Key Features of Rational Expressions (Algebraically)

Key Features of Rational Expressions (Algebraically)

Advanced Algebra

Multiple Choice

What is the end behavior of the graph? (Click on the graph to enlarge)

• $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ 2$ $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\ 2$

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

• $As\ x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ -1$ $As\ x\ \longrightarrow\ \infty,\ f\left(x\right)\ \longrightarrow\ -1$

Rational Functions

What do we need to find algebraically?

1) X-Intercepts

2) Y-Intercepts

3) Vertical Asymptotes

4) Horizontal or Slant Asymptotes

X-Intercepts

To find the X-Intercept, set the numerator of the rational function equal to zero and solve

Remember, your x-intercept has to be a REAL number (No imaginaries)

Multiple Choice

 $f\left(x\right)=\ \frac{x+5}{x^2+2x-3}$  What is the x-intercept(s) of the given function?

•  $\left(-5,0\right)$  

•  $\left(-3,0\right),\ \left(1,0\right)$  

•  $\left(0,-3\right),\ \left(0,1\right)$  

•  $\left(0,-5\right)$  

Y-Intercepts

Y-Intercepts are still found the same way that we have been doing since the beginning of the year.

Plug in 0 for x and solve the equation for f(x) or y (whichever is used)

Multiple Choice

 $f\left(x\right)=\frac{2x-3}{4x^2+2x+1}$  Find the y-intercept of the rational function

•  $\left(\frac{1}{2},0\right)$  

•  $\left(-3,0\right)$  

•  $\left(0,\frac{1}{2}\right)$  

•  $\left(0,-3\right)$  

Vertical Asymptotes

Finding the vertical asymptotes is very similar to finding the x-intercepts with one exception.

When finding vertical asymptotes, set the DENOMINATOR equal to zero and solve.

Vertical asymptotes are always written as x=

Multiple Choice

 $f\left(x\right)=\frac{x^2-4}{x^2+7x+10}$  Find the vertical asymptote(s)

•  $x=2,\ x=-2$  

•  $x=-2,\ x=-5$  

•  $y=-2,\ y=-5$  

•  $y=2,\ y=-2$  

Horizontal/Slant Asymptotes

A problem will either have ONE horizontal asymptote or ONE slant asymptote. NEVER BOTH and NEVER MORE THAN ONE

It is all based off the degree of the numerator vs the degree of the denominator.

Horizontal Asymptote

Two Cases:

First Case - If the degree of the numerator is less than the degree of the denominator, the HA is y=0.

Second Case - If the degree of the numerator is equal to the degree of the denominator, the HA is y= the LC of the numerator / the LC of the denominator

Multiple Choice

 $f\left(x\right)=\frac{3x^2-4x+3}{9x^2+2x+1}$  Find the horizontal asymptote of the rational equation.

•  $y=0$  

• None. It's a Slant Asymptote

•  $x=0$  

•  $y=\frac{1}{3}$  

Slant Asymptote

Slant Asymptotes are found when the degree of the numerator is one larger than the degree of the denominator.

You must use polynomial long division to find the slant asymptote.

When finding the slant asymptote, the remainder DOES NOT MATTER

Multiple Choice

 $f\left(x\right)=\frac{x^2-6x+7}{x+5}$  Find the slant asymptote of the rational function.

• None. It is a horizontal asymptote.

•  $y=0$  

•  $y=x-11$  

•  $y=x+5$  

Poll

Which key features do you not feel good about finding?

• X-Intercept

• Y-Intercept

• Vertical Asymptote

• Horizontal Asymptote

• Slant Asymptote

TOTD

Find the following for the given equation:

1) x-intercept:

2) y-intercept:

3) vertical asymptote:

4) horizontal/slant asymptote: