Graphing Rational Functions

Graphing Rational Functions

Parent function is

 $f\left(x\right)=\frac{1}{x}$  

This is the graph of  $f\left(x\right)=\frac{1}{x}$  

Pay attention to what the branches of your graph are doing as they get closer to the vertical asymptote. 

Multiple Choice

What is the degree on your x variable in the denominator?

• odd

• even

Multiple Choice

What do you think will happen to the function if it changes to

 $f\left(x\right)=\frac{1}{x^3}?$  

• the graph will pretty much look the same

• the graph will reflect over the x-axis

You're right, it pretty much looks the same

However it does flatten out faster as x gets really big or really negative (small).

Multiple Select

Here is a rational function, check all that you think apply to the graph of this function.

 $f\left(x\right)=\frac{1}{\left(x-2\right)}$  

• the vertical asymptote is at x=2

• the horizontal asymptote is at y=0

• the branches keep going to positive infinity as x approaches 2 from the right side

• the branches keep going to negative infinity as x approaches 2 from the left side

• the x-intercept is at (0, 0)

Here is the graph of

 $f\left(x\right)=\frac{1}{x-2}$  

Can you see how the graph behaves near the asymptote?  

All of these odd powered equations would all do the same thing at x=2

•  $f\left(x\right)=\frac{1}{x-2}$  green

•  $f\left(x\right)=\frac{1}{\left(x-2\right)^3}$  purple

•  $f\left(x\right)=\frac{1}{\left(x-2\right)^5}$  blue

Multiple Select

What will happen as the graph approaches the asymptote of x= -3 for

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

• the branches approach the vertical asymptote on opposite sides

• the branches approach the vertical asymptote on the same side.  

Multiple Choice

Which of these graphs match the function

 $f\left(x\right)=\frac{1}{x+4}$  

Multiple Choice

Knowing what you do about reflections and equations which graph MUST match this rational function?

 $f\left(x\right)=\frac{-1}{x+4}$  

Multiple Choice

Knowing what you do about odd power rational functions which graph MUST match this EVEN power rational function?  

 $f\left(x\right)=\frac{1}{\left(x+4\right)^2}$  

Multiple Choice

Knowing what you do about odd power rational functions AND reflections which graph MUST match this function.  

 $f\left(x\right)=\frac{-1}{\left(x+4\right)^2}$  

Multiple Choice

Which of these graphs match the function

 $f\left(x\right)=\frac{1}{x+4}$  

Let's move on to more interesting rational functions

Here is this rational function.

•  $f\left(x\right)=\frac{x-1}{x-2}$  

• This changed the horizontal asymptote but not the branch behavior.  

Multiple Choice

Which equation below fits this graph?

• $f\left(x\right)=\frac{2x-1}{x-3}$

• $f\left(x\right)=\frac{2x-1}{x+3}$

Here is the graph for this rational function

 $f\left(x\right)=\frac{x}{x+3}$  

• I did not expect these branches to do this!  

• The next slide is excellent.  

Multiple Choice

For this function what is the y-value on the right side of the asymptote?

 $f\left(x\right)=\frac{2x-1}{x+2}$  

•  $f\left(-1\right)=-3$  ;  this tell me the graph will be downwards as we approach x=-2 from the right

•  $f\left(-1\right)=3$  ; this tells me the graph will be upwards as we approach x=-2 from the right.

Multiple Choice

Choose:

 $f\left(x\right)=\frac{2x-1}{x+2}$  

Summary

• Get your vertical asymptotes and check odd vs even multiplicity

• Get your horizontal asymptote

• Get your x and y -intercepts

• Check values on the sides of your vertical asymptotes

• Sketch this information and complete your graph

Multiple Select

Check all that you know about this rational function

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

• vertical asymptotes at x=-3 and x=3

• horizontal asymptote at y=1

• horizontal asymptote at y-0

•  $f\left(4\right)=\frac{2}{7}$  

• multiplicity of-3 and 3 is odd so the branches are opposite directions.  

Multiple Select

What else do we know about this rational function

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

• the branches start downward as x approaches 3 from the right side

• the branches start upward as x approaches 3 from the right side

• the graph crosses the horizontal asymptote at x=2

•  $f\left(-4\right)=-\frac{6}{7}\$  

•  $f\left(0\right)=\frac{2}{9}$   

 Graph looks like this

 $f\left(x\right)=\frac{x-2}{\left(x+3\right)\left(x-3\right)}$  ; please notice the opposite ends of the branches on both vertical asymptotes.  

Multiple Choice

Rational Function graphs can cross a horizontal asymptote

• True

• False

Multiple Choice

Rational Function graphs can cross a vertical asymptote

• True

• False

Multiple Choice

Rational Function graphs can cross an oblique asymptote

• True

• False

Sketch this graph by hand; answer on next slide

 $f\left(x\right)=\frac{x+5}{x-6}$  

Try to sketch this one. Pay attention to your denominator and the multiplicity; answer on next slide

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

How'd you do?

please don't worry about the "vertex" of the middle portion. This graph crosses the horizontal asymptote at x=-5 and then stays close to y=0 below the x-axis