Everything Maths Grade 10

 4  3  2  1 1 2 3 4

 4

 3

 2

 1

1

2

3

4

0

h(x) =^1

x

x

h(x)

Forx= 0the functionhis undefined. This is called a discontinuity atx= 0.

y=h(x) =

1

x

therefore we can write thatxy= 1. Since the product of two positive numbersandthe

product of two negative numbers can be equal to 1, the graph lies in the first and third quadrants.

Step 3: Determine the asymptotes

As the value ofxgets larger, the value ofh(x)gets closer to, but does not equal 0. This is a horizontal asymptote,
the liney= 0. The same happens in the third quadrant; asxgets smallerh(x)also approaches the negative
x-axis asymptotically.

We also notice that there is a vertical asymptote, the linex= 0; asxgets closer to 0,h(x)approaches the
y-axis asymptotically.

Step 4: Determine the range

Domain:fx:x 2 R; x̸= 0g

From the graph, we see thatyis defined for all values except 0.

Range:fy:y 2 R; y̸= 0g

Step 5: Determine the lines of symmetry
The graph ofh(x)has two axes of symmetry: the linesy=xandy=x. About these two lines, one half of
the hyperbola is a mirror image of the other half.

Functions of the formy=

a

x

+q EMA4R

Investigation: The effects ofaandqon a hyperbola.

On the same set of axes, plot the following graphs:

1.y 1 =

1

x

2

2.y 2 =

1

x

1

Chapter 6. Functions 169