Overview
This lecture provides a comprehensive review of AP Pre-Calculus Units 1-3 (excluding Unit 4), summarizing key concepts, function types, transformations, trigonometry, and introduction to polar coordinates. The focus is on understanding function behavior, modeling, and the foundational tools needed for success on the AP exam.
Functions and Rates of Change
- A function maps each input (domain, x) to exactly one output (range, y).
- Increasing functions have outputs that rise as inputs rise; decreasing functions have outputs that fall as inputs rise.
- The slope of a graph represents its rate of change.
- Concave up graphs have increasing slopes (U-shaped); concave down have decreasing slopes (∩-shaped).
- The equation y = mx + b represents a line, where m is the slope and b is the y-intercept.
- Average rate of change between two points: (y₂ - y₁) / (x₂ - x₁).
- In linear functions, rate of change is constant; in quadratics, it varies linearly.
Polynomial and Rational Functions
- A polynomial has only non-negative integer exponents and no division or imaginary coefficients.
- Degree of a polynomial is its highest exponent; number of zeros equals the degree.
- Zeros, roots, and x-intercepts are values where y=0; real zeros are seen on the graph, complex include i.
- Odd multiplicity: graph passes through zero; even multiplicity: graph bounces off zero.
- Even functions: symmetric about the y-axis; odd: symmetric about the origin.
- End behavior of polynomials and rationals depends on degree and leading coefficient.
- Rational functions are the ratio of polynomials and can have vertical (undefined points) and horizontal/slant asymptotes.
- Real zeros of rationals come from numerator; holes occur where numerator and denominator share factors.
Function Transformations and Modeling
- Vertical/horizontal shifts and dilations change function graphs: f(x) + k (up/down), f(x + h) (left/right), af(x) (vertical stretch), f(bx) (horizontal stretch).
- Practice identifying transformations in equations.
- Use regression and R values on calculators to find best-fit models (linear, quadratic, cubic, quartic, exponential, log).
- Residuals measure model error; random residuals mean a good model fit.
Exponential and Logarithmic Functions
- Arithmetic sequences add a constant (linear); geometric sequences multiply (exponential).
- Exponential functions: f(x) = ab^x, with a > 0, b > 0, b ≠ 1; growth if b > 1, decay if 0 < b < 1.
- Exponent properties: product, power, negative exponents, and roots.
- e (~2.718) is the base for continuous growth: f(x) = ae^x.
- Logarithms are inverses of exponentials: log_b(c) = a means b^a = c; domain is x > 0.
- Key log properties: product, quotient, power, and natural log ln(x).
- Change-of-base formula: log_b(x) = log_k(x) / log_k(b).
- Model log functions with input/output pairs and systems of equations.
- Semi-log plots help identify exponential vs. log models.
Trigonometric and Polar Functions
- Periodic functions repeat patterns at regular intervals (period = cycle length).
- Unit circle defines sine (y-coordinate), cosine (x-coordinate), tangent (y/x); radians: full circle = 2π.
- Special triangles (30-60-90, 45-45-90) help compute exact trig values.
- Sine is odd, cosine is even; both have amplitude (height) and midline.
- Sinusoidal functions: f(θ) = a sin(b(θ - c)) + d; period = 2π/b, amplitude = |a|, phase shift = c, midline = d.
- Tangent has vertical asymptotes; period is π.
- Inverse trig functions have restricted domains to be functions.
- Reciprocal identities: cosecant = 1/sin, secant = 1/cos, cotangent = 1/tan.
- Pythagorean identities, sum/difference, and double-angle identities are essential for simplifying trig expressions.
- Polar coordinates: points (r, θ), where r is radius, θ is angle; convert from (x, y) using r = √(x² + y²), θ = arctan(y/x).
- Basic polar graphs: circles (no A), cardioids (A = B), limaçons (B > A), and roses (multiple petals).
Key Terms & Definitions
- Domain — Set of input values (x) for a function.
- Range — Set of output values (y) for a function.
- Slope — Measure of a line's steepness; change in y over change in x.
- Asymptote — Line a graph approaches but never touches.
- Multiplicity — Number of times a zero is repeated.
- Residual — Difference between observed and predicted values in a model.
- Exponential Function — f(x) = ab^x, b > 0, b ≠ 1.
- Logarithm — Inverse of an exponential; log_b(x).
- Sinusoidal Function — Periodic function based on sine or cosine.
- Polar Coordinates — System using (r, θ) instead of (x, y).
Action Items / Next Steps
- Memorize all key identities, function properties, and skeleton equations.
- Practice modeling and transforming functions both algebraically and graphically.
- Use calculators for regression analysis and function modeling.
- Ensure calculators are set to radians for trigonometry.
- Review special triangles and unit circle values for trigonometric functions.
- Complete any assigned problems or revisit particularly challenging concepts for mastery.