1.7 Rational Functions and End Behavior

A rational function is a quotient of two polynomials, and its end behavior depends on comparing the degrees of the numerator and denominator. Compare the leading terms to find whether the graph has a horizontal asymptote at $y = 0$, a horizontal asymptote equal to the ratio of leading coefficients, or a slant asymptote.

Why This Matters for the AP Precalculus Exam

Rational functions are a core part of Unit 1, which carries the largest weight on the exam. End behavior questions ask you to read a function in different forms (equation, graph, or table) and describe what happens as x grows large in either direction. On the free-response section, you may need to justify your conclusion, so being able to explain why the higher-degree polynomial controls the long-run behavior is just as important as getting the right asymptote. This topic also sets up later work with vertical asymptotes, zeros, and holes.

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Key Takeaways

• A rational function is a quotient of two polynomials, and it compares the size of the numerator to the size of the denominator at each input in its domain.
• For inputs of large magnitude, each polynomial behaves like its leading term, so end behavior comes from the quotient of the leading terms.
• If the denominator has the greater degree, the graph has a horizontal asymptote at y = 0.
• If the numerator and denominator have equal degree, the horizontal asymptote is the ratio of the leading coefficients.
• If the numerator degree is exactly one more than the denominator degree, the graph has a slant (oblique) asymptote.
• A horizontal asymptote at y = b means the outputs approach b as x increases or decreases without bound, written as a limit.

Rational Functions: A Quotient of Two Polynomials

A rational function can be written as the quotient of two polynomial functions. In other words, it is a ratio of two polynomials, where both the numerator and the denominator are polynomial expressions.

The function gives a measure of the relative size of the numerator polynomial compared to the denominator polynomial for each value in its domain. The degree of the numerator and the degree of the denominator are what drive the function's end behavior.

For example, the expression below is a rational function:

$r(x) = \frac{x^2 + 5}{x + 2}$

Here the numerator is the degree 2 polynomial $x^2 + 5$ and the denominator is the degree 1 polynomial $x + 2$.

A quick preview of what end behavior tells you:

• If the numerator degree is less than the denominator degree, the outputs approach zero as the inputs get large in magnitude.
• If the numerator degree is greater than the denominator degree, the outputs increase or decrease without bound as the inputs get large in magnitude.

End Behavior of Rational Functions

The end behavior of a rational function describes what the outputs do as the inputs increase or decrease without bound. You find it by comparing the degree of the polynomial in the numerator and the denominator.

The polynomial with the higher degree has the greatest influence on the overall behavior. As inputs grow in magnitude, the leading term of the higher-degree polynomial outweighs the leading term of the other polynomial, so it controls the behavior of the whole rational function.

The fastest way to analyze this is to look at the quotient of the leading terms of the numerator and denominator.

Case 1: Numerator Polynomial Has the Greater Degree

If the leading term of the numerator outweighs the leading term of the denominator, then the quotient of the leading terms is a nonconstant polynomial. The original rational function has the same end behavior as that polynomial.

If the quotient of the leading terms is a linear polynomial (this happens when the numerator degree is exactly one more than the denominator degree), then the graph has a slant asymptote parallel to that line.

Case 2: Neither Polynomial Has the Greater Degree

If neither polynomial outweighs the other for inputs of large magnitude (the degrees are equal), look at the quotient of the leading terms. This quotient is a constant, and that constant is the location of a horizontal asymptote for the graph. The asymptote is the ratio of the leading coefficients.

Case 3: Denominator Polynomial Has the Greater Degree

If the denominator outweighs the numerator for inputs of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and a nonconstant polynomial in the denominator. The graph has a horizontal asymptote at y = 0.

As inputs grow in magnitude, the denominator outgrows the numerator, so the overall value of the function approaches zero.

Limits and Horizontal Asymptotes

When a rational function $r$ has a horizontal asymptote at $y = b$, where $b$ is a constant, the outputs get arbitrarily close to $b$ and stay close to $b$ as the inputs increase or decrease without bound.

In limit notation:

$\lim_{x \to \infty} r(x) = b \quad \text{or} \quad \lim_{x \to -\infty} r(x) = b$

On a graph, the curve approaches the line $y = b$ but does not have to touch or cross it for large magnitudes of x.

How to Use This on the AP Precalculus Exam

Problem Solving

• Identify the degree and leading coefficient of the numerator and the denominator first. That comparison decides which case you are in.
• Denominator degree larger: horizontal asymptote at y = 0.
• Degrees equal: horizontal asymptote at y = (leading coefficient of numerator) / (leading coefficient of denominator).
• Numerator degree larger by exactly one: slant asymptote. Use polynomial long division to find the line.
• Write the end behavior using limit notation when asked, such as $\lim_{x \to \infty} r(x) = b$.

Free Response

• When a question asks you to justify, state the reason: for inputs of large magnitude each polynomial behaves like its leading term, so the quotient of the leading terms gives the end behavior.
• Keep your variables and notation precise. Clear setup and correct notation make your reasoning easy to follow.
• Match your answer to the form requested. If the prompt gives a graph or table, connect your degree comparison to what the representation shows.

Common Trap

• A horizontal asymptote describes end behavior only. A graph can cross its horizontal asymptote in the middle region, so do not assume the curve never touches it.

Common Misconceptions

• A horizontal asymptote and a slant asymptote do not happen at the same time. A graph has one type of end behavior on each side, decided by the degree comparison.
• The horizontal asymptote for equal degrees is the ratio of the leading coefficients, not y = 1 and not y = 0.
• "The curve never touches its asymptote" is only guaranteed near the ends. A rational function can cross its horizontal asymptote for smaller inputs.
• A larger numerator degree does not always give a slant asymptote. You only get a slant asymptote when the numerator degree is exactly one more than the denominator degree. If it is two or more higher, the end behavior follows a higher-degree polynomial instead.
• End behavior is about large magnitudes of x, not about vertical asymptotes or holes. Those come from the denominator's zeros and are handled in later topics.

Related AP Precalculus Guides

• 1.9 Rational Functions and Vertical Asymptotes
• 1.11 Equivalent Representations of Polynomial and Rational Expressions
• 1.4 Polynomial Functions and Rates of Change
• 1.8 Rational Functions and Zeros
• 1.12 Transformations of Functions
• 1.3 Rates of Change in Linear and Quadratic Functions