﻿ FCPS Program of Studies
Mathematics Curriculum
Precalculus

Precalculus includes all the topics of Trigonometry and a thorough treatment of functions through the study of polynomials, rational functions, logarithmic, exponential, and inverse functions.  Topics include continuity, maximum and minimum points, an intuitive approach to limits, and rates of change.  A transformational approach to graphing is used with families of related graphs.  Numerical, graphical, and algebraic solutions are considered for all problems as applicable.  Emphasis will be placed on problem solving techniques.  Graphing utilities, especially graphing calculators, are integral to the course.

Last Updated: 08/17/12 10:28 AM

MTH.PC
Standard 1
CONNECT CIRCULAR FUNCTION WITH TRIG FUNCTION DEFINITIONS

Benchmark 1.a
Connect Circular Function with Trig Function Definitions

 Indicator 1.a.1 Define the six triangular trigonometric functions of an angle

 Indicator 1.a.2 Define the six circular trigonometric functions of an angle

 Indicator 1.a.3 Make the connection between the trig function and unit circle

 Indicator 1.a.4 Recognize and draw an angle in standard position

 Indicator 1.a.5 Determine a triangle from a point on the terminal side of an angle

 Indicator 1.a.6 Find angles that are coterminal with a given angle.

MTH.PC
Standard 2
FIND THE VALUES OF TRIGONOMETRIC FUNCTIONS

Benchmark 2.a
Given Value of One Trig Function, Find Values of Others

 Indicator 2.a.1 Find all related trig function values, given the value of one

 Indicator 2.a.2 Label the special angles and quadrantal angles on the unit circle

 Indicator 2.a.3 Use circular function definitions and unit circle to solve problems

 Indicator 2.a.4 Make connections between unit circle and coordinate geometry

 Indicator 2.a.5 Make connections between unit circle & cosine/sine values

 Indicator 2.a.6 Make connections between unit circle & special right triangles

MTH.PC
Standard 3
FIND EXACT TRIG VALUES OF SPECIAL ANGLES IN DEGREES AND RADIANS

Benchmark 3.a
Find Trig Values of Special Angles Without A Calculator

 Indicator 3.a.1 Find trig function values of special angles in degrees and radians

 Indicator 3.a.2 Apply the properties of the unit circle without using a calculator

 Indicator 3.a.3 Convert from radians to degrees and from degrees to radians

 Indicator 3.a.4 (Optional) Find the arc length, linear speed, and angular velocity

MTH.PC
Standard 4
APPROXIMATE VALUES OF TRIG FUNCTIONS AND INVERSE TRIG FUNCTIONS

Benchmark 4.a
Use A Calculator To Approximate Trig Values and Inverse Trig Values

 Indicator 4.a.1 Use a calculator to approximate trig values in degrees and radians

 Indicator 4.a.2 Define inverse trigonometric functions

 Indicator 4.a.3 Use inverse trig functions to find angle measures

MTH.PC
Standard 5
VERIFY BASIC TRIGONOMETRIC IDENTITIES

Benchmark 5.a
Verify Basic Trig Identities Using Substitution

 Indicator 5.a.1 Verify trig identities using reciprocal and Pythagorean identities

 Indicator 5.a.2 Verify trig identities using sum and difference identities

 Indicator 5.a.3 Verify trig identities using double-angle identities

 Indicator 5.a.4 Verify trig identities using half-angle identities

 Indicator 5.a.5 Verify trig identities using quotient identities

MTH.PC
Standard 6
FIND DOMAIN, RANGE, AMPLITUDE, PERIOD AND GRAPH TRIG FUNCTIONS

Benchmark 6.a
Determine The Domain And Range Of Trig Functions

 Indicator 6.a.1 State the domain and range of trig functions

Benchmark 6.b
Determine Amplitude, Period, Phase Shift, Vertical Shift, & Asymptoles

 Indicator 6.b.1 Determine amplitude, period, phase shift, vertical shift, & asymptotes

 Indicator 6.b.2 Describe the effects of changing parameters in a trig equation

 Indicator 6.b.3 Write equations of trig functions

 Indicator 6.b.4 Determine and graph trig functions from real-world situations

Benchmark 6.c
Sketch The Graph Of A Trig Function Using Transformations

 Indicator 6.c.1 Sketch the graph of a trig function for at least two periods or cycle

 Indicator 6.c.2 Sketch the graph of the parent functions, relating to the unit circle

MTH.PC
Standard 7
IDENTIFY THE DOMAIN, RANGE, AND GRAPHS OF INVERSE TRIG FUNCTIONS

Benchmark 7.a
Identify The Domain, Range, And Graphs Of Inverse Trig Functions

 Indicator 7.a.1 Find the domain and range of inverse trig functions

 Indicator 7.a.2 Find the values of inverse trig functions

 Indicator 7.a.3 Identify the graphs of inverse trig functions

 Indicator 7.a.4 Evaluate expressions of composite trig and inverse trig functions

 Indicator 7.a.5 Graph inverse trig functions using transformations

MTH.PC
Standard 8
SOLVE TRIG EQUATIONS AND INEQUALITIES

Benchmark 8.a
Solve Trig Equations With Infinite Solutions and Restricted Domains

 Indicator 8.a.1 Solve trig equations over a restricted domain

 Indicator 8.a.2 Solve trig equations with infinite solutions

 Indicator 8.a.3 Verify algebraic solutions and check reasonableness

MTH.PC
Standard 9
SOLVE PRACTICAL PROBLEMS USING THE LAW OF SINES AND COSINES

Benchmark 9.a
Use The Pythagorean Theorem, The Law Of Sines, And The Law Of Cosines

 Indicator 9.a.1 Create a real-world problem involving triangles

 Indicator 9.a.2 Solve a real-world problem involving triangles

 Indicator 9.a.3 Solve problems using Law of Sines, Cosines, and Pythagorean Theorem

 Indicator 9.a.4 Identify a solution technique that could be used with a given problem

 Indicator 9.a.5 Prove + and - formulas for sine/cosine/tangent & use to solve problem

 Indicator 9.a.6 Find the area of triangles using Heron's formula and the sine formula

MTH.PC
Standard 10
IDENTIFY CHARACTERISTICS AND GRAPH POLYNOMIAL AND RATIONAL FUNCTIONS

Benchmark 10.a
Identify Characteristics And Graph Polynomial And Rational Functions

 Indicator 10.a.1 Identify a polynomial function, given an equation or graph

 Indicator 10.a.2 Identify a rational function, given an equation or graph

 Indicator 10.a.3 Identify characteristics of a function given its graph

 Indicator 10.a.4 Sketch the graph of a polynomial function

 Indicator 10.a.5 Sketch the graph of a rational function

 Indicator 10.a.6 Use a graphing utility to investigate characteristics of a function

 Indicator 10.a.7 Identify chars. of polynomial/rational functions given equation

 Indicator 10.a.8 Find the intersection points of two or more functions

 Indicator 10.a.9 Determine whether a function is even, odd, or neither

 Indicator 10.a.10 Solve polynomial or rational equations and inequalities

 Indicator 10.a.11 Decompose a rational expression into partial fractions

MTH.PC
Standard 11
FIND COMPOSITIONS AND INVERSES OF FUNCTIONS

Benchmark 11.a
Find The Composition Of Functions And The Inverse Of A Function

 Indicator 11.a.1 Find the composition of two or more functions

 Indicator 11.a.2 Find the inverse of a function, if it exists

 Indicator 11.a.3 Find the domain and range of the composition of functions

 Indicator 11.a.4 Find the domain and range of the inverse of a function

 Indicator 11.a.5 Model problems using composition of functions

MTH.PC
Standard 12
INVESTIGATE AND DESCRIBE THE CONTINUITY OF FUNCTIONS

Benchmark 12.a
Describe The Continuity Of Functions From A Graph

 Indicator 12.a.1 Identify absolute value, step, and piece-wise functions

 Indicator 12.a.2 Describe the continuity of a function

 Indicator 12.a.3 Investigate the continuity of various functions

 Indicator 12.a.4 Use transformations to sketch the graphs of various functions

 Indicator 12.a.5 Verify graphs of functions using a graphing utility

 Indicator 12.a.6 Graph absolute value, piece-wise-defined and step functions

 Indicator 12.a.7 Use transformations to sketch piece-wise-defined functions

MTH.PC
Standard 13
FIND THE LIMIT OF FUNCTIONS GRAPHICALLY, NUMERICALLY AND ALGEBRAICALLY

Benchmark 13.a
Find The Limit Of A Function, If It Exists

 Indicator 13.a.1 Find limits of algebraic functions

 Indicator 13.a.2 Find a limit algebraically and verify graphically

 Indicator 13.a.3 Find a limit numerically and verify graphically

 Indicator 13.a.4 Use limit notation when describing end behavior of a function

 Indicator 13.a.5 Use the formal definition of continuity to solve problems

 Indicator 13.a.6 Recognize a limit doesn't exist when evaluating a 1 or 2 sided limit

MTH.PC
Standard 14
IDENTIFY CHARACTERISTICS AND GRAPH EXPONENTIAL AND LOG FUNCTIONS

Benchmark 14.a
Identify Characteristics And Graph Exponential And Log Functions

 Indicator 14.a.1 Identify exponential functions

 Indicator 14.a.2 Identify logarithmic functions

 Indicator 14.a.3 Define Euler's constant, e

 Indicator 14.a.4 Write log equations in exponential form and vice versa

 Indicator 14.a.5 Identify common and natural logarithms

 Indicator 14.a.6 Solve exponential equations & use of law exponents

 Indicator 14.a.7 Solve logarithmic equations

 Indicator 14.a.8 Model real-world problems using exponential and logarithmic functions

 Indicator 14.a.9 Graph exponential/logarithmic functions & identify key characteristics

MTH.PC
Standard 15
EXPAND BINOMIALS USING THE BINOMIAL THEOREM AND PASCAL'S TRIANGLE

Benchmark 15.a
Expand Binomials Using The Binomial Theorem And Pascal's Triangle

 Indicator 15.a.1 Expand binomials having positive integral exponents

 Indicator 15.a.2 Use the binomial theorem, combinations, and Pascal's triangle

MTH.PC
Standard 16
SOLVE PROBLEMS INVOLVING SEQUENCES AND SERIES

Benchmark 16.a
Solve Problems Involving Arithmetic and Geometric Sequences and Series

 Indicator 16.a.1 Use and interpret notation of sequences and series

 Indicator 16.a.2 Find the nth term of a sequence

 Indicator 16.a.3 Find the sum of a series

 Indicator 16.a.4 Model and solve problems using sequences and series

 Indicator 16.a.5 Distinquish between a convergent and divergent series

 Indicator 16.a.6 Discuss convergent series in relation to the concept of a limit

MTH.PC
Standard 17
PROVE FORMULAS AND STATEMENTS USING MATHEMATICAL INDUCTION

Benchmark 17.a
Prove Formulas And Statements Using Induction

 Indicator 17.a.1 Compare inductive and deductive reasoning

 Indicator 17.a.2 Prove formulas/statements, using mathematical induction

MTH.PC
Standard 18
IDENTIFY CHARACTERISTICS AND GRAPH POLAR EQUATIONS

Benchmark 18.a
Identify Characteristics And Graph Polar Equations

 Indicator 18.a.1 Recognize the type of polar graph, given an equation or graph

 Indicator 18.a.2 Determine the effects of changing parameters in polar equations

 Indicator 18.a.3 Convert complex numbers between rectangular & polar form

 Indicator 18.a.4 Find the intersection points of two or more polar graphs

 Indicator 18.a.5 Identify and plot polar coordinates

 Indicator 18.a.6 Find equivalent polar coordinates

 Indicator 18.a.7 Convert coordinates between rectangular form and polar form

 Indicator 18.a.8 (Optional) Apply DeMoivre's theorem

 Indicator 18.a.9 (Optional) Identify rectangular graphs from polar graphs & vice versa

 Indicator 18.a.10 (Optional) Graph polar equations by hand

MTH.PC
Standard 19
MODEL AND SOLVE PROBLEMS PARAMETRICALLY

Benchmark 19.a
Model And Solve Problems Parametrically

 Indicator 19.a.1 Graph parametric equations

 Indicator 19.a.2 Model motion using parametric equations

 Indicator 19.a.3 Solve parametric equations

 Indicator 19.a.4 Compare solution methods

 Indicator 19.a.5 Eliminate parameters and sketch curves defined by parametrics

MTH.PC
Standard 20
PERFORM OPERATIONS ON VECTORS AND SOLVE PRACTICAL PROBLEMS

Benchmark 20.a
Perform Operations on Vectors and Solve Real-World Problems

 Indicator 20.a.1 Use vector notation

 Indicator 20.a.2 Perform operations on vectors

 Indicator 20.a.3 Graph vectors and resultant vectors

 Indicator 20.a.4 (Optional) Express complex numbers in vector notation

 Indicator 20.a.5 Define unit vector and find unit vectors

 Indicator 20.a.6 Perform vector addition, scalar multiplication and find dot products

 Indicator 20.a.7 Find vector components

 Indicator 20.a.8 Find the norm of a vector

 Indicator 20.a.9 (Optional) Use vectors in geometric proofs

 Indicator 20.a.10 (Optional) Perform operations with vectors in space

MTH.PC
Standard 21
INVESTIGATE FUNDAMENTAL PRINCIPLES OF DIFFERENTIAL CALCULUS

Benchmark 21.a
Investigate Principles Of Calculus

 Indicator 21.a.1 Investigate derivative definition graphically and algebraically

 Indicator 21.a.2 Find derivatives using the definition and apply to problems

 Indicator 21.a.3 Find derivatives using the power, product, quotient and chain rules

 Indicator 21.a.4 Determine the differentiability of functions

 Indicator 21.a.5 (Optional) Find second derivatives and apply to problems

Essential - Standard, benchmark, or indicator from the VDOE Standards of Learning document. In the absence of VDOE standards for a given course, content subject to testing such as AP and IB can be labeled Essential.
Expected - Standard, benchmark, or indicator added by the FCPS Program of Studies to provide a context, a bridge, or an enhancement to the Essential SBIs.
Extended - Standard, benchmark, or indicator added by the FCPS Program of Studies generally used to differentiate instruction for advanced learners (Honors/GT)