Understanding Engineering Mathematics

half the curve. This may reduce considerably the amount of work we have to do in

sketching the curve.

G.Look for points where the curve crosses the axes, or‘gateways’. It crosses they-axis
at points wherex=0, i.e. at the value(s)y=f( 0 ). It crosses thex-axis at points
wherey=0, i.e. at solutions of the equationf(x)=0.

R.Consider anyrestrictionson either the domain or the range of the function – that is,
forbidden regions where the curve cannot exist. The simplest such case occurs when
we havediscontinuities. Thus, for example

y=

1

x+ 1

does not exist at the pointx=−1 – there is a break in the curve at this point. See

Figure 10.5.

x

y

1

− 1 0

y =x 1 + 1

Figure 10.5Asymptotes at a discontinuity.

Any rational function will have a number of such points equal to the number of real

roots of the denominator. Irrational functions such as

√

1 −x^2 exhibit whole sets of

values thatxcan’t take, because for the square root to exist 1−x^2 must be positive,

hence the curvey=

√

1 −x^2 does not exist for|x|>1. Thus, the curve is confined

to the region|x|≤1.

A.Look forasymptotes. These are lines to which the curve becomes infinitely close as

xorytend to infinity. Thus, the functiony=

1

x+ 1

has thex-axis as a horizontal

asymptote, sincey→0asx→±∞. It also has the linex=−1 as an asymptote to

which the curve tends asy→±∞. Of course the curve can never coincide with the

line because of the restriction (see above)x=−1.

Also, look for the behaviour of the curve for very small and very large values ofxor

y. This is often very instructive. For example, the curvey=x^3 +xbehaves like the

curvey=xfor small values ofx, allowing us to approximate the curve near the origin

by a straight line at 45°to thex-axis. For very large (positive or negative) values of

xit behaves like the cubic curvey=x^3.

P.Consider any specialpoints, apart from ‘gateways’ in the axes. They may literally

be simply specific points you plot to determine which side of an asymptote the curve

approaches from, for example.