#AP Pre-Calculus: Linear & Exponential Functions - The Night Before 🚀

Hey there! Let's get you prepped for the exam. We're going to break down linear and exponential functions, focusing on what you really need to know. Let's make sure you're feeling confident and ready to ace this thing! 💪

#2.2 Change in Linear and Exponential Functions

#Arithmetic Sequence Lookalikes

Let's start with the basics: linear functions and how they relate to arithmetic sequences. Remember, arithmetic sequences are like a set of discrete points, while linear functions connect those points into a smooth line. Let's dive in!

#Linear Functions

A linear function has the form: $f(x) = b + mx$

• $b$ = y-intercept (where the line crosses the y-axis)

• $m$ = slope (rate of change, steepness, and direction)

  • Positive slope: line goes up ↗️
  • Negative slope: line goes down ↘️
  • Zero slope: horizontal line ↔️

#Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. It's defined as:

$a_n = a_0 + dn$

• $a_0$ = first term
• $d$ = common difference

*Image Courtesy of Math and Multimedia*

They're both great for modeling real-world scenarios with constant change, like population growth or simple interest. 📈

#1️⃣ Similar Ways of Expression, Part 1

Just like arithmetic sequences, linear functions can be expressed with a known value and a constant rate of change. Here's how they match up:

• Arithmetic Sequence: $a_n = a_k + d(n-k)$ ($a_k$ is a known term, $d$ is the common difference)
• Linear Function (Point-Slope Form): $f(x) = y_i + m(x - x_i)$ ($(x_i, y_i)$ is a known point, $m$ is the slope)