Frameworks

Polynomial and Rational Functions

CONTENT STANDARD 1.

PRF.1.PCT.1

Investigate and sketch, with and without appropriate technology, the graphs of polynomia l and rational functions using the characteristics of domain and range, upper and lower bounds , maximum and minimum points, asymptotes and end behavior , zeros , multiplicity of zeros , y-intercepts, and symmetry.

PRF.1.PCT.2

Solve, with and without appropriate technology, polynomial equations utilizing techniques such as Descartes' Rule of Signs, upper and lower bounds, Intermediate Value Theorem and Rational Root Theorem.

PRF.1.PCT.3

Describe, with and without appropriate technology, the fundamental characteristics of rational functions: zeros, discontinuities (including vertical asymptotes ), and end behavior (including horizontal asymptotes).

PRF.1.PCT.4

Apply the concepts of polynomial and rational functions to model real world situations using appropriate technology when needed.

Exponential and Logarithmic Functions

CONTENT STANDARD 2.

ELF.2.PCT.2

Develop and apply the laws of logarithms and the change-of-base formula to simplify and evaluate expressions

ELF.2.PCT.3

Solve graphically, algebraically and numerically, with and without appropriate technology, equations and real world problems involving exponential and logarithmic expressions

ELF.2.PCT.4

Find, with and without appropriate technology, the domain, range, intercepts, and asymptotes of logarithmic and exponential functions

ELF.2.PCT.5

Draw and analyze, with and without appropriate technology, graphs of logarithmic and exponential functions

Conics

CONTENT STANDARD 3.

C.3.PCT.1

Identify, graph, write, and analyze equations of conic sections , using properties such as symmetry, intercepts, foci, asymptotes, and eccentricity , and when appropriate, use technology

C.3.PCT.2

Solve, with and without appropriate technology, systems of equations and inequalities involving conics and other types of equations

C.3.PCT.3

Solve, with and without appropriate technology, real world problems involving conic sections

Sequences and Series

CONTENT STANDARD 4.

SS.4.PCT.1

Develop, with and without appropriate technology, a representation of sequences recursively

SS.4.PCT.2

Define and discriminate between arithmetic and and series and use appropriate technology when needed

SS.4.PCT.3

Solve, with and without appropriate technology, problems involving the sum (including Sigma notation ) of finite and infinite sequences and series

SS.4.PCT.4

Determine the n th term of a sequence given a rule or specific terms and use appropriate technology when needed

SS.4.PCT.5

Use, with and without appropriate technology, sequences and series to solve real world problems

Trigonometric Functions

CONTENT STANDARD 5.

Students will use different perspectives to develop and apply the definitions of the six trigonometric functions.
They will sketch and analyze graphs, find inverse functions, and solve real world problems.

TF.5.PCT.1

Define the six trigonometric functions as

• circular functions
• ratios of sides of right triangles
• functions of an angle in standard position when given a point on the terminal side of the angle
  

TF.5.PCT.2

Use degrees and radians interchangeably to represent angle measure

TF.5.PCT.3

Sketch an angle in standard position and determine the reference angle and coterminal angles

TF.5.PCT.4

Find the values of the trigonometric functions given the value of one trigonometric function and an additional piece of qualifying information or given the coordinates of a point on the terminal side of an angle

TF.5.PCT.5

Develop and become fluent in the recall of the exact values of the trigonometric functions for special angles

TF.5.PCT.6

Solve, with and without appropriate technology, real world problems involving applications of trigonometric functions

TF.5.PCT.7

Graph the six trigonometric functions, identify domain, range, intercepts, period , amplitude , and asymptotes as applicable and use symmetry to determine whether the function is even or odd through appropriate technology when needed

TF.5.PCT.8

Determine, with and without appropriate technology, the amplitude, period, phase shift , and vertical shift, and sketch the graph of transformations of the trigonometric functions
 

TF.5.PCT.9

Identify and graph, with and without appropriate technology, the inverse of trigonometric functions including the restrictions on the domain

Oblique Triangles

CONTENT STANDARD 6.

OT.6.PCT.1

Develop and use the Law of Sines and the Law of Cosines to solve oblique triangles and use appropriate technology when needed

OT.6.PCT.2

Solve real world problems applying the Law of Sines and the Law of Cosines and appropriate technology when needed

OT.6.PCT.3

Determine the area of an oblique triangle by using an appropriate formula and appropriate technology when needed

OT.6.PCT.4

Use vectors to solve problems and describe addition of vectors and multiplication of a vector by a scalar , both symbolically and geometrically

OT.6.PCT.5

Use vectors to model situations defined by magnitude and direction and analyze and solve real world problems by using appropriate technology when needed

Trigonometric Equations and Identities

CONTENT STANDARD 7.

TEI.7.PCT.1

Develop the Pythagorean Identities and use to verify other identities and simplify expressions

TEI.7.PCT.2

Develop and use trigonometric formulas including sum and difference formulas and multiple-angle formulas

TEI.7.PCT.3

Solve trigonometric equations algebraically and graphically and use appropriate technology when needed

Polar Coordinates

CONTENT STANDARD 8.

PC.8.PCT.1

Convert polar coordinates to rectangular coordinates and rectangular coordinates to polar coordinates

PC.8.PCT.2

Represent equations given in rectangular coordinates in terms of polar coordinates

PC.8.PCT.3

Graph polar equations and use appropriate technology when needed

PC.8.PCT.4

Apply polar coordinates to real world situations and use appropriate technology when needed

TOP

Rules and Theorems

• Descartes' Rule of Signs
• Intermediate Value Theorem
• Rational Root Theorem
• laws of logarithms
• Change of Base
• Even and Odd Functions

Pre-Calculus including Trigonometry GLOSSARY

A

Amplitude

In the equation y = A sin x or y = A cos x , the amplitude is given by the .

Arithmetic

The mathematics of integers, rational numbers, real numbers, or complex numbers under addition, subtraction, multiplication, and division.

Arithmetic Series

The indicated sum of the terms of an arithmetic sequence

Asymptote

A line to which a graph becomes arbitrarily close as the value of x or y increases or decreases without bound (vertical, horizontal, slant)

C

Circular Functions

The six basic trigonometric functions defined using a unit circle

Conic Section

Any figure that can be formed by slicing a double cone with a plane

Coterminal Angles

Two angles in standard position having the same terminal side

D

Discontinuity

A point in the domain of a function at which the function is not continuous

E

Eccentricity

For a conic, the ratio of the distance of a point from a fixed point to its distance from a fixed line

End Behavior

A reference to the graph of a polynomial function as rising or falling to the right and rising or falling to the left

Exponential Functions

A function in which variable(s) occur in exponent(s)

F

Finite Sequence

A finite sequence with n terms is a function whose domain is the set of integers {1, 2, 3, …, n }

Finite Series

The indicted sum of a finite sequence

G

Geometric Sequence

A sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio, r

Geometric Series

The indicated sum of the terms of a geometric series

H

Horizontal Asymptote

A horizontal line to which a graph becomes arbitrarily close as the value of x increases or decreases without bound.

I

Infinite Sequence

An infinite sequence is a function whose domain is the set of positive integers.

Infinite Series

The indicated sum of an infinite series

L

Logarithmic Functions

A function of the form y = log b x, where b>0 and b<1

Lower Bound

A number which is less than or equal to every number in the set

M

Maximum

The greatest value of the function if it has such an extreme value

Minimum

The least value of the function if it has such an extreme value

Multiplicity of Zeros

The number of times that a repeated zero of a function occurs

O

Oblique Triangles

Triangles that have no right angles

P

Period

The interval of the domain over which the function repeats

Phase Shift

The horizontal shift of a trigonometric function

Polar Coordinates

The system of coordinates in which a point is located by its distance from a fixed point and the angle that the line from this point to the given point makes with a fixed line, called the polar axis

Polar Equation

An equation in polar coordinates

Polynomial Functions

A function that can be described by an equation of the form P(x) = , where the coefficients represent real numbers, is not zero, and n represents a nonnegative integer

R

Radian

A central angle subtended in a circle by an arc whose length is equal to the radius of the circle

Rational Functions

An equation of the form , where p(x) and q(x) are polynomial functions and q(x) 0.

Recursive Sequence

When given one or more of the first few terms, all other terms of the sequence are then defined using previous terms.

S

Scalar Multiplication (Vectors)

The product of a scalar a and a vector v is the vector having the same direction as v and of length equal to the product of a and the length of v

Sigma Notation

Notation that uses the symbol to indicate a sum of a series

Standard Position

The horizontal distance from any point on the graph of a function to that point where the graph begins to repeat

Symmetry

A figure has symmetry if the figure and its image coincide after a transformation.

U

Upper Bounds

A number that is greater than or equal to every number in the set

V

Vector

A quantity that is described by both magnitude and direction

Vertical Asymptotes

A vertical line to which a graph becomes arbitrarily close as the value of f(x) increases or decreases without bound

Z

Zeros

For any function f(x) , if f(a) = 0, then a is a zero of the function.