Higher Engineering Mathematics

FUNCTIONS AND THEIR CURVES 209

C

4

6

y

2

− 4 − 2 2 4 x

y =

x

x^2 + 1

y = x

x^2 + 1

y = x − 4

− 6

− 2

0

Figure 19.35

19.8 Brief guide to curve sketching

The following steps will give information from
which the graphs of many types of functionsy=f(x)
can be sketched.

(i) Use calculus to determine the location and

nature of maximum and minimum points (see

Chapter 28)

(ii) Determine where the curve cuts thex- andy-

axes

(iii) Inspect the equation for symmetry.

(a) If the equation is unchanged when−xis

substituted forx, the graph will be sym-

metrical about they-axis (i.e. it is aneven

function).

(b) If the equation is unchanged when−yis

substituted fory, the graph will be symmet-

rical about thex-axis.

(c) Iff(−x)=−f(x), the graph is symmet-

rical about the origin (i.e. it is an odd

function).

(iv) Check for any asymptotes.