PrecalculusQ&A by dot point

Topic 1.1 Change in Tandem: describe how the output values of a function change as the input values change, using increasing or decreasing behavior, concavity, and the relationship shown in a graph, table or context.

Topic 1.11 Equivalent Representations of Polynomial and Rational Expressions: rewrite polynomial and rational expressions in equivalent forms using factoring, the binomial theorem and polynomial long division to reveal zeros, asymptotes and end behavior.

Topic 1.14 Function Model Construction and Application: construct a polynomial or rational function model from a context, restricted domain or data set, and apply it to make predictions and solve problems.

Topic 1.13 Function Model Selection and Assumption Articulation: select an appropriate type of function to model a data set or context, and articulate the assumptions and limitations of the chosen model.

Topic 1.5 Polynomial Functions and Complex Zeros: relate the real and non-real complex zeros of a polynomial to its factored form, degree and graph, including the effect of multiplicity.

Topic 1.6 Polynomial Functions and End Behavior: determine the end behavior of a polynomial from its degree and leading term, and express that behavior using limit notation.

Topic 1.4 Polynomial Functions and Rates of Change: relate the degree of a polynomial to its number of local extrema and points of inflection, and analyze where its rate of change is positive, negative or zero.

Topic 1.3 Rates of Change in Linear and Quadratic Functions: characterize linear functions by their constant rate of change and quadratic functions by their constant rate of change of the rate of change.

Topic 1.2 Rates of Change: compute and interpret the average rate of change of a function over an interval, and estimate the rate of change at a point.

Topic 1.7 Rational Functions and End Behavior: determine the end behavior of a rational function by comparing the degrees of its numerator and denominator, identifying horizontal or slant asymptotes.

Topic 1.10 Rational Functions and Holes: identify removable discontinuities (holes) of a rational function from factors common to numerator and denominator, and find the coordinates of each hole.

Topic 1.9 Rational Functions and Vertical Asymptotes: locate the vertical asymptotes of a rational function from the zeros of the denominator that do not cancel, and describe the behavior with one-sided limits.

Topic 1.8 Rational Functions and Zeros: determine the real zeros of a rational function from the zeros of its numerator, accounting for values excluded by the denominator.

Topic 1.12 Transformations of Functions: construct and analyze additive and multiplicative transformations (translations, dilations and reflections) of a function and their effect on the graph, domain and range.

Topic 2.1 Change in Arithmetic and Geometric Sequences: define arithmetic sequences by a constant common difference and geometric sequences by a constant common ratio, and write their explicit and recursive forms.

Topic 2.2 Change in Linear and Exponential Functions: contrast linear functions, which change by a constant amount, with exponential functions, which change by a constant percentage or factor over equal-length input intervals.

Topic 2.6 Competing Function Model Validation: compare competing function models for a data set by analyzing residuals, and validate or reject a model based on the pattern and size of its residuals.

Topic 2.7 Composition of Functions: form the composition of two functions, evaluate it, decompose a composite function, and determine the domain of a composition.

Topic 2.13 Exponential and Logarithmic Equations and Inequalities: solve exponential and logarithmic equations and inequalities using inverse operations, the logarithm properties, and checks for extraneous solutions.

Topic 2.5 Exponential Function Context and Data Modeling: construct an exponential model from a context or data set, interpret the initial value and growth or decay factor, and use the model to make predictions.

Topic 2.4 Exponential Function Manipulation: rewrite exponential expressions using the product, power, negative-exponent and rational-exponent properties to reveal equivalent forms.

Topic 2.3 Exponential Functions: define exponential functions, describe how the base and initial value determine growth or decay, and analyze the domain, range and horizontal asymptote of the graph.

Topic 2.8 Inverse Functions: determine whether a function has an inverse, find the inverse by swapping input and output, and verify an inverse using composition and the reflection over the line y = x.

Topic 2.10 Inverses of Exponential Functions: construct the inverse of an exponential function as a logarithmic function, and relate the graph, domain and range of each to the other.

Topic 2.9 Logarithmic Expressions: define a logarithm as the exponent that produces a given value, and evaluate logarithmic expressions by rewriting them in exponential form.

Topic 2.14 Logarithmic Function Context and Data Modeling: construct a logarithmic model from a context or data set, interpret its parameters, and use it to make predictions.

Topic 2.12 Logarithmic Function Manipulation: rewrite logarithmic expressions using the product, quotient, power and change-of-base properties to expand or condense them.

Topic 2.11 Logarithmic Functions: analyze the parent logarithmic function and its transformations, including its domain, range, vertical asymptote, and increasing or decreasing behavior.

Topic 2.15 Semi-log Plots: use a semi-log plot to determine whether an exponential model is appropriate, and interpret the slope and intercept of the resulting line.

Topic 3.12 Equivalent Representations of Trigonometric Functions: use the Pythagorean, sum and difference, and double-angle identities to rewrite trigonometric expressions in equivalent forms.

Topic 3.9 Inverse Trigonometric Functions: define arcsine, arccosine and arctangent on restricted domains, and evaluate and interpret their outputs.

Topic 3.1 Periodic Phenomena: identify a periodic relationship, and describe its period, amplitude and key features from a graph, table or context.

Topic 3.13 Trigonometry and Polar Coordinates: locate points using polar coordinates and convert between polar and rectangular coordinates.

Topic 3.14 Polar Function Graphs: construct and interpret the graph of a polar function r = f(theta), including circles, roses, limacons and spirals.

Topic 3.15 Rates of Change in Polar Functions: analyze how r changes as theta increases, using the average rate of change to describe whether the curve moves toward or away from the pole.

Topic 3.4 Sine and Cosine Function Graphs: construct and interpret the graphs of sine and cosine, identifying period, amplitude, midline, zeros and concavity.

Topic 3.3 Sine and Cosine Function Values: determine sine and cosine values using the unit circle, reference angles, symmetry and the Pythagorean identity.

Topic 3.2 Sine, Cosine, and Tangent: define sine, cosine and tangent using the unit circle and right-triangle ratios, and evaluate them at key angles.

Topic 3.7 Sinusoidal Function Context and Data Modeling: construct a sinusoidal model from a periodic context or data, and use it to make and interpret predictions.

Topic 3.6 Sinusoidal Function Transformations: describe how changing each parameter transforms a sinusoid, and combine vertical and horizontal stretches, reflections and shifts.

Topic 3.5 Sinusoidal Functions: write a sinusoidal function in the form a*sin(b(x - c)) + d (or with cosine) and relate amplitude, period, phase shift and vertical shift to the parameters.

Topic 3.11 The Secant, Cosecant, and Cotangent Functions: define the reciprocal trigonometric functions and identify their periods, asymptotes, ranges and graphs.

Topic 3.8 The Tangent Function: define the tangent function, graph it, and identify its period, vertical asymptotes, zeros and behavior between asymptotes.

Topic 3.10 Trigonometric Equations and Inequalities: solve trigonometric equations and inequalities, using inverse functions, symmetry and periodicity to find all solutions.

Topic 4.6 Conic Sections: identify and analyze parabolas, ellipses, circles and hyperbolas from their equations, and describe their key features.

Topic 4.5 Implicitly Defined Functions: interpret a relation given by an equation in x and y, and analyze its graph even when it is not a function of x.

Topic 4.12 Linear Transformations and Matrices: represent a linear transformation of the plane by a matrix, and identify the matrices for scalings, reflections and rotations.

Topic 4.13 Matrices as Functions: interpret a matrix as a function from vectors to vectors, and relate matrix multiplication to composition and the inverse matrix to the inverse function.

Topic 4.14 Matrices Modeling Contexts: use matrices to model transitions between states, and apply repeated multiplication to project the state forward in time.

Topic 4.10 Matrices: represent data with a matrix, and add, subtract, scale and multiply matrices, including multiplying a matrix by a vector.

Topic 4.3 Parametric Functions and Rates of Change: compute the average rates of change of x and y with respect to t, and use them to describe the direction and relative speed of motion.

Topic 4.2 Parametric Functions Modeling Planar Motion: use a parametric function to model the position of a moving point over time, and describe its path, direction and position at a given time.

Topic 4.1 Parametric Functions: define a parametric function giving x and y as functions of a parameter t, and graph and interpret the curve it traces.

Topic 4.4 Parametrically Defined Circles and Lines: write and interpret parametric equations for circles and lines, controlling radius, center, direction and starting point.

Topic 4.7 Parametrization of Implicitly Defined Functions: find parametric equations that trace an implicitly defined curve, and verify the parametrization satisfies the implicit equation.

Topic 4.11 The Inverse and Determinant of a Matrix: compute the determinant and inverse of a 2x2 matrix, and use them to determine invertibility and solve matrix equations.

Topic 4.9 Vector-Valued Functions: interpret a vector-valued function whose output is a position vector, and relate it to parametric motion and velocity.

Topic 4.8 Vectors: represent a vector by components, compute its magnitude and direction, and add, subtract and scale vectors.