AP Precalculus Course and Exam Description Study Notes

Core Principles and Academic Integrity of the Advanced Placement Program

Clarity and Transparency: The AP Program maintains public course frameworks and sample assessments to ensure teachers and students have clear expectations for classroom work.
Encounter with Evidence: AP courses emphasize independent thinking, the scientific method, and drawing conclusions based on evidence.
Opposition to Censorship: AP protects intellectual freedom; if a school bans required topics, the AP designation is removed (e.g., evolution is essential to AP Biology).
Opposition to Indoctrination: Students are expected to analyze multiple perspectives. No points are awarded for agreeing with a specific viewpoint, and students are not asked to subscribe to specific cultural or political values.
Open-Mindedness: Grounded in primary sources, courses require the study of diverse nationalities, cultures, religions, and ethnicities.
Respect and Debate: Classrooms are expected to cultivate respectful debate of ideas while prohibiting personal attacks. Students are encouraged to evaluate arguments, not one another.
Informed Choice: Enrolling in an AP course is a choice representing agreement between parents, students, and educators to adhere to these college-level principles.

AP Precalculus Course Framework and Mathematical Practices

Course Structure: The course is divided into four units. Units 1, 2, and 3 are required for the AP Exam, while Unit 4 is elective based on local requirements.
Practice 1: Procedural and Symbolic Fluency: * 1.A: Solve equations and inequalities represented analytically, with and without technology. * 1.B: Express functions, equations, or expressions in analytically equivalent forms useful for specific contexts. * 1.C: Construct new functions using transformations, compositions, inverses, or regressions.
Practice 2: Multiple Representations: * 2.A: Identify information from graphical, numerical, analytical, and verbal representations. * 2.B: Construct equivalent representations (graphical, numerical, analytical, and verbal).
Practice 3: Communication and Reasoning: * 3.A: Describe function characteristics with varying levels of precision. * 3.B: Apply numerical results in mathematical or applied contexts. * 3.C: Support conclusions or choices with a logical rationale or appropriate data.

Unit 1: Polynomial and Rational Functions

Change in Tandem (1.1): * Function definition: A relation mapping a domain (input) to a range (output) where each input is uniquely mapped to one output. * Increasing/Decreasing: A function is increasing on an interval if for all $a < b$ , $f(a) < f(b)$ ( $f(a) > f(b)$ for decreasing). * Concavity: Concave up occurs when the rate of change is increasing; concave down occurs when the rate of change is decreasing.
Rates of Change (1.2–1.3): * Average Rate of Change: The ratio of output change to input change over an interval: $\text{AROC} = \frac{f(b)-f(a)}{b-a}$ . It represents the slope of the secant line. * Linear vs. Quadratic: Linear functions have a constant rate of change. Quadratic functions have average rates of change over equal intervals that form a linear progression.
Polynomial Characteristics (1.4–1.6): * General Form: $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$ . * Zeros and Multiplicity: If $(x - a)^n$ is a repeated linear factor, the zero $a$ has multiplicity $n$ . Even multiplicity causes the graph to be tangent (bounce) at the x-axis. * Complex Conjugates: If $a + bi$ is a non-real zero, then $a - bi$ is also a zero. * End Behavior: Determined by the leading term ( $a_n x^n$ ). Notation includes $\lim_{x \to \infty} p(x)$ and $\lim_{x \to -\infty} p(x)$ .
Rational Functions (1.7–1.11): * End Behavior: Analyzed by the ratio of leading terms. If degrees are equal, a horizontal asymptote exists at the ratio of coefficients. If the numerator degree is one higher, a slant asymptote exists. * Asymptotes vs. Holes: A vertical asymptote at $x = a$ occurs if the denominator's zero multiplicity exceeds the numerator's. A hole occurs at $x = c$ if the numerator's zero multiplicity is greater than or equal to the denominator's. * Function Modeling: Rational functions model inversely proportional quantities (e.g., gravitational force vs. squared distance).

Unit 2: Exponential and Logarithmic Functions

Sequences and Comparisons (2.1–2.2): * Arithmetic Sequence: $a_n = a_0 + d n$ (constant difference). * Geometric Sequence: $g_n = g_0 r^n$ (constant ratio). * Linear vs. Exponential: Linear involves repeated addition; Exponential involves repeated multiplication of a growth factor.
Exponential Functions (2.3–2.5): * General Form: $f(x) = a b^x$ , where $a \neq 0$ and b > 0, b \neq 1. * Asymptotic Behavior: General form exponential functions have a horizontal asymptote at $y = 0$ . * Natural Base: The constant $e \approx 2.718$ .
Functional Composition and Inverses (2.7–2.8): * Composition: $(f \circ g)(x) = f(g(x))$ . The range of $g$ must fall within the domain of $f$ . Composition is non-commutative. * Identity Function: $f(x) = x$ ; serves as the identity for composition: $f(g(x)) = g(f(x)) = g(x)$ . * Inverses: $f(f^{-1}(x)) = x$ . Inverse graphs are reflections over $y = x$ .
Logarithmic Functions (2.9–2.12): * Definition: $\log_b c = a \iff b^a = c$ . * Properties: $\log_b(xy) = \log_b x + \log_b y$ , $\log_b(x^n) = n \log_b x$ , $\log_b x = \frac{\log_a x}{\log_a b}$ . * Asymmetry: Logarithmic functions have vertical asymptotes at $x = 0$ and are inverses of exponential functions.
Semi-Log Plots (2.15): * Scaling: When the y-axis is logarithmically scaled, exponential data appears linear: $y = (\log b) x + \log a$ .

Unit 3: Trigonometric and Polar Functions

Periodic Phenomena (3.1): * Definition: A function is periodic if $f(x + k) = f(x)$ for a positive constant period $k$ .
The Unit Circle and Basic Trig (3.2–3.4): * Coordinates: On a unit circle, $(\cos \theta, \sin \theta)$ . On a circle of radius $r$ , $(r \cos \theta, r \sin \theta)$ . * Tangent: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ , representing the slope of the terminal ray. * Radians: Ratio of arc length to radius ( $s = r \theta$ ).
Sinusoidal Functions (3.5–3.7): * General Form: $f(\theta) = a \sin(b(\theta + c)) + d$ or $g(\theta) = a \cos(b(\theta + c)) + d$ . * Parameters: Amplitude ( $|a|$ ), Period ( $\frac{2\pi}{|b|}$ ), Vertical Shift ( $d$ ), Phase Shift ( $-c$ ).
Other Trig Functions (3.8–3.11): * Identities: $\sin^2 \theta + \cos^2 \theta = 1$ , $1 + \tan^2 \theta = \sec^2 \theta$ . * Asymptotes: $\tan \theta$ and $\sec \theta$ have asymptotes where $\cos \theta = 0$ ( $\theta = \frac{\pi}{2} + k\pi$ ). $\csc \theta$ and $\cot \theta$ have asymptotes where $\sin \theta = 0$ ( $\theta = k\pi$ ).
Polar Coordinates (3.13–3.15): * Ordered Pairs: $(r, \theta)$ . * Conversion: $x = r \cos \theta$ , $y = r \sin \theta$ , $r^2 = x^2 + y^2$ , $\theta = \arctan(\frac{y}{x})$ . * Rate of Change: If $r = f(\theta)$ is positive and increasing, the point moves farther from the origin.

Unit 4: Parameters, Vectors, and Matrices (Elective)

Parametric Functions (4.1–4.4): * Defined by $x(t)$ and $y(t)$ . Models planar motion where $t$ is the parameter. * Horizontal direction involves $x'(t)$ , vertical involves $y'(t)$ .
Conic Sections (4.5–4.7): * Analytic Forms: Parabola ( $y - k = a(x - h)^2$ ), Ellipse ( $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ ), Hyperbola ( $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ ).
Vectors (4.8–4.9): * Components: $\langle a, b \rangle$ . Magnitude: $|v| = \sqrt{a^2 + b^2}$ . Dot Product: $\mathbf{a} \cdot \mathbf{b} = a_1a_2 + b_1b_2$ . * Geometry: $\mathbf{a} \cdot \mathbf{b} = |a||b| \cos \theta$ . Orthogonal if dot product equals zero.
Matrices (4.10–4.14): * Determinant of $2 \times 2$ matrix $A = \left(\begin{smallmatrix} a & b \ c & d \end{smallmatrix}\right)$ : $\text{det}(A) = ad - bc$ . * Inverse: Exists if $\text{det}(A) \neq 0$ . * Linear Transformations: Multiplying transformation matrix $A$ by vector $\mathbf{v}$ results in output vector $\mathbf{A}\mathbf{v}$ . * Rotation Matrix: $\left(\begin{smallmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{smallmatrix}\right)$ .

Exam Structure and Scoring Guidelines

Section I: Multiple Choice (62.5%): * Part A: 28 questions, 80 minutes, no calculator. * Part B: 12 questions, 40 minutes, graphing calculator required.
Section II: Free Response (37.5%): * Part A: 2 questions, 30 minutes, graphing calculator required. (Topics: Function Concepts and Modeling Non-Periodic Contexts). * Part B: 2 questions, 30 minutes, no calculator. (Topics: Modeling Periodic Contexts and Symbolic Manipulations).
Formatting/Accuracy: All decimal approximations must be accurate to three decimal places. Radians must be the default mode for all trig calculations.
Task Verbs: * Construct: Develop an analytical representation. * Describe: Develop a verbal representation. * Interpret: Relate math to contextual meaning (include units). * Solve: Find solutions to equations/inequalities.