#AP Pre-Calculus: Rates of Change - The Night Before 🚀

Hey! Let's get you totally prepped for the exam. We're going to break down rates of change in linear and quadratic functions, making sure everything clicks. No stress, just clear understanding.

#1.3 Rates of Change in Linear and Quadratic Functions

#📈 Average Rates of Change

• Linear Functions: The average rate of change is the same everywhere. It's the slope of the line!
• Quadratic Functions: The average rate of change isn't constant. It changes as you move along the curve.

The average rate of change over an interval [a, b] is the slope of the secant line connecting points (a, f(a)) and (b, f(b)).

Caption: A quadratic function's curve. Notice how the steepness changes.

Caption: Secant line (connecting two points) vs. Tangent line (touching one point).

#📐 Change in Average Rates of Change

• Linear Functions: The average rate of change is constant, so it doesn't change. It's like cruise control – steady speed.
• Quadratic Functions: The average rate of change changes, but it changes at a constant rate. This means the function is either accelerating or decelerating consistently.

### 📝 A Note on Concavity

• Concave Up: The function is increasing at an increasing rate (like a smile 😊). The average rate of change is increasing.
• Concave Down: The function is increasing at a decreasing rate (like a frown 🙁). The average rate of change is decreasing.

#Final Exam Focus

• High-Priority Topics:
  • Understanding the difference between linear and quadratic rates of change.
  • Calculating average rates of change over intervals.
  • Interpreting concavity and its relation to rate of change.
• Common Question Types:
  • Multiple-choice questions asking about the rate of change of a function given a graph or equation.
  • Free response questions requiring you to calculate and interpret average rates of change.
  • Questions that combine rate of change with other concepts (e.g., finding the vertex of a quadratic).
• Last-Minute Tips:
  • Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
  • Common Pitfalls: Be careful with negative signs and fractions. Double-check your calculations.
  • Strategies: Always show your work, even if it seems obvious. This can earn you partial credit.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. The average rate of change of the function $f(x) = 2x^2 - 3x + 1$ over the interval [1, 3] is: (A) 5 (B) 7 (C) 8 (D) 10

2. A function is concave down over an interval. Which of the following is true about the average rate of change over equal length intervals? (A) It is constant. (B) It is increasing. (C) It is decreasing. (D) It is sometimes increasing and sometimes decreasing.

#Free Response Question

The function $g(x) = -x^2 + 4x + 2$ represents the height of a ball (in meters) at time x (in seconds).

(a) Calculate the average rate of change of the ball's height between x = 1 second and x = 3 seconds. [2 points] (b) Is the ball accelerating or decelerating over the interval [1,3]? Explain your answer. [2 points] (c) Determine the concavity of the graph of $g(x)$ and explain what it means in the context of the problem. [2 points]

#Answer Key

Multiple Choice:

1. (B)
2. (C)

Free Response:

(a) $g(1) = -(1)^2 + 4(1) + 2 = 5$ and $g(3) = -(3)^2 + 4(3) + 2 = 5$. Average rate of change = $frac{g(3)-g(1)}{3-1} = frac{5-5}{2} = 0$. * 1 point for correct evaluation of g(1) and g(3) * 1 point for correct calculation of average rate of change

(b) The ball is neither accelerating nor decelerating over the interval [1,3]. The average rate of change is 0, indicating that the ball's height is not changing on average over this interval. * 1 point for stating neither accelerating nor decelerating * 1 point for correct explanation

(c) The concavity of the graph of $g(x)$ is concave down. This means that the ball is increasing at a decreasing rate. * 1 point for stating concave down * 1 point for correct explanation

You've got this! Go ace that exam! 💪