Download Pre-calculus

Pre-calculus
I. Functions and Their Graphs
A. Graphs of Equations
1) Sketch graphs of equations
2) Find x- and y- intercepts of graphs of equations
3) Use symmetry to sketch graphs of equations
4) Find equations and sketch graphs of circles
5) Use graphs of equations in solving real-life problems
B. Linear
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Equations in Two Variables
Use slope to graph linear equations in two variables
Find slopes of lines
Write linear equations in two variables
Use slope to identify parallel and perpendicular lines
Use linear equations in two variables to model and solve real-life problems
C. Functions
1) Determine whether relations between two variables are functions
2) Use function notation and evaluate functions
3) Find the domain of functions
4) Use functions to model and solve real-life problems
D. Analyzing Graphs of Functions
1) Use the Vertical Line Test for functions
2) Find the zeros of functions
3) Determine intervals on which functions are increasing or decreasing
4) Identify even and odd functions
E. Graphing Functions
1) Identify and graph linear and quadratic functions
2) Identify and graph cubic, square root, and reciprocal functions
3) Identify and graph step and other piecewise-defined functions
4) Recognize graphs of common functions
F. Translations of Functions
1) Use vertical and horizontal shifts to sketch graphs of functions
2) Use reflections to sketch graphs of functions
3) Use non-grid transformations to sketch graphs of functions
G. Combinations of functions
1) Add, subtract, multiply, and divide functions
2) Find the composition of one function with another function
3) Use combinations of functions to model and solve real-life problems
H. Inverse Functions
1) Find inverse functions informally and verify that two functions are inverse
functions of each other
2) Use graphs of functions to determine whether functions have inverse
functions
3) Use the Horizontal Line Test to determine if functions are one-to-one
4) Find inverse functions algebraically
I. Mathematical Modeling
1) Use mathematical models to approximate sets of data points
2) Write mathematical models for direct variation, nth power, inverse variation,
and joint variation
3) Use the regression feature of a graphing utility to find the equation of a
least squares regression line
II. Polynomial and Rational Functions
A. Quadratic Functions
1) Analyze graphs of quadratic functions
2) Write quadratic functions in standard form and use the results to sketch
graphs of functions
3) Use quadratic functions to model and solve real-life problems
B. Polynomial Functions of Higher Degree
1) Use transformations to sketch graphs of polynomial functions
2) Use the Leading Coefficient Test to determine the end behavior of graphs
and polynomial functions
3) Use zeros of polynomial functions as sketching aids
4) Use the Intermediate Value Theorem to help locate zeros of polynomial
functions
C. Polynomial and Synthetic Division
1) Use long division to divide polynomials by other polynomials
2) Use synthetic division to divide polynomials by binomials of the form (x-k)
3) Use the Remainder Theorem and the Factor Theorem
D. Complex Numbers
1) Use the imaginary unit i to write complex numbers
2) Add, subtract, and multiply complex numbers
3) Write the quotient of two complex numbers in standard form
4) Find complex solutions of quadratic equations
E. Zeros of Polynomial Functions
1) Use the fundamental Theorem of Algebra to determine the number of zeros
of polynomial functions
2) Find rational zeros of polynomial functions
3) Find conjugate pairs of complex zeros
4) Find zeros of polynomials by factoring
5) Use Descartes’ Rule of Signs and the Upper and Lower Bound Rules to find
zeros of polynomials
F. Rational Functions
1) Find the domains of rational functions
2) Find the horizontal and vertical asymptotes of graphs and rational functions
3) Analyze and sketch graphs of rational functions
4) Sketch graphs of rational functions that have slant asymptotes
G. Partial Fractions
1) Recognize partial fraction decompositions of rational expressions
2) Find partial fraction decompositions of rational expressions
III. Exponential and Logarithmic Functions
A. Exponential Functions and Their Graphs
1) Recognize and evaluate exponential functions with base a
2) Graph exponential functions
3) Recognize and evaluate exponential functions with base e
4) Use exponential functions to model and solve real-life applications
B. Logarithmic Functions and Their Graphs
1) Recognize and evaluate logarithmic functions with base a
2) Graph logarithmic functions
3) Recognize and evaluate natural logarithmic functions
4) Use logarithmic functions to model and solve real-life applications
C. Properties of Logarithms
1) Know how to use the change-of-base formula to rewrite and evaluate
logarithmic expressions
2) Use properties of logarithms to evaluate or rewrite logarithmic expressions
3) Use properties of logarithms to expand or condense logarithmic expressions
4) Use logarithmic functions to model and solve real-life applications
D. Exponential and Logarithmic Equations
1) Solve simple exponential and logarithmic equations
2) Solve more complicated exponential and logarithmic equations
3) Use exponential and logarithmic equations to model and solve real-life
applications
E. Exponential and Logarithmic Models
1) Recognize the five most common types of models involving exponential and
logarithmic functions
2) Use Gaussian, logistic growth, and logarithmic functions to model and solve
real-life problems
IV. Matrices and Determinants
A. Matrices and Systems of Equations
1) Write a matrix and identify its order
2) Perform elementary row operations on matrices
3) Use matrices and Gaussian elimination to solve system of equations of
linear equations
4) Use matrices and Gauss-Jordan elimination to solve systems of linear
equations
B. Operations with Matrices
1) Decide whether two matrices are equal
2) Add and subtract matrices and multiply matrices by real numbers
3) Multiply two matrices
4) Use matrix operations to model and solve real-life problems
C. The Inverse of a Square Matrix
1) Verify that two matrices are inverses of each-other
2) Use Gauss-Jordan elimination to find the inverses of matrices
3) Use a formula to find the inverses of 2 x 2 matrices
4) Use inverse matrices to solve systems of linear equations
D. The Determinant of a Square Matrix
1) Find the determinants of 2 x 2 matrices, minors and cofactors of square
matrices, and the determinants of square matrices
E. Applications of Matrices and Determinants
1) Use Cramer’s Rule to solve systems of linear equations
2) Use determinants to find the areas of triangles
3) Use a determinant to test for collinear points and find an equation of a line
passing through two points
V.
Sequences, Series, and Probability
A. Sequences and Series
1) Use sequence notation to write the terms of a sequence
2) Use Factorial Notation
3) Use summation Notation to write sums
4) Find the sum of an infinite series
5) Use sequences and series to model and solve real-life problems
B. Arithmetic and Geometric Sequences and Partial Sums
1) Recognize and write arithmetic and geometric sequences
2) Find an nth partial sum of an arithmetic sequence
3) Use arithmetic and geometric sequences to model and solve real-life
problems
4) Find the sum of finite arithmetic and geometric sequences
5) Find the infinite sum of geometric sequences
C. Find the sums of powers of integers
D. Binomial Theorem
1) Calculate coefficients by combinations and Pascal’s Triangle
2) Expand a binomial using coefficients
E. Counting Principles
1) Solve simple counting problems
2) Use Fundamental Counting Principles to solve counting problems
3) Use permutations and combinations to solve counting problems
F. Probability
1) Find probabilities of events
2) Find probabilities of mutually exclusive and independent events
3) Find the probability of the complement of an event
VI. Topics in Analytic Geometry
A. Lines
1) Find the inclination of a line, the angle between two lines, and the distance
between a point and a line
B. Introduction to Conics: Parabolas
1) Recognize a conic as the intersection of a plane and a double-napped cone
2) Write the standard form of the equation of a parabola
3) Find and equation of a tangent line to a parabola at a given point
4) Use the reflective property of parabolas to solve real-life problems
C. Ellipses
1) Write the standard form of the equation of a hyperbola
2) Use properties of ellipses to model and solve real-life problems
3) Find the eccentricity of an ellipse
D. Hyperbolas
1) Write the standard form of the equation of an hyperbola
2) Find the asymptotes of a hyperbola
3) Use properties of hyperbolas to solve real-life problems
4) Classify a conic from its general equation
E. Rotation of Conics
1) Rotate the coordinate axes to eliminate the xy-term in the equation of a
conic
2) Use the discriminate to classify a conic
F. Parametric Equations
1) Evaluate a set of parametric equations for a given value of the parameter
2) Sketch the curve that is represented by a set of parametric equations and
rewrite the equations as a single rectangle equation
3) Find a set of parametric equations for a graph
G. Polar Coordinates
1) Plot points on the polar coordinate system
2) Convert points from rectangle to polar form and vice versa
3) Convert equations from rectangular to polar form and vice versa
H. Graphs of Polar Equations
1) Graph Polar Equations by point plotting
2) Use symmetry, zeros, and maximum r-values to sketch graphs of polar
equations
3) Recognize special polar graphs
I. Polar Equations of Conics
1) Define conics in terms of eccentricity and write equations of conics in polar
form
2) Use equations of Conics in polar form to model real-life problems