Chapter 18

Graphs reducing non-linear

laws to linear form

18.1 Introduction

In Chapter 17 we discovered that the equation of a
straight line graph is of the formy=mx+c,where
mis the gradient andcis they-axis intercept. This
chapterexplainshowthelawofagraphcanstillbedeter-
mined even when it is not of thelinear formy=mx+c.
The method used is calleddetermination of lawand is
explained in the following sections.

18.2 Determination of law

Frequently, the relationship between two variables, say
xandy, is nota linear one;i.e., whenxis plottedagainst
ya curve results. In such cases the non-linear equation
may be modified to the linear form,y=mx+c,sothat
theconstants,and thus thelaw relating thevariables,can
be determined. This technique is called ‘determination
of law’.
Some examples of the reduction of equations to linear
form include

(i) y=ax^2 +bcompares withY=mX+c,where

m=a,c=bandX=x^2.

Hence,yis plotted vertically againstx^2 horizon-

tally to produce a straight line graph of gradient

aandy-axis interceptb.

(ii) y=

a

x

+b,i.e.y=a

(

1

x

)

+b

yis plotted vertically against

1

x

horizontally to

produce a straight line graph of gradientaand

y-axis interceptb.

(iii) y=ax^2 +bx

Dividing both sides byxgives

y

x

=ax+b.

Comparing withY=mX+cshows that

y

x

is

plotted vertically againstxhorizontally to pro-

duce a straight line graph of gradientaand

y

x

axis interceptb.

Here are some worked problems to demonstrate deter-

mination of law.

Problem 1. Experimental values ofxandy,

shown below, are believed to be related by the law

y=ax^2 +b. By plotting a suitable graph, verify

this law and determine approximate values of

aandb

x 1 2 3 4 5

y 9. 8 15. 2 24. 2 36. 5 53. 0

Ifyis plotted againstxa curve results and it is not

possible to determine the values of constantsaandb

from the curve.

Comparingy=ax^2 +bwithY=mX+cshows that

yis to be plotted vertically againstx^2 horizontally. A

table of values is drawn up as shown below.

x 1 2 3 4 5

x^21491625

y 9. 8 15. 2 24. 2 36. 5 53. 0

DOI: 10.1016/B978-1-85617-697-2.00018-1