Everything Maths Grade 10

Thex-intercept:

Every point on thex-axis has ay-coordinate of 0, therefore to calculate thex-intercept, lety= 0.

For example, thex-intercept ofg(x) =

2

x

+ 2is given by settingy= 0:

y=

2

x

+ 2

0 =

2

x

+ 2

2

x

= 2

x=

2

2

= 1

This gives the point(1; 0).

Asymptotes

There are two asymptotes for functions of the formy=

a

x

+q.

The horizontal asymptote is the liney=qand the vertical asymptote is always they-axis, the linex= 0.

Axes of symmetry

There are two lines about which a hyperbola is symmetrical:y=x+qandy=x+q.

Sketching graphs of the formy=

a

x

+q EMA4T

In order to sketch graphs of functions of the form,y=f(x) =

a

x

+q, we need to determine four characteristics:

1.sign ofa

2.y-intercept

3.x-intercept

4.asymptotes

Worked example 10: Sketching a hyperbola

QUESTION

Sketch the graph ofg(x) =

2

x

+ 2. Mark the intercepts and the asymptotes.

SOLUTION

Step 1: Examine the standard form of the equation

We notice thata > 0 therefore the graph ofg(x)lies in the first and third quadrant.

172 6.4. Hyperbolic functions