Understanding Engineering Mathematics

y

− 4 − 3 − 2 − 1 0 1 2 3 4 x

6

3

4

5

2

1

− 1

− 2

− 3

− 4

− 5

(−2,5) (4,5)

y = x^2 − 2 x − 3

(−1,0) (3,0)

(1,−4)

(0,−3) (2,−3)

Figure 3.2The graph ofy=x^2 − 2 x−3.

to givex=3andx=−1 as the crossing points (‘gateways’ through the

x-axis), as we observe from the graph. Also, we can complete the square

to get:

x^2 − 2 x− 3 =(x− 1 )^2 − 4

which tells us that the minimum value is −4 and occurs atx=1.

Again, we see this from the graph. (Note that any quadratic curve – or

parabola– is symmetric about its minimum or maximum point.)

3.2.3 Formulae

➤

88 108➤

Perhaps the most common way you have been exposed to functions before is in the use
offormulaeto express the value of some variable in terms of other given quantities. For
example for an ideal gas pressureP can be expressed in terms of the volumeV and