Rational Functions and End Behavior - AP PreCalculus Study Guide

The Core Idea: Rational Functions and End Behavior

A rational function is fundamentally a function formed by the division of two polynomial functions, expressed as $r(x)=\frac{p(x)}{q(x)}$. The central concept of this topic is to understand the end behavior of these functions. End behavior describes what happens to the output values, $r(x)$, as the input values, $x$, become infinitely large in either the positive or negative direction (as $x\to \infty$ or $x\to -\infty$).

The key insight is that for extreme values of $x$, the behavior of a polynomial is overwhelmingly dominated by its leading term (the term with the highest power of $x$). Therefore, the end behavior of the entire rational function can be precisely determined by analyzing the ratio of the leading term of the numerator, $p(x)$, to the leading term of the denominator, $q(x)$. This analysis allows us to identify horizontal asymptotes, which are horizontal lines that the function's graph approaches as $x$ moves towards infinity or negative infinity, and to describe the function's unbounded growth or decay when no such asymptote exists.

Key Rules for End Behavior

The end behavior of a rational function $r(x)=\frac{p(x)}{q(x)}$ is determined by comparing the degree of the numerator, $n$, with the degree of the denominator, $m$. Let the leading term of $p(x)$ be $a{x}^n$ and the leading term of $q(x)$ be $b{x}^m$. The end behavior of $r(x)$ is the same as the end behavior of the function $y=\frac{a{x}^n}{b{x}^m}$.

There are three distinct cases that arise from this comparison:

Case 1: Degree of Numerator is Less Than Degree of Denominator ($n<m$)

• The horizontal asymptote is the line $y=0$ (the x-axis).

• The limit of the function as $x$ approaches positive or negative infinity is zero.

• In limit notation: $$\underset{x\to \infty }{\lim }r(x)=0\text{and}\underset{x\to -\infty }{\lim }r(x)=0$$

Case 2: Degree of Numerator is Equal to Degree of Denominator ($n=m$)

• The horizontal asymptote is the line $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.

• The limit of the function as $x$ approaches positive or negative infinity is the ratio of the leading coefficients.

• In limit notation: $$\underset{x\to \infty }{\lim }r(x)=\frac{a}{b}\text{and}\underset{x\to -\infty }{\lim }r(x)=\frac{a}{b}$$

Case 3: Degree of Numerator is Greater Than Degree of Denominator ($n>m$)

• There is no horizontal asymptote.

• The function's output values will grow without bound, approaching either positive or negative infinity.

• The end behavior of $r(x)$ is the same as the end behavior of the polynomial quotient $y=\frac{a{x}^n}{b{x}^m}=\frac{a}{b}{x}^{n-m}$.

• In limit notation: $$\underset{x\to \pm \infty }{\lim }r(x)=\pm \infty$$

  The specific signs of infinity depend on the signs of $\frac{a}{b}$ and the behavior of $x^{n-m}$ as $x\to \infty$ or $x\to -\infty$.

Understanding the Dominance of Leading Terms

The rules for determining the end behavior of rational functions are not arbitrary; they are a direct consequence of how polynomials behave for very large input values. Consider a polynomial like $p(x)=3x^4-50x^3+1000$. When $x$ is a small number, like $x=2$, all terms contribute significantly. However, when $x$ is a very large number, like $x=1,000,000$, the value of the leading term, $3x^4$, becomes so immense that it dwarfs the contributions of all other terms. The terms $-50x^3$ and $+1000$ become negligible in comparison.

This principle of "leading term dominance" is the foundation for analyzing rational functions. When we examine $r(x)=\frac{p(x)}{q(x)}$ as $x\to \pm \infty$, we are essentially asking what happens to the ratio of two very large numbers. Since the leading terms of $p(x)$ and $q(x)$ are the only parts that matter at these extremes, we can simplify our analysis by focusing exclusively on them. The end behavior of $r(x)=\frac{a{x}^n+\dots }{b{x}^m+\dots }$ is effectively the same as the end behavior of the much simpler function $y=\frac{a{x}^n}{b{x}^m}$. This simplification directly leads to the three cases: if the denominator's power ($m$) is greater, the fraction goes to zero; if the powers are equal ($n=m$), the fraction stabilizes at the ratio of coefficients; and if the numerator's power ($n$) is greater, the fraction grows without bound.

Core Concepts & Rules

• Definition: A rational function is a ratio of two polynomial functions, $r(x)=\frac{p(x)}{q(x)}$, where $q(x)\ne 0$.

• End Behavior Focus: The end behavior of a rational function describes its value as the input `x$ approaches $\infty$ and $-\infty$.

• Leading Term Ratio: The end behavior of $r(x)$ is determined entirely by the ratio of the leading term of the numerator to the leading term of the denominator.

• Degree Comparison is Key: To find the end behavior, first identify the degree of the numerator ($n$) and the degree of the denominator ($m$).

• Rule for $n<m$: If the denominator's degree is higher, the function approaches zero. The horizontal asymptote is $y=0$.

• Rule for $n=m$: If the degrees are equal, the function approaches the ratio of the leading coefficients. The horizontal asymptote is $y=\frac{a}{b}$.

• Rule for $n>m$: If the numerator's degree is higher, the function does not have a horizontal asymptote. Its output values increase or decrease without bound, mimicking the end behavior of the polynomial $y=\frac{a}{b}{x}^{n-m}$.

Step-by-Step Example 1: Equal Degrees

Problem: Determine the end behavior of the function $f(x)=\frac{8x^3-5x+2}{2-x^2+4x^3}$ and express it using limit notation.

Step 1: Identify the Numerator and Denominator Polynomials

• Numerator: $p(x)=8x^3-5x+2$

• Denominator: $q(x)=4x^3-x^2+2$ (It is crucial to write the denominator in standard form to easily identify the leading term).

Step 2: Determine the Degree and Leading Coefficient of Each Polynomial

• For the numerator $p(x)$:

  • The leading term is $8x^3$.

  • The degree is $n=3$.

  • The leading coefficient is $a=8$.

• For the denominator $q(x)$:

  • The leading term is $4x^3$.

  • The degree is $m=3$.

  • The leading coefficient is $b=4$.

Step 3: Compare the Degrees

• Here, $n=3$ and $m=3$. The degrees are equal ($n=m$).

Step 4: Apply the Appropriate Rule

• Since the degrees are equal, the function has a horizontal asymptote at $y=\frac{a}{b}$.

• Calculate the ratio of the leading coefficients: $y=\frac{8}{4}=2$.

• The horizontal asymptote is the line $y=2$.

Step 5: Express the End Behavior Using Limit Notation

• As $x$ approaches both positive and negative infinity, the function's value approaches 2.

• $$\underset{x\to \infty }{\lim }\frac{8x^3-5x+2}{4x^3-x^2+2}=2$$

• $$\underset{x\to -\infty }{\lim }\frac{8x^3-5x+2}{4x^3-x^2+2}=2$$

Step-by-Step Example 2: Exam-Style Application (Degree Greater)

Problem: Describe the end behavior of the function $g(x)=\frac{3x(x+1{)}^2}{2x^2-5}$.

Step 1: Expand the Polynomials to Find the True Leading Terms

• The function is given in a partially factored form. To find the degree and leading coefficient, we must determine what the leading term would be if the numerator were fully expanded. We do not need to expand the entire polynomial, only enough to find the term with the highest power.

• Numerator: $p(x)=3x(x+1{)}^2=3x(x^2+2x+1)$. The highest power term will come from multiplying $3x$ by $x^2$, which gives $3x^3$.

  • Leading term of numerator: $3x^3$

  • Degree of numerator: $n=3$

• Denominator: $q(x)=2x^2-5$. This is already in standard form.

  • Leading term of denominator: $2x^2$

  • Degree of denominator: $m=2$

Step 2: Compare the Degrees

• Here, $n=3$ and $m=2$. The degree of the numerator is greater than the degree of the denominator ($n>m$).

Step 3: Apply the Appropriate Rule and Analyze the End Behavior Model

• Since $n>m$, there is no horizontal asymptote.

• The end behavior of $g(x)$ is the same as the end behavior of the function formed by the ratio of the leading terms:

  $$y=\frac{3x^3}{2x^2}=\frac{3}{2}x$$

• This end behavior model, $y=\frac{3}{2}x$, is a line with a positive slope.

Step 4: Describe the End Behavior

• As $x\to \infty$, the model $y=\frac{3}{2}x$ approaches $\infty$. Therefore, $g(x)\to \infty$.

• As $x\to -\infty$, the model $y=\frac{3}{2}x$ approaches $-\infty$. Therefore, $g(x)\to -\infty$.

Step 5: Express the End Behavior Using Limit Notation

• $$\underset{x\to \infty }{\lim }\frac{3x(x+1{)}^2}{2x^2-5}=\infty$$

• $$\underset{x\to -\infty }{\lim }\frac{3x(x+1{)}^2}{2x^2-5}=-\infty$$

Using Your Calculator

While the rules for end behavior are analytical, a graphing calculator is an excellent tool for verifying your conclusions. It allows you to visualize the end behavior and confirm the presence or absence of a horizontal asymptote.

**To verify the end behavior of a rational function, such as f(x) = \frac{8x^3 - 5x + 2}{4x^3 - x^2 + 2}` from Example 1:** **Method 1: Using the Table** 1. Press the `Y=` button and enter the function into `Y1`. Be careful to use parentheses for the entire numerator and the entire denominator: `Y1=(8X^3-5X+2)/(4X^3-X^2+2)`. 2. Press `2nd` then `TBLSET` (Table Setup). Set `TblStart` to 0 and `ΔTbl` to 1. More importantly, set `Indpnt:` to `Ask`. 3. Press `2nd` then `TABLE`. Now you can enter very large positive and negative values for $x to observe the trend in $y$.

*   Enter $100$. The `Y1` value will be close to 2 (e.g., 2.002).

*   Enter $1000$. The `Y1` value will be even closer to 2 (e.g., 2.0002).

*   Enter $-100$. The `Y1` value will be close to 2 (e.g., 1.997).

*   Enter $-1000$. The `Y1` value will be even closer to 2 (e.g., 1.999).

4. This numerical evidence strongly supports the analytical conclusion that \lim_{x \to \pm\infty} f(x) = 2.

Method 2: Using the Graph

1. Enter the function into Y1 as described above.

2. Press ZOOM and select 6:ZStandard to see an initial view. The end behavior may not be clear.

3. To better see the long-term trend, adjust the WINDOW. Set $Xmin$ to a large negative number (e.g., -100) and $Xmax$ to a large positive number (e.g., 100). You may need to adjust $Ymin$ and $Ymax$ (e.g., -5 to 5) to "zoom in" on the horizontal behavior.

4. Press GRAPH. You should see the function graph flattening out and approaching a horizontal line as it moves to the far left and far right of the screen.

5. To visualize the asymptote, go to Y= and enter your calculated asymptote into Y2. For Example 1, you would enter Y2=2.

6. Press GRAPH again. You will see the graph of the function $f(x)$ getting closer and closer to the line $y=2$ at the edges of the screen, visually confirming the horizontal asymptote.

AP Exam Quick Hit

Common Question Types

• Identifying Horizontal Asymptotes: Given a function $f(x)=\frac{10x^2-3x}{5x^2+1}$, what is the equation of the horizontal asymptote?

  • Answer: Since the degrees are equal ($n=2,m=2$), the asymptote is $y=\frac{10}{5}=2$.

• Evaluating Limits at Infinity: Evaluate ${\lim }_{x\to -\infty }\frac{x-4}{x^3+2x-1}$.

  • Answer: Since the degree of the numerator ($n=1$) is less than the degree of the denominator ($m=3$), the limit is 0.

• Describing Unbounded End Behavior: Which of the following functions has the same end behavior as $y=2x$?

  (A) $f(x)=\frac{2x^2+1}{x^2-1}$

  (B) $g(x)=\frac{4x^3-x}{2x^2+5}$

  (C) $h(x)=\frac{2x+7}{x^2-3}$

  • Answer: (B), because the ratio of its leading terms is $\frac{4x^3}{2x^2}=2x$.

Common Mistakes

• Forgetting to Expand: When given a function like $f(x)=\frac{(x-3)(2x+1)}{x^2+4}$, students might mistakenly use the degree of $(x-3)$ as $n=1$, ignoring the other factor. You must find the leading term of the fully expanded numerator, which is $2x^2$, to correctly determine the degree is $n=2$.

• Misidentifying Leading Terms: In a polynomial not written in standard form, such as $q(x)=1-5x^2+3x$, students might incorrectly identify $1$ as the leading term. Always reorder the polynomial to find the term with the highest degree ($-5x^2$).

• Confusing Rules: A common error is to mix up the rules. For example, applying the ratio of leading coefficients rule ($y=a/b$) when the degree of the denominator is larger (the rule should be $y=0$).

• Sign Errors for $n>m$: When analyzing the end behavior for a function like $g(x)=\frac{-4x^3+1}{x^2+1}$, the end behavior model is $y=\frac{-4x^3}{x^2}=-4x$. Students may forget the negative sign and incorrectly state that the function approaches $\infty$ as $x\to \infty$. The correct behavior is that it approaches $-\infty$.

• "No Asymptote" vs. "Asymptote at y=0": Students sometimes confuse the case $n<m$ (where the asymptote is $y=0$) with having no asymptote at all. An asymptote at $y=0$ is a very specific and important type of end behavior. The only case with no horizontal asymptote is when $n>m$.