Cambridge International Mathematics

394 Introduction to functions (Chapter 19)

The hyperbolic shape is also noticed in the home when a lamp is close to a wall.

The light and shadow form part of a hyperbola on the wall.

There are many situations in which two quantities varyinversely.

They form a relationship which can be described using a reciprocal

function.

For example, the pressure and volume of a gas at room temperature

vary inversely according to the equation P=

77 : 4

V

.

IfPis graphed againstV, the curve is one branch of a hyperbola.

Discovery 2 The family of curves y=

k

x

, k 6 =0

#endboxedheading

In this discovery you should use agraphing packageorgraphics calculatorto draw

curves of the form y=

k

x

where k 6 =0.

What to do:

1 On the same set of axes, draw the graphs of y=

1

x

, y=

4

x

and y=

8

x

:

2 Describe the effect of the value ofkon the graph for k> 0.

3 Repeat 1 for y=

¡ 1

x

, y=

¡ 4

x

and y=

¡ 8

x

:

4 Comment on the change in shape of the graph in 3.

5 Explain why there is no point on the graph whenx=0.

6 Explain why there is no point on the graph wheny=0.

You should have noticed that functions of the form y=kx are

undefined when x=0.

On the graph we see that the function is defined for values ofx

getting closer and closer to x=0, but the function never reaches

the line x=0. We say that x=0is avertical asymptote.

Likewise, as the values ofxget larger, the values ofyget closer

to 0 , but never quite reach 0. We say that y=0is ahorizontal

asymptote.

The graph of y=

2

x¡ 1

alongside is undefined whenx¡1=0,

which is when x=1. It has the vertical asymptote x=1.

As the values ofxget larger, the values ofyapproach, but never

quite reach, 0.

The graph has the horizontal asymptote y=0.

P¡()kPa

5 10 15 20 25 V¡()m^3

25

20

15

10

For1kgofO

at 25°C

2

O

GRAPHING

PACKAGE

y

O x

x

y=k

y

x

O

1

2

• =
  x
  y

x¡=¡1

y¡=¡0

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y:\HAESE\IGCSE01\IG01_19\394IGCSE01_19.CDR Wednesday, 8 October 2008 10:22:07 AM PETER