#AP Pre-Calculus: Unit 1 Study Guide - Polynomial and Rational Functions

Welcome to your ultimate review for AP Pre-Calculus Unit 1! This guide is designed to help you feel confident and fully prepared for your exam. Let's dive in! 🚀

#🧐 What's the Point of AP Pre-Calculus?

AP Pre-Calculus is all about understanding functions and how they model real-world scenarios. This course is designed to prepare you for college-level math and beyond. We'll explore functions through modeling, graphical analysis, and numerical methods. You'll learn to manipulate equations, understand inverses, and apply transformations. The focus is on understanding the relationship between inputs (domain) and outputs (range). This approach will set you up for success in calculus and other STEM fields. 🌎

#🔊 Unit 1 Breakdown: Polynomial and Rational Functions

Unit 1 focuses on polynomial and rational functions. Let's break down what each of these entails:

#Polynomial Functions

A polynomial function is a sum of terms, each consisting of a coefficient and a variable raised to a non-negative integer power. The degree of the polynomial is the highest power of the variable.

• Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

#Rational Functions

A rational function is a ratio of two polynomial functions.

• Form: $f(x) = p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomials.
• Key features include vertical asymptotes, horizontal asymptotes, and sometimes holes.

Image Courtesy of Wikiversity

#Tandem and Rates of Change

This section focuses on how changes in one function affect another and introduces the concept of rate of change.

• Rates of Change: How quickly a function's output changes with respect to its input.
• Linear Functions: Constant rate of change (slope). 📈
• Quadratic Functions: Rate of change varies; it's not constant.

#Polynomial Functions

• Polynomial Functions and Rates of Change: Calculating rates of change using power functions and the power rule.
• Polynomial Functions and Complex Zeros: Understanding how complex zeros affect the behavior of polynomial functions. 🕵️‍♀️

Image Courtesy of Mrs. Hahn

• Polynomial Functions and End Behavior: Determining the leading coefficient and degree to understand the function's end behavior.

#Rational Functions

• Rational Functions and End Behavior: Understanding how rational functions differ from polynomial functions, especially in terms of asymptotes.
• Rational Functions and Zeros: Finding zeros using the Factor Theorem and long division.
• Rational Functions and Vertical Asymptotes: Identifying vertical asymptotes and their impact on the function's behavior.
• Rational Functions and Holes: Recognizing holes in rational functions and their implications. ⛳

Image Courtesy of Sciencing

#Final Exam Focus

Here's what to prioritize for the exam:

• Rates of Change: Master calculating and interpreting rates of change for linear, quadratic, and polynomial functions.
• Polynomial Functions: Focus on end behavior, zeros (real and complex), and the relationship between factors and roots.
• Rational Functions: Understand asymptotes (vertical and horizontal), holes, and how to sketch the graph of a rational function.

#Last-Minute Tips

• Time Management: Don't spend too long on a single question. If you're stuck, move on and come back later.
• Common Pitfalls: Watch out for sign errors and incorrect application of formulas.
• Challenging Questions: Practice combining concepts from different units. AP questions often test multiple skills at once.
• Show Your Work: Always show your steps for partial credit, especially on FRQs.

#Practice Questions

Practice Question

Multiple Choice Questions

1. What is the end behavior of the polynomial function $f(x) = -3x^4 + 2x^3 - x + 5$? (A) As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$ (B) As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$ (C) As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$ (D) As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$ and as $x \rightarrow -\infty$, $f(x) \rightarrow \infty$

2. Which of the following rational functions has a vertical asymptote at $x = 2$ and a hole at $x = -1$? (A) $f(x) = \frac{x-2}{(x-2)(x+1)}$ (B) $f(x) = \frac{(x-2)(x+1)}{(x-2)(x+1)}$ (C) $f(x) = \frac{(x+1)}{(x-2)(x+1)}$ (D) $f(x) = \frac{(x-2)}{(x-2)(x-1)}$

3. What is the average rate of change of the function $f(x) = x^2 + 2x$ over the interval $[1, 3]$? (A) 4 (B) 6 (C) 8 (D) 10

Free Response Question

Consider the rational function $f(x) = \frac{x^2 - 4}{x^2 - x - 2}$.

(a) Find the zeros of the function. (b) Find the vertical asymptotes of the function. (c) Find the horizontal asymptote of the function. (d) Identify any holes in the graph of the function. (e) Sketch a graph of the function, labeling all key features.

Scoring Breakdown

(a) Zeros: $x^2 - 4 = 0 \implies x = \pm 2$ (1 point for each correct zero) (b) Vertical Asymptotes: $x^2 - x - 2 = (x-2)(x+1) = 0 \implies x = 2, x = -1$. (1 point for each correct vertical asymptote) (c) Horizontal Asymptote: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is $y = \frac{1}{1} = 1$ (1 point for correct horizontal asymptote) (d) Holes: $f(x) = \frac{(x-2)(x+2)}{(x-2)(x+1)}$. There is a hole at $x = 2$ (1 point for identifying the hole) (e) Sketch: (1 point for correct shape, 1 point for labeling zeros, asymptotes, and holes)

Answers

Multiple Choice

1. (B)
2. (C)
3. (B)

Free Response (a) $x = -2, 2$ (b) $x = -1$ (c) $y = 1$ (d) Hole at $x = 2$ (e) Graph should show zeros at -2, vertical asymptote at x=-1, horizontal asymptote at y=1, and a hole at x=2.

Good luck with your exam! You've got this! 👍