© annie patton asymptotes next slide. © annie patton aim of lesson next slide to introduce what an...
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© Annie Patton© Annie Patton
Aim of LessonAim of Lesson
Next slide
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
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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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-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.
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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.
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-5
5
10
15
x
y
x= -4
y= 3
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3x-1y=
x+4
© Annie Patton© Annie Patton
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.
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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
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-4
-3
-2
-1
1
2
3
4
5
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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.
© Annie Patton© Annie Patton
-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?
Next slide
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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-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