PC: Precalculus

Precalculus

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Polynomial, rational, exponential, and logarithmic functions are studied in depth, including transformations, representations, and modeling applications. Trigonometric functions, identities, and the unit circle extend into periodic modeling. Conic sections, sequences and series, matrices, and an introduction to limits complete the course.

PC.1Define the constant e in a variety of contexts.PC.10Solve problems involving velocity and other quantities that can be represented by vectors.PC.11Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.PC.12Add and subtract vectors.PC.13Multiply a vector by a scalar.PC.14Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.PC.15Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems, extending to infinite geometric series.PC.16Derive and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).PC.17Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer, n, where x and y are any numbers.PC.18Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated cases, a computer algebra system.PC.19Add, subtract, multiply, and divide rational expressions.PC.2Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.PC.20Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.PC.21Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.PC.22Represent a system of linear equations as a single matrix equation in a vector variable.PC.23Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).PC.24Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.PC.25Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.PC.26Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.PC.27Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.PC.28Find inverse functions.PC.29Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.PC.3Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.PC.30Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.PC.31Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.PC.32Solve application-based problems involving parametric and polar equations.PC.33Use special triangles to determine geometrically the values of sine, cosine, and tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.PC.34Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.PC.35Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.PC.36Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.PC.37Use trigonometric identities to solve problems.PC.4Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.PC.5Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.PC.6Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.PC.7Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.PC.8Explain that vector quantities have both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.PC.9Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.PC.A-APREAlgebra - Arithmetic With Polynomials and Rational ExpressionsPC.A-RWEIAlgebra - Reasoning With Equations and InequalitiesPC.A-SSEAlgebra - Seeing Structure in ExpressionsPC.BFBuilding FunctionsPC.FFunctions - Interpreting FunctionsPC.NQ-CNSNumber and Quantity - The Complex Number SystemPC.NQ-LNumber and Quantity - LimitsPC.NQ-VMQNumber and Quantity - Vector and Matrix QuantitiesPC.TFTrigonometric Functions
Example Problems
Find .
A curve is defined by the parametric equations and .
What is in terms of t?
These are the component forms of vectors and :


Subtract the vectors.
The component form of vector is .
Find .
Evaluate:
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