Higher Engineering Mathematics

Graphs

C

19

Functions and their curves

19.1 Standard curves

When a mathematical equation is known, co-

ordinates may be calculated for a limited range of

values, and the equation may be represented pictori-

ally as a graph, within this range of calculated values.

Sometimes it is useful to show all the characteristic

features of an equation, and in this case a sketch

depicting the equation can be drawn, in which all

the important features are shown, but the accurate

plotting of points is less important. This technique is

called ‘curve sketching’ and can involve the use of

differential calculus, with, for example, calculations

involving turning points.

If, say,ydepends on, say,x, thenyis said to be

a function ofxand the relationship is expressed as

y=f(x);xis called the independent variable andy

is the dependent variable.

In engineering and science, corresponding values

are obtained as a result of tests or experiments.

Here is a brief resumé of standard curves, some

of which have been met earlier in this text.

(i) Straight Line

The general equation of a straight line isy=mx+c,

wheremis the gradient

(

i.e.

dy

dx

)

andcis they-axis

intercept.

Two examples are shown in Fig. 19.1

(ii) Quadratic Graphs

The general equation of a quadratic graph is

y=ax^2 +bx+c, and its shape is that of a parabola.

The simplest example of a quadratic graph,y=x^2 ,

is shown in Fig. 19.2.

(iii) Cubic Equations

The general equation of a cubic graph is

y=ax^3 +bx^2 +cx+d.

The simplest example of a cubic graph,y=x^3 ,is

shown in Fig. 19.3.

(iv) Trigonometric Functions (see Chapter 15,

page 148)

Graphs ofy=sinθ,y=cosθandy=tanθare shown

in Fig. 19.4.

Figure 19.1

Figure 19.2