Asymptopes

Asymptopes

The white elephant in the room.

Asymptope

• An asymptote is a line that a curve approaches (but never actually touches), as it heads towards infinity

• Three types: horizontal, vertical, and oblique.

• The degree of the polynomials will lead us to what type of asymptope we have.

Horizontal Asymptopes: when and where?

• Degree of the numerator is less than the denominator, there will be a horizontal asymptope. (specifically, it will be at y=0). For example:

   $f\left(x\right)=\frac{x^2}{x^3}$  or  $f\left(x\right)=\frac{2x^3+3x^2+5x}{4x^4+5x^3+2x^2+14x}$  

Open Ended

What would be the range of the function

 $f\left(x\right)=\frac{x^2}{x^{3\ }}$  (hint: it never touches y=0!)

Horizontal Asymptopes: when and where?

• Degree of the numerator is equal to the denominator, there will be a horizontal asymptope. (specifically, it will be at the ratio of the coefficients of the highest degree terms.) For example:

•  $f\left(x\right)=\frac{9x^2+5}{3x^2+3}$  will have a horizontal asymptope at y=3.

Horizontal Asymptopes: when and where?

• There will never be a horizontal asymptope when the degree of the numerator is greater then the denominator.

• Range: all real numbers.

Multiple Choice

Find the horizontal asymptote.

• None
• y=-2
• y=2
• y=0

Multiple Choice

Find the horizontal asymptote.

• $y=-2$

• none

• $y=0$

• $y=\frac{1}{2}$

Multiple Choice

What is the horizontal asymptote?

• y = -4
• y = 1
• x = 1
• y = -6

Open Ended

Give me a function that would have a horizontal asymptope at y=0.

Vertical Asymptopes: when and where

• Focus on the denominator!

• Whenever the denominator is 0 (the function is undefined), there will be a vertical asymptope.

• Finding the domain is vital for vertical asymptopes.

• For example  $f\left(x\right)=\frac{1}{x}$  will have a vertical asymptope at x=0. 

Multiple Choice

Find the vertical asymptote.

• x = 2

• none

• x = 0

• x = -1/2

Multiple Choice

What is the Vertical Asymptotes? 

• x= -5
• x= 5
• x= 6
• x= -6

Multiple Choice

Which asymptote(s) are determined by looking at the denominator?

• vertical
• horizontal
• slant
• none

Open Ended

Can a function have two vertical asymptopes??!!

Hint: think about the function

 $f\left(x\right)=\frac{x}{x^2-9}\ =\ \frac{x}{\left(x-3\right)\left(x+3\right)}$