Lesson 19

End Behavior of Rational Functions

  • Let’s explore the end behavior of rational functions.

19.1: Different Divisions, Revisited

Complete all three representations of the polynomial division following the forms of the integer division.

Diagram of integer long division of 2,775 by 11 and the equivalent expression 252 + 3/11.

\( \require{enclose} \begin{array}{r}  2x^2 \phantom{+7x+55} \\ x+1 \enclose{longdiv}{2x^3+7x^2+7x+5} \\    \end{array}\)

\(2775=11(252)+3\)

\(2x^3+7x^2+7x+5=\)

\(\dfrac{2775}{11}=252+\dfrac{3}{11}\)

\(\dfrac{2x^3+7x^2+7x+5}{x+1}=\)

19.2: Combined Fuel Economy

In 2000, the Environmental Protection Agency (EPA) reported a combined fuel efficiency for cars that assumes 55% city driving and 45% highway driving. The expression for the combined fuel efficiency of a car that gets \(x\) mpg in the city and \(h\) mpg on the highway can be written as \(\frac{100xh}{55x+45h}\).

  1. Several conventional cars have a fuel economy for highway driving is that is about 10 mpg higher than for city driving. That is, \(h=x+10\). Write a function \(f\) that represents the combined fuel efficiency for cars like these in terms of \(x\).
  2. Rewrite \(f\) in the form \(q(x)+\frac{r(x)}{b(x)}\) where \(q(x)\), \(r(x)\), and \(b(x)\) are polynomials.

19.3: Exploring End Behavior

function degree
of num.
degree
of den.
rewritten in the form of
\(q(x)+\frac{r(x)}{b(x)}\)
end behavior
\(g(x)=\text-\frac{5}{x+2}\)        
\(h(x)=\frac{7x-5}{x+2}\)        
\(j(x)=\frac{3x^2+7x-5}{x+2}\)        
\(k(x)=\frac{2x^3+3x^2+7x-5}{x+2}\)        
\(m(x)=\frac{x+2}{2x^3+3x^2+7x-5}\)        
  1. Complete the table to explore the end behavior for rational functions.
  2. What do you notice about the end behavior of different types of rational functions?


  1. Graph \(y=j(x)\) and the line it approaches.
  2. Under what conditions would the end behavior of the graph of a rational function approach a line that is not horizontal?
  3. Create a rational function that approaches the line \(y=2x-3\) as \(x\) gets larger and larger in either the positive or negative direction.

Summary

In earlier lessons, we saw rational functions whose end behavior could be described by a horizontal asymptote. For example, we can rewrite functions like \(d(x)=\frac{x+4}{x}\) as \(d(x)=1+\frac{4}{x}\) to see more clearly that as \(x\) gets larger and larger in either the positive or negative direction, the value of \(\frac{4}{x}\) gets closer and closer to 0, which means the value of \(d(x)\) gets closer to 1. We can use similar thinking to understand rational functions that do not have horizontal asymptotes.

For example, consider \(f(x)=\frac{x^2+4x+5}{x-3}\). Using division, the expression can be rewritten as \(f(x)=x+7+\frac{26}{x-3}\). As \(x\) gets larger and larger in either the positive or negative direction, the value of the term \(\frac{26}{x-3}\) gets closer and closer to 0, which means the value of \(d(x)\) gets closer to the value of \(x+7\). This means that the end behavior of \(f\) can be described by the line \(y=x+7\). Here is a graph of \(y=f(x)\), the line \(y=x+7\), and the vertical asymptote of the function at \(x=3\):

Graph of a rational function f(x) with no horizontal asymptotes, two dashed intersecting lines on coordinate plane.

Glossary Entries

  • horizontal asymptote

    The line \(y =c\) is a horizontal asymptote of a function if the outputs of the function get closer and closer to \(c\) as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the \(x\)-axis.

  • rational function

    A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.

  • vertical asymptote

    The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.