

ii)

i) When , ii) When ,

14

(5,3)

(2,0.5)q

p

X

x


a) The diagram above shows part of a line of best fit obtained by plotting a graph of p against q.

The line passes through points (2,0.5) and (5,3).i) Find the equation of the line of best fit ii) Determine the value of

( a ) q when p=0 ( b ) p when q = -1

Find m:

=0.8

Find c: p =0.8q+c

At point (2,0.5), 0.5=0.8(2)+c

c=-1.1


p=0.8q-1.1a) when p=0 , 0=0.8q-1.1

q=

=1.38b) when q=-1, p=0.8(-1)-1.1 =-1.9

15(3,0.5)

(2,2)

t

v

a) i)

a) ii)


b) Two variables, v and t , are known to be linearly related as shown by the line of best fit in the

graph above. The line passes through points (2,2) and (3,0.5).

i) Determine the linear equation relating v and t

ii) Find the value of v when( a) t = 0 ( b) t = 5

b i) Find m:

=-1.5

Find c: v =-1.5t+cAt point (2,2), 2=-1.5(2)+c c=5


v=-1.5t+5

a) when t = 0 , v =-1.5(0)+5 = 5 b) when t = 5, v = -1.5(5)+5 = -2.5

Learning Objective:2.2 Apply linear law to non-linear relations

16

[ Answer : a) i) ii)(a)1.38 (b) -1.9 b) i) ii) (a)5 (b) -2.5 ]

b) ii)


Learning Outcomes:2.2.1 Reduce non-linear relations to linear form

Example:

x 0 1 2 3 4 y -5 -4 -1 4 11

0 252015105

15

10

5

-5

20

0 252015105

15

10

5

-5

20

What do you observe?

Quadratic graphs can be drawn as linear graphs if we change the representation of x- axis, that is x x2 .

b)

x 0.5 1 2 3 4

17

x2 0 1 4 9 16y -5 -4 -1 4 11

2 1 0.5 0.3 0.25

y 5 3 2 1.7 1.5

y

x

×

×

×

×

×

×


y 5 3 2 1.7 1.5

5

4

3

2

1

1 2 3 4

5

4

3

2

1

1 2 3 4

Again, what do you observe?

Reciprocal graphs can be drawn as linear graphs if we change the representation of x-axis, that is x .

Conclusion: A non-linear equation can be reduced to a linear form Y = mX + c.

Linear Form of Equation

18 or , c=0

x

y

0

y

0

×

×

××

×

×

××

×

×

0.5 1 1.5 21x


2. Identify based on the following linear form.Example:

i)

ii)

Fill in the blank with the correct answer :

Linear equation Y m X cy 5 x2 -8

22 xx

y 1 x 2 2

xy

7

7 0

x

y -5 2

xy 8 x 3

y

1

log y log b x log a

3. Reduce the following non linear equation to linear equation in the form of . Hence identify

Example:

19

Variables for the x-axis

Variables for the Y-axiscoefficient=1

m=gradient

Constant (no variable)y-intercept

Y=y m=2 X=x2 c=-3

m=3 X=x2 c=-5


i)

Create the y-intercept

Create coefficient of y=1 & Arrange in the form Y=mX+c

Compare to Y=mX+c

ii)

Create the y-intercept

Arrange in the form Y=mX+c

Compare to Y=mX+c

4. Fill in the blank with correct answer

Example:

Non Linear Equation Linear Form Equation

a) = - 5

20y 2x2 5


Y m X c

2 -5


b)

Y m X c


c) =

Y m X c

8


d) = 7 +

Y m X c

21

y x2

y

x

1 x2 -7

1

x

0

xy x33

8

x


3


e)

Y m X C

Non Linear Equation Linear form equation

f)

Y m X C


g) +

Y m X C

22

xy 7

y

x

-3 x 5

1

y

4

9

1

x

1

loga xlogb


Learning Outcomes:2.2.2 Determine values of constants on non-linear relations given:2.2.2a)Determine values of constants from lines of best fit

1. Find the value of the following situation.

Example:

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 x2 .Find the values of a and b.

xy (4,50) (1,35)

x2

0

From the graph identify the representation of y-axis and x-axis Reduce the equation given to linear form , Y = mX +c:

Compare Y=mX+c


23

logy logb x


Find c , substitute X=1 , Y=35 , m=5 into the equation

mX+c 35=5(1)+c c=30 Find the variables a and b

=5

a) Diagram shows a straight line graph of against

Given that where p and q are constants, find the values of p and q. [p=1, q=2]

From the question identify the representation of y-axis and x-axis

Reduce the equation given to linear form , Y = mX +c:

Compare Y=mX+c

24

0

(4,6)

(1,3)



= 1 Find c , substitute X=1 , Y=3 , m=5 into the equation

mX+c 3 = 1(1)+c c = 2 Find the variables p and q

b) Diagram shows a straight line graph of against .The variables x and y are related by

the equation where a and b are constants. Find the values a and b.

(3,7)

(1,3)

[ a = 1 , b = 2]

From the question identify the representation of y-axis and x-axis

Reduce the equation given to linear form , Y = mX +c:

Compare Y=mX+c

25



Find c , substitute X=1 , Y=3 , m=5 into the equation

mX+c 3 = 2(1)+c c=1

Find the variables a and b

2.2.2b)Determine values of constants from data

1. Find the values of the constants based on the table of values.

Example:

x 1 2 3 4 5y 1.00 2.83 3.81 5.00 5.90

The table above shows experimental values of two variables, x and y. The variables x and y are

known to be related by the equation where h and k are constants.

( i )Plot a graph of against x and draw a line of best fit.

( ii )From the graph obtained in ( b ) , find the values of h and k.

Solution:

From the graph identify the representation of y-axis and x-axisY= , X=x

Construct a new table

X=x 1 2 3 4 51.00 4.00 6.60 10.00 13.19

26


Plot the points and draw the graph

Reduce the equation given to linear form ,

:

Compare

Find m and y-intercept from the graph :

27

y-intercept=c=-2c=k=-2

-2

Graph of y x against x

x

y x

14

12

10

8

6

4

2

1 2 3 4 50

×

×

×

×

×

10 – 2 = 8

4 – 1.4 = 2.6


a) Table 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 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

( i )Plot xy against x2 , by using a scale of 2 cm to 5 units on both axes.Hence , draw the line of best fit.

( ii )Use the graph from ( a ) to find the value of

( a ) p,

( b ) r. [ p = 1.3778 , r = 5.5111 ]

From the graph identify the representation of y-axis and x-axis Y = , X = x2


X=x2 1 4 9 16 25 30.255.5 9.4 15 26 38.5 46.2


28


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35


:

Compare


29

35-4=31

y-intercept=c=4

c= =4

r = 5.5111

xy

x222.5 – 4=18.5


b) Table 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 2 3 4 5 6 7y 8 13.2 20 27.5 36.6 45.5

( i )Plot against x , using a scale of 2 cm to 1 unit on both axes.Hence , draw the line of best

fit.

( ii )Use the graph from ( a ) to find the value of

( a ) p,

( b ) k. ( c )y when x = 1.2 [ p = 0.754 , k = 0.26 ,y = 4.2 ]

From the graph identify the representation of y-axis and x-axis

Y = , X = x


X=x 2 3 4 5 6 7

4 4.4 5 5.5 6.1 6.5


30

y

x


0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7


:

Compare


31

y-intercept=c=

2.7=

p = 0.7763

x=1.2


From the graph :

When x = 1.2 ,

y = 4.08

32

Documents

AM_F5_C2(T)