Pre-calculus

PreCalculus 2021-2022

Prerequisites

Number Sets

Natural Numbers
Whole Numbers
Set of Integers
Rationals
Irrationals
Real Numbers
Pure Imaginary Numbers
Complex Numbers

Notation and Abbreviations

Do you understand the shorthand used by advanced mathematics?
Can you translate verbal to symbolic?
Can you translate symbolic to verbal?
Can you describe specific portions of a graph, a function's domain, or a function's range using advanced notation?

Functions

Do you understand the notation, vocabulary, and abbreviations used to describe a function?

domain of a function
range of a function
end behavior of a function
asymptote of a function
vertical asymptote of a function
horizontal asymptote of a function
slant asymptote of a function
oblique asymptote of a function
reciprocal of a function
vertex of a quadratic function
vertex of an absolute value function

local extremes
local minimums
local maximums
absolute extremes
absolute minimum
absolute maximum
inverse function
one to one function
even function
odd function
continuous function
constant function
piecewise function
step functions
floor function
ceiling function
composition of functions
index of a radical function
radicand of a radical function
base of a logarithmic function
base of an exponential function

Can you perform and interpret the meaning of HLT and the VLT?
- HLT = horizontal line test
- VLT = vertical line test
Can you state the domain and range of a function using a variety of methods?
- verbal
- abbreviations
- symbolic
Can you predict the integer or rational coordinates that lie on functions?
Can you perform operations with functions?
Can you perform composition with functions?
Do you understand the connection between a function and its inverse?
Can you write a piecewise function for an absolute value function or a quadratic function?
Can you determine a function's inverse when possible and be able to state when whether a singe inverse exists or not?

Exponent and Radical Laws

Do you know the exponent laws and definitions?
Do you know the radical laws and definitions?
Can you rewrite expressions that involve exponents?
Can you rewrite expressions that involve radicals?
Can you rewrite an expression with negative exponents?
Can you convert an expression with fractional exponents to its related radical expression?
Can you convert an expression with a radical to its related expression with fractional exponents?
Can you convert an expression with fractional exponents to its related radical expression?

Factoring and Rewriting Expressions

Can you express a linear function in a variety of notations?
- slope intercept
- point slope
- modified point slope
- standard form
Can you express a quadratic function in a variety of notations?
- standard form
- vertex form
- intercept form
Can you factor any quadratic expression?
Do you know the special patterns related to quadratic expressions?
- DOTS-difference of two squares
- PST-perfect square trinomials
- missing constant quadratics
Can you factor specific cubic polynomials?
- DOC-difference of cubes
- SOC-sum of cubes
- Factor by Grouping
- missing constant cubics
- Factoring using Rational Root Theorem
Can you factor specific higher degree polynomials?
- binomials in quadratic form
- binomials in cubic form
- trinomials in quadratic form
- four term polynomials using grouping
- polynomials missing constant
Can you use Pascal's Triangle to expand powers of a binomial?

Rational Expressions

Can you combine two rationals expressions into a single rational expression?
- addition
- subtraction
- mutiplication
- division
Can you determine the domain restrictions of a rational expression?
Can you determine the zeros of a rational expression?
Can you rewrite complex fractions

Graphs and Transformations

Can you graph each of the types of function or a relations regardless of form?
- linear
- quadratic
- cubic
- absolute value
- radical
- logarithmic
- exponential
Can you transform the parent function?
- linear
- quadratic
- cubic
- absolute value
- radical
- logarithmic
- exponential
Do you understand the impact of type and signs related to coefficients, constants, bases, and other related parts on the graph of a function or a relation?
- linear
- quadratic
- cubic
- absolute value
- radical
- logarithmic
- exponential
Can you express the domain and range a function?
Can you determine the intercepts from a function algebraically or with the assistance of technology?
Can you determine the "special" points from a function algebraically or with the assistance of technology?
- intercepts
- integer coordinates
- rational coordinates
- local extremes
Can you determine the "specific" points from a function algebraically or with the assistance of technology?
- given x find y
- given y find x
- tables that increment by a specific amount

Solving Equations

Can you solve an equation algebraically?
- linear
- quadratic
- cubic
- absolute value
- radical
- logarithmic
- exponential
Can you connect a graph of a system to its related equation's solutions?
Can you use the rational root theorem for polynomials?
Can you use tables and the location principle to find an estimate of a solution?
Can you use technology to confirm solutions to equations?
Can you use technology to find solutions to equations?
Do you understand what it means for an equation to not have a solution algebraically?
Do you understand what it means for an equation to not have a solution graphically?

Trigonometry

Angles and Measures

Can you define a radian and support the definition of radian with a picture?
Can you express an angle in a variety of units?
- degrees
- radians
- DMS notation
Can you determine the complement of an angle?
Can you determine the supplement of an angle?
Can you determine the explement of an angle?
Can you express an angle in terms of its rotation?
Can you express a rotation in terms of its angle?
Can you express an angle using positive and negative angles?
Can you use reference angles in each quadrant?
Do you know the special trigonometric values in quadrant one?
Do you know the special trigonometric values in quadrant two?
Do you know the special trigonometric values in quadrant three?
Do you know the special trigonometric values in quadrant four?
Do you know the special trigonometric values on the axes?

Unit Circle

Do you know why the unit circle is called the unit circle?
Do you know the coordinate of all points on the unit circle in rectangle coordinate form?
Do you know the coordinate of all points on the unit circle in polar coordinate form?
Do you know the special coordinates in each quadrant and on the axes related to the unit circle?
Can you place the sixteen special coordinates on the unit circle?
Can you connect the unit circle to the graphs of sine and cosine?
Can you connect the unit circle to the graphs of the other trigonometric functions?

Right Triangle Trigonometry

Determine angles and sides of right triangle using SOHCAHTOA
Determine missing sides of a right triangle using Pythagorean Theorem
Classify a triangle based on side lengths
Classify a triangle based on angle measurement
Use right triangles to find perimeter and area of polygons.
Use right triangles to find surface area and volume of solids.
Apply trigonometric models in real life context

Trigonometric functions of any angle

be able to state all six trigonometric ratios related to any angle
connect basic trigonometric functions to their related reciprocal functions
Use advanced notation to state all angles that have a specific trigonometric ratio.
Connect the points on the graphs of trigonometric functions
- sine to cosine
- sine and cosine to tangent
- sine to cosecant
- cosine to secant
- tangent to cotangent
- secant and cosecant to cotangent
Understand the difference between reciprocal trigonometric function and inverse trigonometric function

Graphs of Trigonometric Functions

Graphing a single period of any trigonometric function
Graphing multiple periods of any trigonometric function
Graphing specific domain of any trigonometric function
Determine the special points that lie on any trigonometric function
- local extremes
- y intercept
- x intercept
- points on the midline
Determine the cause and location of a domain restriction for reciprocal and quotient trigonometric functions
- tangent
- cosecant
- secant
- cotangent
Determine the cause and location for any vertical asymptotes for trigonometric functions
- tangent
- cosecant
- secant
- cotangent

Inverse Trigonometric Functions

Determine the domain and range of inverse trigonometric functions
Determine special points that lie on inverse trigonometric functions
Determine asymptotes related to inverse trigonometric functions
Graph the inverse trigonometric functions

Applications and Models that Involve Trigonometric Functions

Arc Length, Linear Speed, and Angular Speed
Applications of periodic nature
Applications related to a right triangle

Analytic Trigonometry

Fundamental Trigonometric Identities

Can you use the trigonometric identities to rewrite expressions?
- Pythagorean Identities
- Quotient Identities
- Reciprocal Identities
- Even Odd Identities
- Co-function Identities

Verification of Trigonometric Identities

Can you derive trigonometric identities?
- Pythagorean Identities
- Quotient Identities
- Reciprocal Identities
- Even Odd Identities
- Co-function Identities

Solving Trigonometric Equations

Determine an exact and approximate solution of a trigonometric equation algebraically
Determine an exact and approximate solution of a trigonometric equation graphically
Determine all exact and approximate solutions of a trigonometric equation algebraically
Determine all exact and approximate solutions of a trigonometric equation graphically
Determine specific exact and approximate solutions of a trigonometric equation over a given domain algebraically
Determine all exact and approximate solution of a trigonometric equation over a given domain graphically

Trigonometric Formulas

Rewrite trigonometric expressions using trigonometric formulas
- Sum and Difference Formulas
- Double Angle Formulas
- Half Angle Formulas
- Sum to Product Formulas
- Product to Sum Formulas

Additional Topics in Trigonometry

Law of Sines

Be able to solve an oblique triangle when given three pieces of information related to a triangle
Recognition of when to use the Law of Sines to find an angle or a side of an oblique triangle
- ASA (Angle Included Side Angle)
- AAS (Angle Angle Non-Included Side)
- SSA (Side Side Non-Included Side)
Be able to determine when and why the ambiguous case occurs
- SSA -0 Solutions
- SSA- 1 Solution
- SSA- 2 Solutions

Law of Cosines

Be able to solve an oblique triangle when given three pieces of information related to a triangle
Recognition of when to use the Law of Cosines to find an angle or a side of an oblique triangle
- SAS (Side Included Angle Side)
- SSS (Side Side SIde)

Applications of Trigonometry

Be able to use trigonometry to find the area of a triangle using a variety of methods
- Heron's Formula (SSS)
- SAS Triangle Area Formula
Be able to use trigonometry to determine areas and perimeters of regular and irregular polygons
Be able to apply the Laws of Sine and Cosine to solve real-life problems

Vectors in the Plane

basic vector notations and vocabulary

Be able to use proper notation, abbreviation, and vocabulary as it relates to vector mathematics in the plane
Be able to write a direction vector in component form
Be able to write a direction vector in unit vector form
Be able to write a direction vector using polar coordinates
Be able to distinguish the difference between a unit vector and unit vector form
Be able to determine the unit vector related to a direction vector regardless of form
Be able to determine if vectors are parallel or perpendicular
Be able to determine the angle formed with x axis and a vector
Be able to determine the angle formed by two vectors

vector operations

Be able to perform vector operations
- addition
- subtraction
- scalar multiplication
- dot product
Be able to write a vector equation related to a vector triangle or vector polygon

plotting vectors

Be able to plot a given vector on a coordinate grid
Be able to plot a vector that is parallel to a given vector on a coordinate grid
Be able to plot a perpendicular (orthogonal) vector to a given vector on a coordinate grid
Be able to use the parallelogram law to show vector operations
Be able to plot a scalar multiple to a given vector on a coordinate grid
Be able to plot a vector equation as a vector polygon on a coordinate grid

applications of vectors

Be able to apply properties and operations with vectors in a real-life setting
- use vectors to determine speed of an airplane or boat influenced by wind or current.
- use vectors to model basic physics applications

Matrices

basic matrix notations and vocabulary

Know and properly use notation and vocabulary related to matrix
- row of a matrix
- column of a matrix
- dimension of a matrix
- inverse matrix
- scalar of a matrix
- inner dimension of a matrix product
- outer dimension of a matrix product
- square matrix
- rectangular matrix
- identity matrix
- matrix equation
- transpose matrix
- augmented matrix equation

matrix operations

Be able to perform operations with matrices and be able to determine when and why an matrix operation is impossible.
- matrix addition
- matrix subtraction
- scalar multiplication involving a matrix
- matrix multiplication
Be able to find the determinant of square 2 x 2 and 3 x 3 matrices algebraically
Be able to find the determinant of any square matrix using technology
Be able to find the inverse of a 2 x 2 matrix algebraically
Be able to find the inverse of any square matrix
Be able to explain why a matrix product is not possible
Be able to explain why a matrix would have a determinant of 0
Be able to explain why a matrix equation will have no single solution
Be able to explain why a matrix equation will have no solution
Be able to solve 2 x 2 and 3 x 3 matrix equations algebraically
Be able to solve 2 x 2 and 3 x 3 matrix equations using Cramer's Rule and determinants
Be able to solve any n x n matrix equation using matrix inverses

applications of matrices

Be able to set up and use matrices in an inventory scenario
Be able to use coordinates and determinants to find areas of triangles
Be able to use coordinates and matrices to build polynomials from their coordinates

Sequences and Series

sequence and series notation and vocabulary

Be able to use a sequence of no less than three values to write general rule for all terms in sequence
Be able to use proper notation, abbreviations, and vocabulary associated with sequences and series
- common difference
- common ratio
- first term in a sequence
- last term in a finite sequence
- nth term in a sequence
- arithmetic sequence
- geometric sequence
- recursive sequence
- sum of a finite sequence
- sum of an infinite sequence
Be able to classify a sequence properly

arithmetic sequence and series

Be able to determine when a given sequence of no less than three terms is arithmetic
Be able to write an arithmetic sequence rule for a given arithmetic sequence
Be able to determine the number of terms are in a finite sum of an arithmetic sequence

geometric sequence and series

Be able to determine when a given sequence of no less than three terms is geometric
Be able to write a geometric sequence rule for a given geometric sequence
Be able to determine the number of terms are in a finite sum of an geometric sequence
Be able to determine the sum of a geometric sequence of infinite number of terms

recursive series and sequence

Be able to write a recursive rule for an arithmetic sequence
Be able to write a recursive rule for an geometric sequence
Be able to write a recursive rule for for a given sequence

factorials, permutations, and combinations

Be able to use proper vocabulary and notations associated with factorials, permutations, and combinations
Be able to determine the value of factorials
Interpret the meaning of a factorial in relation to arrangement of items
Understand and be able to use the algebraic definitions and notations associated with permutation
Understand and be able to use the algebraic definitions and notations associated with permutation
Be able to distinguish the difference between permutation of n items, permutation of n choose r items, and the combination of n choose r items
Connect permutation, combination, and factorial to probability
Be able to use technology to calculate permutations, combinations, and factorials

summation

Be able to use summation notation to represent finite and infinite sums
Be able to use algebraic methods to determine finite and infinite sums
Be able to use technology to find given sums and general rule for a sequence

Advanced Functions and Relations

Exponential Functions and their Transformations

Determine the domain and range of exponential functions
Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of an exponential function
Be able to graph the parent exponential function related to a specific base with and without technology
Be able to use proper vocabulary related to transformation of exponential functions
Be able to distinguish the difference between exponential functions that have a base that is between 0 and 1 and exponential functions that have a base that is greater than 1
Connect exponential functions to logarithmic functions
Use exponential functions in a real-life scenario

Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a logarithmic function

Logarithmic Functions and their Transformations

Determine the domain and range of logarithmic functions
Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a logarithmic function
Be able to graph the parent logarithmic function related to a specific base with and without technology
Be able to use proper vocabulary related to transformation of logarithmic functions
Be able to distinguish the difference between logarithmic functions that have a base that is between 0 and 1 and logarithmic functions that have a base that is greater than 1
Connect exponential functions to logarithmic functions
Use logarithmic functions in a real-life scenario

Polynomial Functions and their Transformations

Determine the domain and range of polynomial functions
Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a polynomial function
Be able to graph the polynomial function with and without technology
Be able to use proper vocabulary related to transformation of polynomial functions
Use polynomial functions in a real-life scenario

Rational Functions and their Transformations

Determine the domain and range of rational functions
Be able to determine intercepts, asymptotes, integer coordinates, and rational coordinates that lie on the graph of a rational function
Be able to graph the rational function with and without technology
Be able to use proper vocabulary related to transformation of rational functions
Use rational functions in a real-life scenario

Piecewise Functions

Be able to describe the domain and range of a piecewise function
Be able to graph a piecewise function with and with out technology
Be able to classify a piecewise function as continuous or not continuous
Be able to write a piecewise function from its graph
Be able to write a piecewise function for absolute value functions.
Be able to write a piecewise function for quadratic functions.

Conic Sections

Be able to use proper notation and vocabulary related to a conic section
- Center
- points on graph
- vertices
- co-vertices
- focus and foci
- minor axis
- major axis
- conjugate axis
- transverse axis
Be able to write a standard form equation of a conic section based on its parts
Graph a conic section from its standard form
- Parabola
- Circle
- Ellipse
- Hyperbola
Graph a section from its general form
- Parabola
- Circle
- Ellipse
- Hyperbola
Apply conic sections in a real-lef scenario

Parabola

Circle

Ellipse

Hyperbola

Parabola

Circle

Ellipse

Hyperbola

Polar Graphs

Be able to plot points on a polar grid
Be able to use proper notation and vocabulary related to polar graphs
- circles with center at (0,0)
- circles tangent to the x axis
- circles tangent to the y axis
- diagonal lines
- vertical lines
- Limacons
- Cardiods
- Rose Curves
- Lemniscates
Be able to plot a variety of polar graphs with and without technology

Probability

Be able to use proper notation and vocabulary related to single and multiple event probabilities
- single event probability
- multiple event probability
- conditional probability
- simple events
- compound events
- independent events
- dependent events
- mutually exclusive events
- complement of an event
- sample space
- Venn diagram
Be able to determine probabilities algebraically
Be able to determine probabilities with the assistance of technology
Be able to use regression to model data
Connect probability to permutations and combinations
Be able to use the Fundamental Counting Principle.
Be able to read and create Venn Diagrams to aid in the comprehension and determination of probabilities