Chapter 18

Functions and their curves

18.1 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 pictorially 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 equa-

tion 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 beafunc-

tionofxand the relationshipis expressed asy=f(x);x

is called the independent variable andyis 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. 18.1

(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. 18.2.

(iii) Cubic Equations

The general equation of a cubic graph is

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

5

4

3

2

1

0 1 2 3

y 52 x 11

y

x

5

(a)

(b)

4

3

2

1

0 1 2 3

y 5522 x

y

x

Figure 18.1

8

6

4

2

(^2221102) x
y
y 5 x^2
Figure 18.2