Key Features of Graphs of Rational Expressions

Key Features of Graphs of Rational Expressions

Advanced Algebra

Multiple Choice

 $f\left(x\right)=x^2-5x-6$  

Bell Ringer: Factor and Solve the Equation

•  $x=-5,\ x=-6$  

•  $x=5,\ x=6$  

•  $x=-1,\ x=6$  

•  $x=1,\ x=-6$  

Open Ended

You have 60 seconds! Find a partner that is sitting around you. When you have found your partner, submit both of your names as the answer to this question.

REVIEW

 $f\left(x\right)=x^3+x^2-4x-4$  

x-int:
y-int:
Asymptotes:
Domain:
Range:
End Behavior:

 $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\$   $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\$  

Multiple Choice

What was the x-int of the previous problem?

• $\left(-2,0\right),\ \left(-1,0\right),\ \left(2,0\right)$

• $\left(0,-2\right),\ \left(0,-1\right),\ \left(0,2\right)$

• $\left(-4,0\right)$

• $\left(0,-4\right)$

REVIEW

 $f\left(x\right)=2^x-4$  

x-int:
y-int:
Asymptotes:
Domain:
Range:
End Behavior:

 $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\$   $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\$  

Multiple Choice

What was the horizontal asymptote of the previous question?

• $y=-3$

• $y=-4$

• $x=-3$

• $x=-4$

What about this one?

 $f\left(x\right)=\frac{1}{x^2}$  

x-int:
y-int:
Asymptotes:
Domain:
Range:
End Behavior:

 $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\$   $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\$  

Open Ended

What was different about the graph we just saw compared with the two previous graphs?

Welcome to Rational Expressions!

• Definition: Rational Expressions are Fractional Expressions where the Degree of the Denominator is Greater than 0

• What does that mean?

Let's talk about the Asymptotes

• Horizontal Asymptotes (ALWAYS y= )

• Vertical Asymptotes (ALWAYS x= )

• But there are also...

Slant Asymptotes

• Found by Using Long Division

• Written as the equation of a line (y=mx+b)

• Don't worry! We will learn more about these tomorrow!

Let's try an example (without partners)

 $f\left(x\right)=\frac{x}{x+1}$  

x-int:
y-int:
Asymptotes:
Domain:
Range:
End Behavior:

 $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\$   $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\$  

Multiple Choice

What is the Domain and Range of the previous graph?

• 
  

  $D:\ \left(-\infty,\ \infty\right)$ $R:\ \left(-\infty,\ \infty\right)$
  

• $D:\ \left(-\infty,\ -1\right)\cup\left(-1,\ \infty\right)$ $R:\ \left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$

• $D:\ \left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$ $R:\ \left(-\infty,\ -1\right)\cup\left(-1,\ \infty\right)$

Poll

Which key features are still confusing you?

• Intercepts

• Domain/Range

• Asymptotes

• End Behavior

TOTD (Individually)

 $f\left(x\right)=\frac{2x}{x-3}$  

x-int:

y-int:

Asymptotes:

Domain:

Range:

End Behavior:  $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\$   $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\$