Summary
Highlights
Clarifies the term 'initial term' for sequence equations. Highlights the strong connection between arithmetic sequences and linear functions, and geometric sequences with exponential functions. Introduced the skeleton exponential function equation f(x) = a * b^x.
This section introduces functions, input/output values (domain/range), increasing/decreasing functions. It also covers graphs, rate of change (slope), concavity (up/down), x-intercepts (zeros), and the linear equation y = mx + b.
Explains how to calculate the average rate of change using the slope formula. It also demonstrates how to use the average rate of change to predict future points on a graph and distinguishes between positive and negative rates of change.
Discusses how rates of change differ in linear versus quadratic functions. For linear functions, the rate is constant. For quadratics, the average rate of change represents the slope of a secant line, and finding the rate at a specific point involves calculating the slope of a tangent line using very close points.
Defines polynomial functions, including their general form (coefficients and degrees), and restrictions (no negative degrees, imaginary coefficients, or division). It also covers local/global maximums/minimums and points of inflection where concavity changes.
Explains zeros of a function (x-intercepts, roots), differentiating between real and imaginary zeros. It covers how the degree of a polynomial indicates the number of zeros and the concept of multiplicity (odd/even exponents) affecting graph behavior at zeros. Also, defines even and odd functions based on symmetry.
Introduces limit notation to describe end behavior of polynomial functions, explaining how x and y values approach infinity or negative infinity. A trick is provided to determine end behavior from the degree and leading coefficient of the equation.
Defines rational functions as two polynomials over one another, introducing vertical and horizontal asymptotes. It outlines three rules for determining end behavior based on degrees of numerator and denominator: bottom-heavy (limit is 0), same-heavy (limit is ratio of leading coefficients), and top-heavy (no horizontal asymptote, potentially slant/oblique asymptotes found via polynomial long division).
Explains how to find real zeros of rational functions by setting the numerator to zero. Critical to note that any zeros matching between numerator and denominator are holes, not zeros. Provides examples of solving polynomial functions for zeros.
Describes how to find vertical asymptotes by setting the denominator to zero and ensuring these values are not also zeros of the numerator (which would indicate a hole). Defines asymptotes as invisible lines the graph approaches but never touches, and discusses limit notation around them.
Explains that holes occur when a common factor exists in both the numerator and denominator after solving. These are represented by open circles on a graph, indicating points where the function is undefined, affecting domain and range.
Covers constructing polynomial functions from roots, polynomial long division to simplify expressions, and Pascal's triangle for binomial expansion. It also details additive (vertical/horizontal translations) and multiplicative (vertical/horizontal dilations, reflections) transformations of functions.
Discusses identifying appropriate function models (linear, quadratic, cubic, piecewise) for data sets or scenarios, emphasizing graphing as the best method. Stresses the importance of considering real-world constraints for domain and range. Introduces using a calculator's regression features (linear, quadratic, cubic, quartic) to find the best-fitting model by selecting the one with an R-value closest to one.
Introduces arithmetic sequences (linear, common difference) and geometric sequences (proportional change, common ratio). Presents formulas for finding terms in both types of sequences, although noting that simple iterative calculation often suffices.
Describes exponential functions as curves, detailing rules for the initial value 'a' and base 'b' (a ≠ 0, b > 0, b ≠ 1). Explains exponential growth (b > 1) and decay (0 < b < 1), their graph behaviors, domain (all real numbers), lack of inflection points, and parent function characteristics (0,1 point, horizontal asymptote at y=0).
Covers four exponent properties: product (b^m * b^n = b^(m+n), a horizontal translation), power ((b^m)^n = b^(mn), a stretch/shrink), negative exponent (b^-n = 1/b^n), and exponent root (b^(1/k) = k-th root of b). Explains their graphical implications.
Explains how to derive an exponential function from two points using a system of equations. Discusses how exponential functions model interest and compound interest. Introduces 'e' (approximately 2.718) as the base for natural exponential functions, used for continuous growth/decay. Mentions using calculator's exponential regression.
Reviews building linear, quadratic, and exponential equations from data. Introduces residuals as the difference between actual and predicted data points, and how a random residual plot indicates an appropriate model. Emphasizes finding the best fit using calculator regressions and interpreting residuals.
Explains function composition (e.g., f(g(x))) by substituting one function into another. Notes that original functions can be decomposed into simpler functions. Mentioned for its relevance in contexts like the SAT.
Defines inverse functions (notated as f^(-1)(x)), explaining how to find them by swapping x and y and solving for y. A crucial condition for an inverse function to exist is that the original function must be one-to-one (passes the horizontal line test). Also, inverse functions swap domain and range with their original functions.
Introduces logarithmic expressions as the inverse of exponential expressions (log_b(c) = a means b^a = c). Outlines two rules: b must be positive and not equal to one. Explains common logs (base 10) and using a calculator to solve for powers in log form.
Introduces logarithmic functions as inverses of exponential functions. Discusses their graphs, which are reflections of exponential graphs over y=x. Details log parent function properties: domain (x > 0), vertical asymptote at x=0, range (all real numbers), constant increase/decrease, no inflection points. End behavior changes as x approaches infinity and zero (or the vertical asymptote). Emphasizes that transformations from Unit 1 still apply.
Covers three key log properties: product (log(xy) = log(x) + log(y)), quotient (log(x/y) = log(x) - log(y)), and power (log(x^k) = k*log(x)). Explains their implications for horizontal and vertical dilation on graphs. Briefly mentions the change of base formula and introduces natural logs (ln x) as log base e of x.
Demonstrates solving intimidating logarithmic equations by applying log properties to simplify them. Briefly reviews solving for inverse functions by swapping x and y. Highlights the importance of checking solutions as logs of negative numbers are undefined.
Explains how to derive a log function from one or two input-output pairs by rearranging and solving for variables, potentially using systems of equations. Notes that data modeled by log functions shows proportional x changes based on multiplication. Mentions using calculator's natural log regression (ln regression).
Introduces semi-log plots, where one axis is scaled logarithmically and the other linearly. If the x-axis is log-scaled, a log function appears linear; if the y-axis is log-scaled, an exponential function appears linear. This helps confirm whether a function is exponential or logarithmic.
Defines periodic functions as those with continuous, repeating patterns over equal intervals, using sine and cosine graphs as examples. Explains how to calculate the period of a graph (horizontal distance between two consecutive maximums or minimums).
Introduces radians as a new angle measure (2π radians = 360°). Explains the unit circle (radius=1) and how points (x, y) on it correspond to (cosine θ, sine θ). Defines sine (y/r), cosine (x/r), and tangent (y/x or sine/cosine).
Explains how to find sine, cosine, and tangent values for angles like 30°, 45°, and 60° by constructing special right triangles (30-60-90 and 45-45-90) within the unit circle. Emphasizes memorizing the unit circle values.
Discusses inverse trigonometric functions (e.g., inverse sine) and their restricted domains (e.g., inverse sine only exists in quadrants 1 and 4). Explains arc notation as an alternative for inverse functions. Covers negative angles (clockwise rotation) and conversions between rectangular (x, y) and polar (r, θ) coordinates.
Visualizes sine and cosine graphs by plotting unit circle values. Notes their cyclical, wave-like nature, domain (all real numbers), and range (typically [-1, 1] without transformations). Highlights that cosine is a horizontally translated sine graph. Defines sinusoidal functions as periodic functions oscillating between a min and max.
Reviews periods (2π for parent sine/cosine), frequency (1/period), midline (y=0 for parent), and amplitude (vertical distance from midline to max, 1 for parent functions). Discusses changes in concavity at the midline. Identifies sine as an odd function and cosine as an even function based on symmetry.
Explains the skeleton sinusoidal equation (f(θ) = A sin(B(θ - C)) + D or cosine). A is amplitude, B relates to period (B = 2π/period), C is phase shift (horizontal), and D is vertical shift (midline). Provides an example of constructing an equation from a graph.
Works through an AP exam question involving modeling a real-world scenario (tire rotation) with a sinusoidal function. Emphasizes plotting five key points and deriving the equation by finding amplitude, period, phase shift, and midline. Stresses keeping calculators in radian mode for this course.
Explores the tangent graph, noting its vertical asymptotes where cosine is zero (π/2, 3π/2). The tangent graph has a period of π. The skeleton equation is similar to sinusoidal functions, but B = π/period for tangent. Introduces how to find the 'amplitude' for tangent by looking at points within a quarter period from an x-intercept.
Revisits inverse trigonometric functions, emphasizing their restricted domains to ensure they are one-to-one. Specifically, inverse sine and tangent are restricted to quadrants 1 and 4, while inverse cosine is restricted to quadrants 1 and 2. Differentiates solving inverse functions (restricted domain) from solving general trigonometric equations (all possible solutions using +2πk or +πk).
Provides examples of solving trigonometric equations, both by hand and using a calculator. Demonstrates isolating the trigonometric function and then finding angles on the unit circle or using inverse functions on a calculator. Stresses adding +2πk (or +πk for tangent) to solutions to represent all periodic possibilities.
Introduces cosecant (1/sine), secant (1/cosine), and cotangent (1/tangent or cosine/sine) as reciprocal identities. Explains how to find their values. Visualizes their graphs by overlaying them with sine/cosine graphs, showing how their U-shapes align with the extrema of their reciprocals. Briefly mentions the general shape of cotangent relative to tangent.
Explores Pythagorean identities (e.g., sin^2 x + cos^2 x = 1) and their use in rewriting expressions. Provides an AP exam example of simplifying a trigonometric expression using these identities. Introduces sum identities (e.g., sin(A+B)) and double angle identities, emphasizing which ones are critical to memorize for the exam.
Introduces polar functions using (r, θ) coordinates instead of (x, y). Explains plotting points by rotating to an angle (θ) and moving out a radius (r). Discusses 'crossing the pole' for negative 'r' values. Shows how to convert between Cartesian and polar coordinates using r = sqrt(x^2 + y^2) and θ = arctan(y/x) (with adjustments for negative x).
Describes polar graphs in the form r = f(θ), where θ is the input (angle) and r is the output (radius). Explains how domain restrictions on θ can cut off parts of the graph. Categorizes common polar graph shapes: circles, cardioids, limacons, and roses, detailing the conditions for each (e.g., cardioid when a=b in r = a + b sin θ).
Discusses what 'increasing' and 'decreasing' mean for polar functions. If r is positive and increasing, the point moves away from the origin; if negative and decreasing, it also moves away. Conversely, if r is positive and decreasing or negative and increasing, the point moves toward the origin. Explains average rate of change for polar functions as how fast the radius (r) changes per unit of angle (θ).