#AP Pre-Calculus: Function Modeling - Your Night-Before Guide 🚀

Hey there! Let's get you totally prepped for your exam. This guide is designed to be your quick, go-to resource, focusing on the most important stuff and making sure you feel confident and ready. Let’s dive in!

#1.13 Function Model Selection and Assumption Articulation

#What is a Function Model? 🔗

A function model is a mathematical representation of a real-world situation. Think of it as a simplified version of reality that helps us understand and predict outcomes. It’s all about finding the right function (linear, quadratic, etc.) that fits the data or scenario. 🌎

!famfunc1.jpg

Caption: A curved graph showing a typical function model, like the path of a ball.

#Linear Functions 🔗

#Basics

Linear functions are your go-to for situations with a constant rate of change. They follow the form y = mx + b, where:

• m is the slope (rate of change)
• b is the y-intercept (starting point)

#Example: Farmer's Crops 🧑‍🌾

A farmer tracks their earnings based on acres planted:

• Acres (x): 0, 10, 20, 30, 40
• Earnings (y): 0, 800, 1600, 2400, $3200

To create a linear model:

1. Find the slope (m): Using points (0, 0) and (10, 800): $$m = \frac{800 - 0}{10 - 0} = 80$$
2. Find the y-intercept (b): In this case, it's 0 since the line passes through the origin.

So, the model is y = 80x. If the farmer plants 50 acres, the model predicts earnings of y = 80(50) = $4,000. 💰

#Quadratic Functions 🔗

#Basics

Quadratic functions are used for situations with a changing rate of change or when you see a symmetrical shape w...