© Annie Patton© Annie Patton

AsymptotesAsymptotes

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−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

© Annie Patton© Annie Patton

Aim of LessonAim of Lesson

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To introduce what an asymptote is, the difference in a horizontal and vertical asymptote and how to find these.

© Annie Patton© Annie Patton

What is an asymptote?What is an asymptote?

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An asymptote is a line, to which a curve gets closer and closer without touching.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

Horizontal

Vertical

© Annie Patton© Annie Patton

How to find a Vertical AsymptoteHow to find a Vertical Asymptote

This is a value of x for which y is undefined, that is

when the denominator equals zero.

Note it will be a line.

3 1.

1

x

x

For example for the curve y=

When x-1= 0, the denominator is undefined.

x =1 is the verticle asymptote.

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-4 -3 -2 -1 1 2 3 4 5 6

-10

10

x

y

© Annie Patton© Annie Patton

How to find a Horizontal AsymptoteHow to find a Horizontal Asymptote

This is the line that y approaches as x becomes

greater and greater, that is as y goes to infinity.

Note it will be a line.

3 1.

1For example for the curve y=

x

x

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133 1

lim lim lim 311 1

x x x

x xyx

x

y=3 is the horizontal asymptote.

-4 -3 -2 -1 1 2 3 4 5 6

-10

10

x

y

© Annie Patton© Annie Patton

Do all curves have asymptotes?Do all curves have asymptotes?

No

( )

( )Only those of the form y= , where f(x) 0.

g x

f x

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© Annie Patton© Annie Patton

1.

4 8Find the equations of the two asymptotes of y=

x

x

Vertical Asymptote

4x-8=0

4x=8

x=2

111 1

lim lim84 8 44

1

4

x x

x xx

x

y

Horizontal Asymptote

is the Horizontal Asymptote

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© Annie Patton© Annie Patton

Draw a rough sketch of the curve with vertical

asymptote x= -4 and horizontal asymptote y=3.

-10 10

-5

5

10

15

x

y

x= -4

y= 3

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3x-1y=

x+4

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3.2

Find the point of intersection of the 2 asymptotes of y=x

x

Vertical Asymptote x=2

x x

Horizontal Asymptote

31+x+3 xlim = lim =1

2x-2 1-x

y=1 is the horizontal asymptote.

-5 5 10

5

10

x

y

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Point (2,1)

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Vertical asymptote x-1=0, therefore x=1.

1 1lim lim .

11 11Horizontal Asymptote

Horizontal Asymptote y=1

x x

x

xx

-6 -4 -2 2 4 6 8

-4

-3

-2

-1

1

2

3

4

5

6

x

y

1

xy

x

1

xy

x

1x

1y

Start clicking when you want to see the answer.

Leaving Certificate 2005 Higher Level Paper 1 no 6(c)(ii)

,1

x

x

The equation of a curveis y= where x 1.

Write down the equations of the asymptotes and hence sketch the curve.

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To verify which quadrants the asymptotes are in, substitute in a point, for example x=4.

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-5 5

-4

-3

-2

-1

1

2

3

4

x

y

(1,1)

(x, y)

,1

The equation of a curveis y= where x 1.

Show that the curve is its own image under the symmetry

in the point of intersection of the asymptotes.

x

x

(1,1)

(x, y)

Leaving Certificate 2005 Higher Level Paper 1 no 6(c)(iii)

Start clicking when you want to see the answer.

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(2-x,2-y)

Take the point (x,y) on the curve,

the image of it under a central

symmetry on (1,1) the point of

intersection of the

asymptotes is(2-x, 2-y).

12

22 12 2

2 21 12 2 2 2

21 1 1 1

x

xx

yxx x

y yx xx x x x x

yx x x x

Check to see if (2-x, 2-y) is on the curve y=

Therefore

Therefore(2-x,2-y) is on the curve.

© Annie Patton© Annie Patton

HomeworkHomework

2

2

4.

4

2

2

2

2

2

Find the Vertical and Horizontal Asymptotes of the folowing curves:

x1. y=

x-4x

2. y=x +1x +1

3. y=x

Find the point of intersection of

the asymptotes of the curve y= and

draw a rough sketch of the

x

x

curve.

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© Annie Patton© Annie Patton

Revision. What is an Revision. What is an asymptote?asymptote?

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An asymptote is a line, to which a curve gets closer and closer without touching.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

Horizontal

Vertical

© Annie Patton© Annie Patton

Revision. How to find a Vertical Revision. How to find a Vertical AsymptoteAsymptote

This is a value of x for which y is undefined, that is

when the denominator equals zero.

Note it will be a line.

3 1.

11

For example for the curve y=

=0, the denominator is undefined.

x=1 is the verticle asymptote.

x

xx

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-4 -3 -2 -1 1 2 3 4 5 6

-10

10

x

y

© Annie Patton© Annie Patton

Revision. How to find a Horizontal Revision. How to find a Horizontal AsymptoteAsymptote

This is the line that y approaches as x becomes

greater and greater, that is as y goes to infinity.

Note it will be a line.

3 1.

1For example for the curve y=

x

x

133 1

lim lim lim 311 1

x x x

x xyx

x

y=3 is the horizontal asymptote.

-4 -3 -2 -1 1 2 3 4 5 6

-10

10

x

y