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)

Geometric Sequence Lookalikes

Now, let's shift gears to exponential functions and their relationship with geometric sequences. Get ready for some growth! 🚀

Exponential Functions

An exponential function has the form: $f(x) = ab^x$

• $a$ = initial value
• $b$ = base (a positive number not equal to 1)

This function shows exponential growth (if $b > 1$) or decay (if 0 < b < 1).

Geometric Sequences

A geometric sequence is a list of numbers with a constant ratio between consecutive terms. It's defined as:

$g_n = g_0 * r^n$

• $g_0$ = first term
• $r$ = common ratio

*Image Courtesy of Quizlet, W. H. Freeman & Company*

They're both used to model exponential growth or decay, like compound interest or radioactive decay. ☢️

2️⃣ Similar Ways of Expression, Part 2

Like geometric sequences, exponential functions can be expressed using a known value and a constant ratio. Here's the breakdown:

• Geometric Sequence: $g_n = g_k * r^(n-k)$ ($g_k$ is a known term, $r$ is the common ratio)
• Exponential Function: $f(x) = y_i * r^(x-x_i)$ ($(x_i, y_i)$ is a known point, $r$ is the ratio)

⚠️ Caveats of Sequence vs. Function: Domains!

For example, the sequence $a_n = 2n$ has a domain of positive integers, but the function $f(x) = 2x$ has a domain of all real numbers.

*Image Courtesy of Jed Q on [Desmos](https://www.desmos.com/calculator)*

Sequences can also be defined recursively, which may further restrict their domain. Remember that functions can be defined more broadly using explicit formulas. 🌎

Similarities and Differences

Let's recap the key similarities and differences between linear and exponential functions:

1️⃣ Rate of Change:

• Linear: Output values change at a constant rate (constant slope). 📏
• Exponential: Output values change proportionally (constant ratio). 📈

2️⃣ Analytical Expression:

• Both linear ($f(x) = b + mx$) and exponential ($f(x) = ab^x$) functions can be expressed with an initial value and a constant involved with change.
  • Linear: Initial value = y-intercept ($b$), constant change = slope ($m$).
  • Exponential: Initial value = $a$, constant change = base ($b$).
  • Key Difference: Linear functions use addition, while exponential functions use multiplication. ➕✖️

3️⃣ Determined by Two Values:

• Both linear and exponential functions (and their sequence counterparts) can be determined by two distinct values:
  • Arithmetic: first term and common difference.
  • Linear: a point on the line and the slope.
  • Geometric: first term and common ratio.
  • Exponential: a point on the function and the base.

Final Exam Focus

Okay, here's the lowdown on what to focus on for the exam:

• High-Priority Topics:

  • Identifying linear vs. exponential growth/decay.
  • Writing equations for linear and exponential functions given different conditions (points, slope, etc.).
  • Understanding the domains of sequences and functions.
  • Applying these concepts to real-world problems.

• Common Question Types:

  • Multiple-choice questions testing your understanding of slope, y-intercept, base, and ratio.
  • Free-response questions requiring you to write equations and interpret graphs.
  • Questions that combine linear and exponential concepts.

• 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 domain restrictions and the difference between addition and multiplication.
  • Strategies: Always double-check your work, especially when dealing with calculations. If you're unsure, try plugging in values to check your equations.

Practice Questions

Okay, let's test your knowledge with some practice questions! These are designed to mimic what you might see on the AP exam. Let's do this! 💯

Practice Question

Multiple Choice Questions

1. A population of bacteria doubles every hour. If the initial population is 500, which of the following functions models the population $P(t)$ after $t$ hours? (A) $P(t) = 500 + 2t$ (B) $P(t) = 500(2)^t$ (C) $P(t) = 1000t$ (D) $P(t) = 500(t^2)$

2. A line passes through the points (1, 5) and (3, 13). What is the equation of the line in point-slope form? (A) $y - 5 = 4(x - 1)$ (B) $y - 1 = 4(x - 5)$ (C) $y - 5 = 4(x + 1)$ (D) $y - 1 = 4(x + 5)$

Free Response Question

A car is purchased for $25,000. The value of the car depreciates by 15% each year.

(a) Write a function $V(t)$ that models the value of the car after $t$ years. (b) What is the value of the car after 5 years? (c) After how many years will the car be worth less than $10,000?

Scoring Rubric

(a) 2 points * 1 point for correctly identifying the initial value (25000). * 1 point for correctly identifying the decay factor (0.85). * Correct Answer: $V(t) = 25000(0.85)^t$

(b) 2 points * 1 point for correctly substituting t=5 into the function. * 1 point for correct calculation. * Correct Answer: $V(5) = 25000(0.85)^5 = 11092.63$

(c) 3 points * 1 point for setting up the inequality 25000(0.85)^t < 10000. * 1 point for using logarithms or guess and check to solve for t. * 1 point for correct answer with proper rounding. * Correct Answer: $t > 5.64$, so after 6 years.

You've got this! Remember to stay calm, take deep breaths, and trust in your preparation. You're ready to rock this exam! 🎉