Horizontal Asymptote

Asymptotes

Introduction to Rational Functions

We discussed this graph ( 1/x ) in class on Monday. One thing we noticed is that the graph approaches 0 but never reaches it. We have talked about asymptotes while graphing log functions, and we will analyze the asymptotes of this function.

What is an asymptote?

Look at the definition of an asymptote : it is a line that stops a graph from reaching a certain point. Note that the graph gets extremely close to the asymptote but never actually touches it

Horizontal vs Vertical

• Horizontal asymptotes are horizontal lines that stop the graph from approaching some y value. Example: y = 1

• Vertical asymptotes are vertical lines that stop the graph from approaching some x value. Example : x = 3

Multiple Choice

What y value can this graph not reach because of a horizontal asymptote?

• -1

• 1

• 3

• 0

Multiple Choice

What x value can this graph not reach because of a vertical asymptote?

• 1

• -15

• 3

• 15

So what about 1/x ?

I know there is a horizontal asymptote at y = 0 because the graph is approaching that y value but it never reaches it

Multiple Choice

Is there a vertical asymptote for 1/x ?

• yes

• no

Multiple Choice

What is the equation of the vertical asymptote?

• x = 0

• y = 0

• x = 1

• there is no vertical asymptote

Asymptotes of y = 1/x

Multiple Choice

What are the asymptotes of y = 1/x ?

• y = 1 and x = 1

• y = -1 and x = -1

• y = 15 and x = 15

• y = 0 and x = 0

Final Points

There is a horizontal and vertical asymptote at 0 ( x = 0 and y = 0 ).

Why does this occur?

If we put in 0 for our x value, we get an ERROR in our calculator. We cannot divide a number by 0.

If we made our y into 0 we would also get an ERROR because we cannot get an answer of 0 from 1/x no matter what values we put in for x.