1.0 Unit 1 Overview: Polynomial and Rational Functions

Welcome to Fiveable’s AP Precalculus study guides! 🥳

Before we even jump into the content itself, let’s look at the big, big, big picture of this course:

🧐 What’s the Point of AP Pre-Calculus?

AP Precalculus is all about understanding how functions work in real-world situations. It's designed to help you get ready for college-level math and other classes that use math. In this class, we're going to look at a lot of different types of functions that are important for careers in math, science, and other fields. Regardless of whether this is your last math class before college or you’re looking for a refresher of advanced algebra, we’ll make sure you have a really solid understanding of what you’ll be learning! 🌎

We're going to spend a lot of time working on real-world problems and using math tools to solve them. This is called "modeling," allowing us to understand functions better by looking at different examples. We'll also look at functions in different ways, like graphically and numerically, so you can get a good sense of how they work.

Throughout the class, we'll work on math skills like manipulating equations and solving problems. Similarly, you'll learn how different types of functions and their different parts, like inverses and transformations, can be understood in different ways.

One of the main things we're going to focus on is understanding what functions are and how they work. A function is just a way to show how different inputs (or "domain") are related to different outputs (or "range"). There are different ways to classify functions based on how they work. We're going to look at how functions change over time and how different variables are related. Research shows that understanding functions like this is the best way to prepare for calculus.

We'll also spend time working with data and making models to better understand the functions. We'll learn how to use the models to understand different kinds of information, like how different variables change over time. And we'll learn that all models have their own limitations and assumptions. By looking at functions from different perspectives, you'll develop a deeper understanding of not only the specific types of functions we're studying, but functions in general. This will help you in this class and in other classes in the future.

Phew! Sounds like a lot, huh? Don’t worry—you’ll learn these concepts in the span of four units throughout the year. Let’s start with Unit 1! ⭐

🔊 Unit Breakdown

AP Precalculus Unit 1 covers the topics of polynomial and rational functions:

A polynomial function involves a sum of powers in one or more variables. The powers are non-negative integers and the coefficients can be any real number. The degree of a polynomial is the highest power of the variable in the polynomial. 👊

A polynomial function can be represented in the form of $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where n is the degree of the polynomial, x is the variable, and $a_n$, $a_{n-1}$, ..., $a_1$, $a_0$ are the coefficients.

A rational function is a function that can be defined as the ratio of two polynomial functions. It is represented by a polynomial function in the numerator and a non-zero polynomial function in the denominator.

Image Courtesy of Wikiversity

A rational function can be represented in the form of $f(x) = p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomial functions. Rational functions have certain behavior patterns like vertical asymptotes, horizontal asymptotes, and sometimes holes.

Tandem and Rates of Change

The unit starts with learning about the relationship between two or more functions and how changes in one function can affect the other. Then, it moves on to Rates of Change, where you’ll learn about the rate at which a function is changing and how to calculate it.

This is followed by Rates of Change in Linear and Quadratic Functions, where you’ll learn how to calculate the rate of change for linear and quadratic functions and how it relates to their graph. 📈

Polynomial Functions

The unit then covers the properties of polynomial and rational functions.

In Polynomial Functions and Rates of Change, you’ll learn how to calculate the rate of change for polynomial functions, including the use of power functions and the power rule.

In Polynomial Functions and Complex Zeros, students learn about the concept of complex zeros and how they relate to the behavior of polynomial functions. 🕵️‍♀️

Image Courtesy of Mrs. Hahn

The section also covers Polynomial Functions and End Behavior, where students learn how to determine the leading coefficient and degree of a polynomial function.

Rational Functions

In Rational Functions and End Behavior, students learn about rational functions and how they are different from polynomial functions.

In Rational Functions and Zeros, students learn how to find the zeros of rational functions, including the use of the Factor Theorem and long division.

The section also covers Rational Functions and Vertical Asymptotes, where you’ll learn about the concept of vertical asymptotes and how they relate to the behavior of rational functions.

Finally, the unit goes over Rational Functions and Holes, where you’ll learn about holes in rational functions and how they relate to the behavior of the function. ⛳

Image Courtesy of Sciencing

🎁 Wrapping Up Unit 1 Overview

Overall, Unit 1 provides a thorough understanding of polynomial and rational functions and their properties, including rates of change, zeros, end behavior, and transformations. It also covers the process of selecting and constructing appropriate function models and applying them to real-world situations. Woo-hoo! 😊