Math Honors 2
Key Learning Objectives
Unit 1: Linear Algebra and Matrices
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Write and solve linear equations in one variable
Solve a literal equation for the indicated variable
Interpret and use set notation and interval notation to describe sets of discrete or continuous numbers
Write, solve, and graph linear inequalities and compound linear inequalities in one variable
Solve and graph absolute value equations and inequalities
Express a linear equation in slope-intercept form, standard form, and point-slope form
Write an equation for a line that contains a given point and is parallel or perpendicular to a given line
Classify a system of equations as inconsistent, consistent and independent, or consistent and dependent
Determine whether an ordered pair is a solution to a system of equations
Solve a system of two linear equations in two variables graphically (by hand and by using the calculator)
Solve a system of linear equations algebraically (by substitution and by elimination)
Represent data in a matrix, and use matrix dimensions to determine whether matrices can be added, subtracted, or multiplied
Multiply by transformational matrices to apply translations, reflections, and size changes to a figure with coordinate vertices
Find the determinant of a square matrix, and use it to establish whether the matrix has an inverse
Find sums, differences, and products of matrices, the product of a scalar and a matrix, and the inverse of a square matrix
Write a system of linear equations as a matrix equation, and use the inverse of the coefficient matrix to solve the system
Use the determinant of the coefficient matrix to classify a system of equations as independent, or dependent or inconsistent
Unit 2: Functions and Quadratics
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State the domain and range of a relation, and state whether it is a function
Evaluate functions at particular values, and find compositions and inverses of functions
Use the horizontal line test to determine whether the inverse is a function
Recognize and graph constant, linear, quadratic, polynomial, square root, reciprocal, and exponential functions
Define and graph piecewise functions, absolute-value functions, and step functions
Identify and graph transformations of functions
Describe the key features of the graph of a quadratic function given in root form, standard form, and vertex form
Graph a quadratic function
Use factoring to solve a quadratic equation and to express a quadratic function in root form to find its real roots
Use completing the square to solve a quadratic equation and to express a quadratic function in vertex form to find its vertex
Express a quadratic function in standard form and use the quadratic formula to find its zeros
Determine the number of real solutions for a quadratic equation by using the discriminant
Add, subtract, multiply, divide, and graph complex numbers, and find and graph the conjugate of a complex number
Write, solve, and graph a quadratic inequality in one variable or in two variables
Unit 3: Polynomials and Rational Functions
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Identify polynomials and classify them by degree and by number of terms
Evaluate, add, and subtract polynomials, writing answers in standard form
Use polynomial functions to solve real-world applications (e.g. cost and profit functions, annuity investments, etc.)
Graph a polynomial function and describe its general shape and important characteristics (end behavior, etc.)
Find the coordinates of the local maxima and the local minima in the graph of a polynomial function
Determine the intervals over which the graph of a polynomial function is increasing and/or decreasing
Multiply and factor polynomials, and divide one polynomial by another using long division and synthetic division
Solve polynomial equations and find the real, rational, and complex zeros of polynomial functions by using a graph, the
Location Principle, the Rational Root Theorem, the Remainder Theorem, the Factor Theorem, the Complex Conjugate Root
Theorem, long division or synthetic division, variable substitution, and factoring, and state the multiplicity of each zero
Use the Fundamental Theorem of Algebra to write a polynomial function given sufficient information about its zeros
Find the domain of a rational function, and graph a rational function by determining the coordinates of its intercepts and
holes, and the equations of its vertical and horizontal or slant asymptotes
Identify and evaluate rational functions, and write a rational function given its intercepts, holes, and asymptotes
Factor, multiply, divide, add, subtract, and simplify rational expressions, including complex fractions
Solve a rational equation or a rational inequality using algebra, a table, and a graph, and discarding any extraneous solutions
Unit 4: Power, Radical, Exponential, and Logarithmic Functions
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Simplify and evaluate expressions involving powers by using the laws of exponents
Find the domains of radical functions, and analyze the graphs of radical functions by describing the applied transformations
Find the inverse of a quadratic function by completing the square or by using the quadratic formula
Add, subtract, multiply, simplify, and evaluate radical expressions by using the laws of radicals
Divide and simplify radical expressions by rationalizing the denominator
Solve radical equations by using algebra and checking for extraneous solutions, or by using graphing technology
Solve radical inequalities by using graphing technology
Classify exponential functions as representing exponential growth or exponential decay, and determine their multipliers
Write and evaluate expressions that model exponential growth and decay situations (e.g. growth of investments, etc.)
Write equivalent forms for exponential and logarithmic equations
Simplify and evaluate logarithmic expressions by using the laws of logarithms, including the change-of-base formula
Define the common logarithmic, the natural exponential, and the natural logarithmic functions to evaluate expressions
Solve exponential and logarithmic equations by using algebra, or by using graphing technology
Unit 5: Trigonometric and Inverse Trigonometric Functions
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Solve right triangles using the Pythagorean theorem and the basic trigonometric ratios
Solve non-right triangles using the law of sines (including the ambiguous case) and the law of cosines
Find the area of a triangle given the measures of two sides and the measure of the angle in between those sides
Determine the number of triangles that can be formed from a given set of side and angle measures
Find the exact values of the six trigonometric functions of acute angles and angles in standard position
Convert between degree measure and radian measure, and find the length of an arc and the area of a sector
Find co-terminal and reference angles, and hence find exact values of trigonometric functions of special angles
Use reference angles and the unit circle to solve for special angles in given trigonometric ratios
Graph the sine, cosine, and tangent functions, their reciprocal and inverse functions, and transformations on these graphs
Determine the domain, range, amplitude, period, frequency, and horizontal and vertical shifts in given trigonometric graphs
Evaluate trigonometric expressions involving inverses, including composite trigonometric functions
Use fundamental trigonometric identities to rewrite trigonometric expressions and to formally prove trigonometric identities
Evaluate, simplify, and rewrite expressions using the sum/difference identities and the double-angle identities
Solve trigonometric equations and inequalities using various algebraic techniques, or by graphing on the calculator
Unit 6: Statistics and Probability
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Find the mean, median, mode, range, quartiles, interquartile range, variance, and standard deviation of a data set
Estimate the mean and standard deviation from an ungrouped or grouped frequency table of data
Make a stem-and-leaf plot, histogram, circle graph, box-and-whisker plot, and cumulative frequency graph from a data set
Represent a sample space as a list, a lattice diagram, a Venn diagram, or a tree diagram
Calculate probabilities of combined events involving AND, OR, or NOT for mutually exclusive events and inclusive events
Recognize the meanings of probabilities of 0 and 1, and understand how to prove that two events are mutually exclusive
Understand that the sum of all probabilities equals 1, and solve problems using the probability of the complement of an event
Find the probability of two independent events occurring, and understand how to prove that two events are independent
Use Venn diagrams and tree diagrams to solve problems involving combined events and/or conditional probability