MyAdditional Mathemat

icsModuleForm 5

Topic 13:

Linear LawSapematter/Sapsapsui

by

NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)

IMPORTANT NOTES:

1. Line of Best Fit

* A straight line drawn that passes through as many points as possible.

* The number of points that do not lie on the straight line drawn should be more or less the same on

both sides of the straight line.

y

+ + + + + +

0 x

2. Non-linear Function

* A function that has one or more variables, x or y, which are not in the first degree.

* A non-linear function that consists of variables x and y (not in a straight-line graph) can be

reduced or converted to the linear form, Y = mX + c, where X and Y represent the functions of

x or y or both (with a straight-line graph).

3. Steps to Find Values of Constants in a Non-linear Function

* Reduce or convert the non-linear function with variables x and y to the linear form, Y = mX +c,

where X and Y represent the functions of x or y or both.

* Prepare a table for the values of X and Y.

* Choose a suitable scale to draw the graph as large as possible and label both axes.

* Plot the graph of Y against X and draw the line of best fit.

* Construct a right-angled triangle on the drawn line of best fit, to calculate the gradient of the

straight line.

y

+ + (x2, y2) + + + + (x1, y1)

0 x * Determine the Y-intercept, which is represented by c, from the straight-line graph.

4. To Determine Variables of x or y

* The values of certain variables, either x or y, can be determined;

(i) from the graph of the line of best fit, or

(ii) from the equation of the line of best fit that is formed.

Gradient, m =

Exercise 13.1: Line of Best Fit

1. (a) Draw the line of best fit for y against x on a graph paper from the data shown on the following table.(b) From the line of the best fit you have drawn;

(i) find the value of y when x = 18,(ii) find the value of x, when y = 40,(iii) form a straight-line equation.

Answer:

(i)

(ii)

(iii)

2. (a) Draw the line of best fit for y against x on a graph paper from the data shown on the following table.

(b) From the line of the best fit you have drawn;(i) find the value of y when x = 0.4,(ii) find the value of x, when y = 10,(iii) form a straight-line equation.

x -2 -1 0 1 2 3

y 1 4 6 8 11 13

Answer:

(i)

(ii)

(iii)

3. (a) Draw the line of best fit for y against x on a graph paper from the data shown on the following table.(b) From the line of the best fit you have drawn;

(i) find the value of y when x = 0.3,(ii) find the value of x, when y = 40,(iii) form a straight-line equation.

x 0.2 0.4 0.6 0.8 1.0 1.2

y 66 60 54 49 43 36

Answer:

(i)

(ii)

(iii)

x 5 10 15 20 25

y 16 28 36 50 62

Exercise 13.2: Applications of Linear Law to Non−Linear Functions

1. Express the following non-linear equation to the linear form Y = mX + c. Hence, state the Y, m, X and c.

No. Non-linear Equation Linear Form Y m X c

1. y2 = x

3 + 4

2. y = 2x2 – 5x

3. y = 10

4. y = a +

5. y =

6. ax2 + by 2 = x

7. y = ab x

8. ay = bx + x2

9. y = ax n

10. y = ax +

2. The following straight-line graph drawn to represent the equation y = ax2 + bx, where a and b are constant. Find the value of a and b.

(1, 4)

0 x (5, 0)

3. The following straight-line graph drawn to represent the equation y = ax + , where a and b are

constant. Find the value of a and b. xy

5

(4, 3)

0 x2

4. The following straight-line graph drawn to represent the equation y = + , where a and b are

constant. Find the value of a and b.

xy

(4, 7)

(2, 3)

0 1/x

5. The following straight-line graph drawn to represent the equation y = abx, where a and b are constant.

Find the value of a and b.

log y

(9, 7)

(1, 3)

0 x

Exercise 13.3: Problem Solving I

1. The following table shows the experimental values of two variables, x and y. It is known that x and y

are related by an equation ax + by = x2, where a and b are constants.

(a) Draw the graph of against x.

(b) From the graph, find(i) the values of a,(ii) the value of b,(iii) the value of y when x = 3.5.

x 1 2 3 4 5 6

y −0.50 −0.33 0.50 1.99 4.17 7.01

2. The following table shows the experimental values of two variables, x and y. It is known that x and y

are related by an equation y = px + , where p and q are constants.

(a) Draw the graph of xy against x2.(b) From the graph, find

(i) the values of p,(ii) the value of q,(iii) the value of y when x = 5.7.

x 1 2 3 4 5 6

y 7.2 8.4 10.9 13.8 16.8 19.9

3. The following table shows the experimental values of two variables, x and y. It is known that x and y

are related by an equation y = , where a and b are constants.

(a) Draw the graph of against x.

(b) From the graph, find(i) the values of a,(ii) the value of b,(iii) the value of x when y = 1.8

x 2 4 6 8 10 12

y 3.20 2.44 1.96 1.64 1.41 1.23

4. The following table shows the experimental values of two variables, x and y. It is known that x and y are related by an equation y = axb, where a and b are constants.(a) Convert the equation into linear form, hence draw the linear graph.(b) From the graph, find

(i) the values of a, (ii) the value of b,

x 2 3 4 5 6

y 11.3 20.8 32.0 44.7 58.8

1. The following straight-line graph is obtained by plotting log3 y against x.

log3 y (a) Express log3 y in term of x.

(3, 10) (b) Express y in term of x. 4 (c) Find the value of y when x = -1

0 x

2. The following straight-line graph is obtained by plotting against .

(a) Express in term of x.

(b) Find the value of y when x = 3. 6

0 4

Exercise 13.4: Problem Solving II

Exercise 13.5: Past Years SPM Papers

1. The 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) The following diagram shows the straight line obtained by plotting log10 y against log10 x. Find the value of;

log10 y (i) log10 k,

(2, h) (ii) h. (4 marks) SPM 2005/Paper 1)

(0, 3) 0 log10 x

Answer: (a) …………………..……………..

(b) (i) .……………………………..

(ii) ……………………………..

2. The following diagram shows a straight line graph of against x. Given that y = 6x – x2, calculate

the value of k and of h. (3 marks)

(SPM 2004/Paper

1)

(2, k) (h, 3)

0 x 1

Answer: k = …………………..…………...

h = ..……………………………..

3. The variables x and y are related by the equation y = px2 + qx, 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)

(SPM 2003/Paper 1)

(2, 9)

(6, 1) Answer: p = …………………..…………… 0 x

q = ..……………………………..

4. Diagram 4(a) shows the curve y = −3x2 + 5. Diagram 4(b) shows the straight line graph obtained when y = −3x2 + 5 is expressed in the linear

form Y = 5X + c. Express X and Y in terms of x and /or y. (3 marks)

(SPM 2006/Paper 1)

y Y

y = -3x2 + 5 X x 0

0 -3 DIAGRAM 4(a) DIAGRAM 4(b)

Answer: X = …………………….…………

Y = ..……………………………..

5. The variables x and y are related by the equation , where m is a constant. The following

diagram shows the straight line graph obtained by plotting log10 y against x. (3 marks)

log10 y SPM2008/Paper1

(a) Express the equation in its linear

form used to obtain the straight line graph. x 0

(b) Find the value of m. (0, -4)

Answer: (a) ...................................................

(b) ...................................................

6. The variables x and y are related by equation y2= 4x(10 – 2x). A straight line graph is obtained by

plotting against x, as shown in the diagram below. Find the values of p and q. (3

marks)

(SPM2007/Paper

1)

(3, q)

0 x (p, 0) Answer: (a) …………………..

……………..

(b) ...……………………………...

7. Use the graph paper provided to answer this question. The following table shows the values of two variables, x and y, obtained from an experiment. The

variables x and y are related by the equation y = px + , where p and r are constants.

x 1.0 2.0 3.0 4.0 5.0 5.5

y 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) (SPM 2005/Paper

2)

8. Use the graph paper provided to answer this question.

The following table shows the values of two variables, x and y, obtained from an experiment. It is known that x and y are related by the equation

2

y = pk , where p and k are constants.

x 1.5 2.0 2.5 3.0 3.5 4.0

y 1.59 1.86 2.40 3.17 4.36 6.76

(a) Plot log10 y against x,2 . Hence draw the line of best fit (5 marks) (b) Use the graph in (a) to find the value of

(i) p, (ii) k, (5 marks)

(SPM 2003/Paper 2)

9. Use the graph paper provided to answer this question.The following table shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation y = pkx, where p and k are constants.

x 2 4 6 8 10 12

y 3.16 5.50 9.12 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 log10 y-axis. Hence, draw the line of best fit. (4 marks)

(b) Use the graph from (a) to find the value of(i) p,(ii) k, (6

marks) (SPM 2004/Paper

2)

10. Use the graph paper provided to answer this question. Table 2 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 6

y 4.0 5.7 8.7 13.2 20.0 28.8

TABLE 2

(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 your graph from 7(a) to find the value of(i) p, (ii) k, (5

marks) (SPM 2006/Paper

2)

11. Use the graph paper to answer this question.Table 8 shows the values of two variables, x and y, obtained from an experiment. The variables x

and y are related by the equation , where k and p are constants. SPM2009/Paper 2

x 1.5 2.0 3.0 4.0 5.0 6.0

y 2.502 0.770 0.465 0.385 0.351 0.328

Table 8

(a) Based on Table 8, construct a table for the values of (2

marks)

(b) Plot using a scale of 2 cm to 0.1 unit on the -axis and 2 cm to 0.5 unit on the

Hence, draw the line of best fit. (3

marks)(c) Use the graph in 11(b) to find the value of

(i) k,(ii) p. (5 marks)

12. Use graph paper to answer this question.

The table below shows the values of two variables s and y, obtained from an experiment. Variables x and y are related by the equation y = hk2x , where h and k are constants. SPM2008/Paper 2

(a) Based on the table, construct a table for the values of log10 y. [1 mark]

(b) Plot log10 y against x, using a scale of 2 cm 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]

(d) Use the graph in (b) to find the value of(i) x when y = 4.8,(ii) h,(iii) k. [5

marks]

x 1.5 3.0 4.5 6.0 7.5 9.0

y 2.51 3.24 4.37 5.75 7.76 10.00