Precalculus

Unit 1

• F.BF.3 – Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
• F.BF.4.a – Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2x^3 or f(x) = (x+1)/(x-1) for x ≠ 1.
• F.BF.4.b – Verify by composition that one function is the inverse of another.
• F.BF.4.c – Read values of an inverse function from a graph or a table, given that the function has an inverse.
• F.BF.5 – Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
• F.IF.3 – Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
• F.IF.7.a – Graph linear functions and show intercepts, maxima, and minima.
• F.IF.7.b – Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Unit 2

• F.IF.7.c – Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
• F.IF.7.d – Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
• F.IF.8.a – Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
• N.CN.3 – Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
• N.CN.4 – Represent complex numbers on the complex plane in rectangular (including real and imaginary numbers).
• N.CN.8 – Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
• N.CN.9 – Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
• F.IF.4 – For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
• F.IF.7.a – Graph quadratic functions and show intercepts, maxima, and minima.

Unit 3

• A.REI.6 – Solve systems of linear equations exactly and approximately (e.g.,with graphs), focusing on pairs of linear equations in two variables.
• A.REI.7 – Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
• A.REI.8 – Represent a system of linear equations as a single matrix equation in a vector variable.
• A.REI.9 – Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
• A.REI.SPS.10 – Find determinants of differing matrix sizes, and use these to complete varying application processes.

Unit 4

• G.GPE.A.1 – Derive the equation of circle.
• G.GPEA.A.2 – Derive the equation of a parabola given a focus and directrix.
• G.GPE.A.SPS.1 – Derive the equation of an ellipse
• G.GPE.A.SPS.2 – Derive the equation of a hyperbola and it’s asymptotes.
• G.GPE.A.SPS.3 – Classify conics from their general equations.
• G.GPE.A.SPS.4 – Evaluate and graph parametric equations.
• G.GPE.A.SPS.5 – Find sets of parametric equations.
• G.GPE.A.SPS.6 – Plot polar coordinates.
• N.CN.4 – Represent complex numbers on the complex plane in rectangular (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
• G.GPE.A.SPS.7 – Convert equations and points from rectangular to polar form and vice versa.
• G.GPE.A.SPS.8 – Graph polar equations.
• G.GPE.A.SPS.9 – Write and graph equations of conics in polar form.

Unit 5

• A.APR.SPS.5 – Plot points in 3-D system and find distances and midpoints between them.
• A.APR.SPS.6 – Write the equations of spheres.
• A.APR.SPS.7 – Find component forms of 3-dimensional vectors; their magnitudes, dot products, angles between, and determining whether two are parallel.
• A.APR.SPS.8 – Find 3-dimensional cross products and triple scalar products to find volumes of parallel pipeds.
• A.APR.SPS.9 – Find parametric and symmetric equations of lines in space. Sketch lines in space and find distance between points and planes.

Unit 6

• A.APR.SPS.1 – Evaluate the limits of polynomial, rational, and trigonometric functions.
• A.APR.SPS.3 – Use the limit definition to find the slope of a tangent line, hence the derivative.
• A.APR.SPS.2 – Evaluate indefinite integrals by using the properties and rules of integration.
• A.APR.SPS.4 – Find limits of summations.