Introduction

In the world of precalculus, rational functions are like the tried-and-true middle path between simple and complex. These functions appear as a quotient of two polynomials. It’s vital to master the end behavior of these mathematical functions. The end behavior reveals how a function acts as the input values grow towards positive or negative infinity. This understanding is crucial when graphing or analyzing rational functions, especially as you prepare for exams and tackle 1.7a Rational Functions and End Behavior in AP® Precalculus.

What are Rational Functions?

Rational functions are expressions formed by dividing one polynomial by another. Think of them as sophisticated fractional equations. The general form looks like this:

f(x) = \frac{p(x)}{q(x)}

Here, p(x) and q(x) are polynomials, and q(x) is not zero. Degrees and leading terms play crucial roles. The degree of a polynomial is the highest power of its variable (like x), and the leading term is the term with that highest power. Understanding these helps determine the end behavior of rational functions, or how the graph behaves as x tends toward infinity.

Understanding End Behavior of Rational Functions

End behavior describes what happens to the graph of a function as x approaches either positive or negative infinity. In simple terms, it shows the trend of the function’s graph at its “ends.” In rational functions, it’s essential because it dictates how the graph behaves outside its central area. When delving into this topic, limit notation is often used to describe end behavior:

• \lim\limits_{x \to \infty} f(x) and \lim\limits_{x \to -\infty} f(x)

These expressions detail the function’s value as x moves towards positive infinity and negative infinity.

Key Concepts of End Behavior of Rational Functions

The end behavior of rational functions heavily relies on the degrees of its numerator and denominator. This is determined by looking at the degree of a polynomial.

• Case 1: The Numerator Dominates
  • If the degree of the numerator is greater than the degree of the denominator, the rational function behaves like its numerator’s leading term.
• Case 2: The Denominator Dominates
  • Conversely, if the degree of the denominator is higher, the function behaves more like 0 as x moves toward infinity.
• Case 3: Degrees Are Equal
  • When the degrees, or powers, are equal, the function approaches a constant. This constant is found by looking at the leading term of a polynomial. In fact, it is the ratio of the leading coefficients of the numerator and denominator

Numerator Dominates

Let’s explore: f(x) = \frac{2x^3 + 5x}{x^2 - x + 1}

1. Compare Degrees:

 Numerator’s degree = 3; Denominator’s degree = 2.

1. Identify End Behavior:

 Numerator dominates.

1. Determine Behavior:

 The end behavior is like 2x^3/x^2 = 2x, implying f(x) tends to infinity as x \to \infty.

End Behavior of Rational Functions: The Horizontal Asymptote

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

1. Compare Degrees:

 Both numerator and denominator have degree 2.

1. Calculate Horizontal Asymptote:

 The end behavior approaches \frac{3}{2}, as x \to \pm \infty.

Steps:

• Leading coefficient of numerator = 3
• Leading coefficient of denominator = 2
• Horizontal asymptote = y = \frac{3}{2}

Denominator Dominates

Suppose h(x) = \frac{x + 1}{x^2 - 4}

1. Compare Degrees:

 Numerator’s degree = 1; Denominator’s degree = 2.

1. Identify End Behavior:

 Denominator dominates; hence, as x \to \pm \infty, h(x) approaches 0.

End Behavior of Rational Functions: The Horizontal and Slant Asymptotes

• Horizontal Asymptotes: When degrees are equal, the function settles around a constant value, defining its horizontal asymptote. For instance, in f(x) = \frac{a_nx^n}{b_nx^n}, the horizontal asymptote is y = \frac{a_n}{b_n}.
• Slant Asymptotes: These occur when the numerator’s degree is exactly one higher than the denominator’s, like y = mx + b. Use polynomial long division to find the slant asymptote.

Incorporating these concepts when sketching graphs makes understanding and predicting function behaviors more intuitive.

Quick Reference Chart

Conclusion

Spotting the end behavior of rational functions requires attention to the degrees and leading terms of the polynomials involved. Being able to do this effectively will not only help in graphing functions but will also deepen an understanding of how different functions act. Practice is the key to mastering these concepts. Start with simpler functions and gradually move on to more complex ones.

Sharpen Your Skills for AP® Precalculus

Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

Need help preparing for your AP® Precalculus exam?

Albert has hundreds of AP® Precalculus practice questions, free response, and an AP® Precalculus practice test to try out.