Revise of Important Concept – Straight Line


(A) Equation of a straight Line 

 An equation of straight line is given by y = mx + c.
The variables x and y are linearly related
The term c is known as y-intercept. It represents the y value where the line cuts the y-axis
The term m is the gradient of the straight line and its value is constant



(B) Gradient of a Straight Line
If the points A ( x 1 , y 1 )  and B ( x 2 , y 2 )   lie on the straight line y = m x + c , then gradient of ,the straight line
m = y 2 y 1 x 2 x 1 o r y 1 y 2 x 1 x 2



(C) Mid Point 



Mid Point AB is given by M = ( x 1 + x 2 2 , y 1 + y 2 2 )


(D) Distance Between Two Points
Distance between point A ( x 1 , y 1 ) and point is B ( x 2 , y 2 ) given by ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2

Equations Of Line Of Best Fit


A set of two variables are related non linearly can be converted to a linear equation.  The line of best fit can be written in the form

Y = mX + c


where
X and Y are in terms of x and/or y
m is the gradient,
c is the Y-intercept

Recall: To find equation of straight line
(1)  Equation of a straight line if gradient (m) and one points ( x 1 , y 1 ) are given:
Y y 1 = m ( X x 1 )
(2)  Equation of a straight line if gradient (m) and y-intercept (c) are given:
Y = m X + c

Reduce Non-Linear Function To Linear Function – Examples (A) To (F)

Examples:
Reduce each of the following equations to the linear form. Hence, state the gradient and the Y-intercept of the linear equations in terms of  a and b.
(a)  y = a x 3 + b x 2
(b)  y = a x + b x
(c)  y = a x b x 2
(d)  x y = p x + q x
(e)  y = a x + b x
(f)  a y = b x + 1

[Note :
X and Y cannot have constant, but can have the variables (for example x and y)
 m and c can only have the constant(for example a and b), cannot have the variables x and y]

Solution:







The Line Of Best Fit

2.2 The Line of Best Fit

Line of best fit 
has 2 characteristics:
(i) it passes through as many points as possible,
(ii) the number of points which are not on the line of best fit are equally distributed on the both sides of the line.

Example
Check whether the following graph has the characteristic of the line of best fit

(a)


Yes! This is the Line of Best Fit
.
Notice that you have 3 points passes through the straight line,1 points above the line and 1 point below the line. The number of points which are not on the line of best fit are equally distributed on the both sides of the line.

(b)


No. This is NOT the Line of Best Fit
.
Notice you have more points above the line than below the line. The number of points which are not on the line of best fit are NOT equally distributed on the both sides of the

SPM Practice 3 (Linear Law) – Question 2


Question 2 (10 marks):
Use a graph to answer this question.
Table shows the values of two variables, x and y, obtained from an experiment. A straight line will be obtained when a graph of y 2 x  against  1 x is plotted.


(a) Based on Table, construct a table for the values of 1 x  and  y 2 x .  
( b ) Plot  y 2 x  against  1 x , using a scale of 2 cm to 0.1 unit on the  1 x -axis   and 2cm to 2 units on the  y 2 x -axis.   Hence, draw the line of best fit.

(c) Using the graph in 11(b)
(i) find the value of y when x = 2.7,
(ii) express y in terms of x.

Solution:
(a)


(b)



(c)(i)
When x=2.7,  1 x =0.37 From graph, y 2 x =5.2 y 2 2.7 =5.2 y=3.75


(c)(ii)

Form graph, y-intercept, c = –4 gradient, m= 16( 4 ) 0.80 =25 Y=mX+c y 2 x =25( 1 x )4 y= 254x


SPM Practice Question 10 – 12

Question 10:
The sequence –11, –5, 1,… is an arithmetic progression. State the three consecutive terms of this arithmetic progression where the sum of these three terms is 93.

Solution:



Question 11:
An arithmetic series, with the first term 12 and common difference d, consists of 23 terms. Given that the sum of the last 3 terms is 5 times the sum of the first 3 terms, find
(a) the value of d,
(b) the sum of the first 19 terms.

Solution:




Question 12:
The sum of n terms of an arithmetic progression is given by the formula S n = n 2 (53n) . Find
(a) the first term,
(b) the common difference,
(c) the tenth term.

Solution:


SPM Practice Question 7 – 9

Question 7:
Find the sum of all the multiples of 7 between 100 and 500.

Solution:




Question 8:
If log 10 p,  log 10 pq and  log 10 p q 2 are the first three terms of a progression, show that it forms an arithmetic progression.

Solution:




Question 9:


Show that the volumes of the cylinders in the above diagram form an arithmetic progression and state its common difference.

Solution:


SPM Practice Question 1 – 3


Question 1:
The third and eighth terms of an arithmetic progression are –5 and 15 respectively. Find
(a) The first term and the common difference
(b) The sum of the first 10 terms.

Solution:
(a)
T 3 =5    Use  T n =a+( n1 )d a+2d=5( 1 ) T 8 =15 a+7d=15( 2 ) ( 2 )( 1 ),  5d=20 d=4 Substitute d=4 into ( 1 ), a+2( 4 )=5    a=13

(b)
S 10 = 10 2 [ 2( 13 )+9( 4 ) ]Use  S n = n 2 [ 2a+( n1 )d ] =50


Question 2:
The first three terms of an arithmetic progression are 2k, 3k + 3, 5k + 1. Find
(a) the value of k,
(b) the sum of the first 15 terms of the progression.

Solution:




Question 3:
Given an arithmetic progression p + 9, 2p + 10, 7p – 1,… where p is a constant. Find
(a) value of p,
(b) the sum of the next five terms.

Solution: