Rational Functions and Zeros

Alice White

6 min read

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Study Guide Overview

This study guide covers rational functions and their zeros. Key topics include finding real zeros by factoring the numerator and checking the domain, analyzing the role of zeros as endpoints and asymptotes, and using interval analysis to solve inequalities involving rational functions. The guide also provides practice questions and exam tips focusing on time management and common pitfalls like forgetting to check the domain.

#AP Pre-Calculus Study Guide: Rational Functions and Zeros

Hey there, future AP superstar! Let's break down rational functions and their zeros. This guide is designed to be your go-to resource the night before the exam. Let's get started! 💪

#1.8 Rational Functions and Zeros

# 😎 Real Zeros of Rational Functions

Key Concept

The real zeros of a rational function are the same as the real zeros of its numerator, but only if those zeros are in the domain. Remember, a rational function is a fraction of two polynomials. 🌳

Think of it this way: a fraction is zero only when its top part (numerator) is zero. The bottom part (denominator) just tells us where the function is defined.

Steps to Find Real Zeros:

Factor: Factor both the numerator and the denominator.
Domain Check: Identify and exclude values where the denominator equals zero. These are not in the domain of the function.
Simplify: Cancel out common factors if possible.
Numerator Zeros: Set the simplified numerator equal to zero and solve for x. These are your real zeros! 0️⃣

Example:

Let's look at the function:

$r(x) = \frac{x^2 - 4}{x - 2}$

Factor: $r(x) = \frac{(x - 2)(x + 2)}{x - 2}$
Domain Check: x cannot be 2, because the denominator would be zero.
Simplify: $r(x) = x + 2, \quad x \ne 2$
Numerator Zeros: Set x + 2 = 0. This gives us x = -2. So, the real zero is -2. 🎉

Memory Aid

Numerator = Zeros: Rememb...

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