Rucete ✏ AP Precalculus In a Nutshell
1. Rates of Change
This chapter introduces the foundational ideas of functions, their graphs, domains, intercepts, and how rates of change help describe how quantities vary in relation to one another. It prepares students to analyze both linear and nonlinear functions, both visually and algebraically.
- Relations vs. Functions
- A relation links inputs (domain) with outputs (range); not every input must have one output.
- A function assigns exactly one output to each input.
- Functions can be represented with equations, tables, graphs, and ordered pairs.
- Example: Taxi fare equation → y = 2.00 + 1.50x, where x = time and y = cost.
- Function Notation and Evaluation
- Standard notation: f(x), where x is input and f(x) is output.
Examples:
- f(x) = 10x − 2 → f(3) = 28
- g = {(−1, 1), (0, 3), …} → g(0) = 3
- v(t) = t² + 5 → v(c + 2) = (c + 2)² + 5
- Graphs of Functions
- Plotted on xy-plane; x is independent, y is dependent.
- Continuous functions can be graphed with connected points.
- A function’s domain and range can be visually interpreted from its graph.
- Closed/Open endpoints determine if boundary values are included.
- Algebraic Domain Rules
Exclude x-values that make:
Denominators = 0
Even roots of negative numbers
- Intercepts and Zeros
- x-intercepts (zeros): where graph touches x-axis → f(x) = 0
- y-intercept: where x = 0 → calculate f(0)
- A function can have multiple x-intercepts, but only one y-intercept
- Vertical Line Test
- A graph represents a function if no vertical line touches it in more than one point.
- Increasing and Decreasing Intervals
- Increasing: as x increases, y increases
- Decreasing: as x increases, y decreases
- A graph can contain both
- Concavity
- Concave up: graph bends upwards, rate of change increases
- Concave down: graph bends downwards, rate of change decreases
- Average Rate of Change
- Formula: (change in y) / (change in x)
- Equivalent to slope between two points (secant line)
- Used to describe how one quantity changes relative to another
- Positive rate → both values increase or decrease together
- Negative rate → one increases while the other decreases
- Instantaneous Rate (Intro)
- Approximated by using smaller intervals around a point
- Tangent line slope approximated with nearby secant slopes
- Linear Functions
- Have constant rate of change
- Slope = average rate of change = coefficient of x in y = mx + b
- Graph is a straight line → neither concave up nor down
- Quadratic Functions
- Form: y = ax² + bx + c
- Average rate of change is not constant
- Rate of change itself changes at a constant rate → graph is parabola
- Concave up if a > 0, concave down if a < 0
- Examples Using Tables and Graphs
- Compare tables for constant rate (linear) vs changing rate (nonlinear)
- Determine concavity from pattern in rate of change
In a Nutshell
Functions describe how two quantities relate. Linear functions have constant rates of change and appear as straight lines. Quadratic functions have changing rates of change, resulting in curved graphs (parabolas). Rates of change help analyze how quickly values grow or shrink and whether graphs bend up or down. Mastering these ideas is essential before diving into deeper calculus.