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Program of Study for High School Mathematics

This course is designed for advanced students who are capable of a more rigorous course at an accelerated pace. Algebra 2 and Trigonometry (313736) contains all of the content in Algebra 2 Honors (313536) and Trigonometry (315000). Students are expected to master algebraic mechanics and to understand the underlying theory in order to apply concepts to real-world situations in a meaningful way. A thorough treatment of advanced algebraic concepts is provided through the study of functions, polynomials, rational expressions, complex numbers, matrices, sequences and series, permutations and combinations, triangular and circular functions, trigonometric identities and equations, inverse trigonometric functions, and selected topics in discrete mathematics. Emphasis is on modeling, logic, and interpretation of results. A transformational approach to graphing is used with families of related graphs. Numerical, graphical, and algebraic solutions should be considered for all problems where appropriate. Graphing utilities, especially graphing calculators, are integral to the course.

All Virginia Standards of Learning for Algebra 2 and Trigonometry are addressed in this course. The number in parentheses following each Essential Knowledge and Skill refers to the related SOL objective. Required Program of Studies indicators that are not addressed in the Algebra 2 and Trigonometry Virginia SOL are in italics type.

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Required Basal Textbooks
  • Schultz, et al. Algebra 2. Holt, Rinehart, & Winston. 2003.
  • Collins, et al. Glencoe Algebra 2: Integration, Applications, Connections.
    Glencoe/ McGraw-Hill. 2001.
  • Larson, Boswell, Kanold, & Stiff. McDougal-Littell Algebra 2. McDougal-Littell. 2001.

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Benchmark

Students develop techniques for solving rational and radical equations that model real-world situations.

Key Terms

complex conjugate complex fraction complex number
constant of variation direct variation domain
extraneous root horizontal asymptote imaginary number
index (indices) inverse variation linear function
power radical radical equation/expression
radicand range rational equation/expression
rational number real number vertical asymptote

Essential Knowledge and Skills/Indicators

  1. State the domain and range of a rational function. (AII/T.9)
  2. Identify the vertical and horizontal asymptotes of a rational function. (AII/T.9)
  3. Sketch the graph of a rational function. (AII/T.9)
  4. Identify a variation as direct, inverse, or joint. (AII/T.20)
  5. Solve practical problems involving joint variation. (AII/T.20)
  6. Simplify rational algebraic expressions. (AII/T.7)
  7. Compare simplifying rational algebraic expressions to simplifying fractions. (AII/T.7)
  8. Add, subtract, multiply, and divide rational algebraic expressions including complex fractions. (AII/T.2)
  9. Solve rational equations algebraically and graphically. The graphing calculator will be used as a primary tool for solution and for checking the algebraic solution. (AII/T.7)
  10. Sketch the graph of a rational function using zeros, “holes,” and vertical and horizontal asymptotes.
  11. Simplify radical expressions with a variety of indices (rational and integral) until:
    • the index, n, is as small as possible
    • the radicand contains no factors that are the nth powers of an integer or polynomial
    • the radicand contains no fractions
    • no radicals appear in the denominator. (AII/T.3)
  12. Rationalize the denominator of a rational expression that contains a radical expression in the denominator. (AII/T.3)
  13. Add, subtract, multiply, and divide radical expressions. (AII/T.3)
  14. Write expressions in radical form as expressions with rational exponents. (AII/T.3)
  15. Evaluate expressions in either exponential or radical form. (AII/T.3)
  16. Solve equations containing radical expressions and equations of nth roots algebraically and graphically. (AII/T.7)
  17. Check solutions for extraneous roots. (AII/T.7)
  18. Recognize a complex number as a number that can be written as a + bi where a and b are real numbers and i is the principal square root of –1. (AII/T.17)
  19. Recognize pure imaginary numbers. (AII/T.17)
  20. Simplify square roots with negative arguments. (AII/T.17)
  21. Represent a complex number geometrically in the coordinate plane. (AII/T.17)
  22. Add, subtract, and multiply complex numbers. (AII/T.17)
  23. Compare and contrast adding, subtraction, and multiplying complex numbers with operating on real numbers. (AII/T.17)
  24. Simplify powers of i and generalize the pattern. (AII/T.17)
  25. Simplify rational expressions with complex numbers in the denominator by using complex conjugates. (AII/T.17)
  26. Place the following sets of numbers in a hierarchy: complex numbers, pure imaginary numbers, real numbers, rational and irrational numbers, integers, whole numbers, and natural numbers. Venn diagrams may be used. (AII/T.1)

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II. Systems of Equations and Inequalities

Benchmark

Students develop techniques for solving systems of equations and inequalities that model real-world situations.

Key Terms

coefficient matrix identity matrix matrix application
compound linear inequality inverse quadratic-quadratic system
constraints linear programming scalar
dimensions (order) linear-quadratic system variable matrix
feasible region matrix

Essential Knowledge and Skills/Indicators

  1. Organize data into matrices and identify the dimensions of the matrix. (AII/T.11)
  2. Investigate commutative and associative properties of matrix addition. Compare and contrast matrix addition with addition of real numbers. (AII/T.1)
  3. Multiply matrices using a calculator or a computer with matrix capability. (AII/T.11)
  4. Solve problems that require matrix multiplication. (AII/T.11)
  5. Investigate commutative and associative properties of matrix multiplication. Compare and contrast matrix multiplication with multiplication of real numbers. (AII/T.1)
  6. Find the determinant of a square matrix. (AII/T.12)
  7. Identify the identity matrix (I). (AII/T.12)
  8. For a matrix A, find the inverse matrix A-1 (if it exists) such that A * A-1 = A-1 * A = I. (AII/T.12)
  9. Use an inverse matrix to solve matrix equations. Use the graphing calculator or a computer application with matrix capabilities. (AII/T.12)
  10. Compare and contrast solving matrix equations and linear equations. (AII/T.1)
  11. Represent a system of equations as a matrix equation where the coefficient matrix times the variable matrix equals the constant matrix. (AII/T.11)
  12. Solve systems of linear equations using inverse matrices. Use the graphing calculator or a computer application with matrix capabilities. (AII/T.12)
  13. Represent real-world situations with a 2 or 3 - variable system of linear equations and solve the system using an appropriate method (i.e. graphs, substitution, linear combination, or inverse matrices).
  14. Solve a system of linear inequalities by graphing. (AII/T.13)
  15. Identify examples of the properties of inequality and order that occur while solving inequalities. (AII/T.13)
  16. Find the maximum and minimum values of a function over a region (linear programming). (AII/T.13)
    • Identify the constraints in a practical situation and model them as inequalities.
    • Graph the system of inequalities and identify the area of intersection as the feasible region. The feasible region contains all solutions possible.
    • The maximum and minimum values of the function occur at the vertices of the feasible region. Substitute the coordinates of each vertex of the feasible region into the function to determine which vertex yields the maximum (or minimum) value of the function.
    • Describe the results of a linear programming problem orally and in writing.
  17. Solve linear-quadratic systems of equations algebraically and identify the set of ordered pairs that is the solution to the system. (AII/T.14)
  18. Solve linear-quadratic systems of equations graphically and identify the set of ordered pairs that is the solution to the system. (AII/T.14)
  19. Solve quadratic-quadratic systems of equations algebraically and identify the set of ordered pairs that is the solution to the system. (AII/T.14)
  20. Solve quadratic-quadratic systems of equations graphically and identify the set of
    ordered pairs that is the solution to the system. (AII/T.14)

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III. Relations and Functions

Benchmark

Students apply a transformational approach with graphing calculators to analyze, describe, and compare a variety of functions and relations.

Key Terms

absolute value hole root
axioms of equality horizontal shift rotation
circle hyperbola scatter graph (plot)
composite function image solution
conic sections inverse function step function
curve of best fit logarithmic function symmetry
dilation natural logarithm (ln) transformation
directrix major axis translations
e minor axis vertex
exponential function parent function/graph vertical shift
focus (foci) quadratic function zero
(h,k) form reflection

Essential Knowledge and Skills/Indicators

  1. Recognize the graphs and equations of parent functions such as y = x, y = x2, y = x3, y = |x| y = ax, step, and other polynomial functions. (AII/T.8, AII/T.15)
  2. Apply transformations (translations, reflections, dilations, and rotations) and combinations of transformations to parent graphs. (AII/T.8, AII/T.15)
  3. Given an image graph of a function, describe the transformations that were performed on the pre-image and the order in which they could have occurred. (AII/T.8, AII/T.15)
  4. Given the equation of a parent graph, vary the coefficients and constants of the equation, observe the changes in the graph of the parent, and generalize the changes to the graphs of other functions. (AII/T.8, AII/T.15)
  5. Build a strong connection between the algebraic and geometric representations of functions. (AII/T.8, AII/T.15)
  6. Graph linear, absolute value, quadratic, step, piecewise, and exponential functions and convert between a graph, a table, and symbolic form, and use these in real-world situations. (AII/T.8, AII/T.15)
  7. Collect data and display it in a scatter graph. (AII/T.19)
  8. Analyze data using measures of central tendency (mean, median, and mode), the range of the data, and box-and-whiskers plots. (AII/T.19)
  9. Determine the equation of the curve of best fit using the graphing calculator. Consider the graphs of the parent functions when determining which curve might be appropriate. Use the equation to make predictions. (AII/T.19)
  10. Find the probability of an event using probability principles and use these to answer questions about real-world problems.
  11. Investigate the commutative and associative properties of combinations of transformations. (AII/T.1)
  12. Given the graph or the equation of a function, identify the domain and range of the function. Include functions with discontinuities. (AII/T.9, AII/T.15)
  13. Find the value of a function, or composition of multiple functions, for a given element in the domain. (AII/T.9)
  14. Find the composition of functions algebraically and graphically. (AII/T.9)
  15. Find the inverse of a function, or composition of multiple functions, algebraically and graphically. (AII/T.9)
  16. Explain how composition of functions and finding the inverse of a function affects the domain and range of the functions. (AII/T.9)
  17. Demonstrate that the exponential and logarithmic functions are inverse functions. (AII/T.9)
  18. Factor polynomials completely (difference of two squares, perfect square trinomials, general trinomials, sum and difference of cubes). (AII/T.5)
  19. Find the roots of a function algebraically and graphically. Quadratic equations may be solved using a variety of techniques, which include but are not limited to factoring to use the zero product property, square roots, the quadratic formula, completing the square, and graphing. (AII/T.6)
  20. Solve exponential equations (including e) and logarithmic equations (including ln). (AII/T.9)
  21. Investigate the relationship between the solutions of an equation, zeros of a function, x-intercepts, and factors of a polynomial. (AII/T.10)
  22. Identify examples of the field properties of real numbers and the properties of equality that occur while solving equations. (AII/T.1)
  23. Given the roots of a polynomial, write an equation for the polynomial function. (AII/T.6)
  24. Identify the relationship between polynomial functions and their graphs, and write an appropriate equation when given the properties of a polynomial graph.
  25. Use synthetic division and synthetic substitution to find all roots (both real and complex) of a polynomial function.
  26. Define absolute value. (AII/T.4)
  27. Evaluate expressions that contain absolute value. (AII/T.4)
  28. Recognize that the value of the argument for absolute value must be greater than or equal to zero and explain why. (AII/T.4)
  29. Solve absolute value equations algebraically and graphically. (AII/T.4)
  30. Solve absolute value inequalities algebraically and graphically. (AII/T.4)
  31. Identify examples of the properties of inequality and order that occur while solving inequalities. (AII/T.1)
  32. Recognize the graphs of the conic sections defined as any figure that can be formed by slicing a double cone (parabola, ellipse, circle, and hyperbola). (AII/T.18)
  33. Given an equation, identify the conic section and graph it using transformations. (AII/T.18)

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IV. Sequences and Series

Benchmark

Students investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems.

Key Terms

arithmetic sequence

common ratio

geometric series

arithmetic series

 geometric mean nth term

common difference

geometric sequence

S notation

Essential Knowledge and Skills/Indicators

  1. Find the next term in a sequence by looking for a pattern. (AII/T.16)
  2. Find the nth term of an arithmetic sequence and find the position of a given term in an arithmetic sequence. (AII/T.16)
  3. Find arithmetic means. (AII/T.16)
  4. Differentiate between a sequence and a series. (AII/T.16)
  5. Find the sum of an arithmetic series. (AII/T.16)
  6. Find specific terms in an arithmetic series. (AII/T.16)
  7. Use sigma (S) notation to denote sums. (AII/T.16)
  8. Compare and contrast arithmetic and geometric sequences. (AII/T.16)
  9. Find the nth term of a geometric sequence. (AII/T.16)
  10. Find geometric means. (AII/T.16)
  11. Find the sum of a geometric series. (AII/T.16)
  12. Find specific terms in a geometric series. (AII/T.16)
  13. Find the sum of an infinite geometric series.
  14. Expand binomials having positive integral exponents through the use of the binomial theorem, combinations, and Pascal’s triangle.
  15. Identify and classify recursive functions and use recursive functions to model real-world phenomena.

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V. Optional Topics

“Other Topics in Algebra 2” are not assessed on the SOL assessment.

Key Terms

combinations

parametric equations

infinite geometric series

permutations

Essential Knowledge and Skills/Indicators

  1. Count the number of permutations and combinations possible in a given situation. Use counting techniques in binomial experiments to determine binomial probabilities.
  2. Graph points and equations in three dimensions.
  3. Study parametric equations and the graphs of parametric equations.

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VI. Triangular & Circuluar Functions

Benchmark

Students apply algebraic techniques to analyze, describe, and compare triangular and circular functions.

Key Terms

angular velocity

initial side

sine

circular functions

 linear velocity special angles

cosecant

quadrantal angle

standard position
cosine radian tangent
cotangent reference angle terminal side
coterminal angle secant unit circle
degree

Essential Knowledge and Skills/Indicators

  1. Define the six triangular trigonometric functions of an angle in a right triangle. (AII/T.21)
  2. Define the six circular trigonometric functions of an angle in standard position. (AII/T.21)
  3. Make the connection between the triangular and circular trigonometric functions. (AII/T.21)
  4. Recognize and draw an angle in standard position. (AII/T.21)
  5. Show how a point on the terminal side of an angle determines a reference triangle. (AII/T.21)
  6. Given one trigonometric function value, find the other five trigonometric function values. (AII/T.22)
  7. Develop the unit circle, using both degrees and radians. (AII/T.22)
  8. Solve problems, using the circular function definitions and the properties of the unit circle. (AII/T.22)
  9. Recognize the connections between the coordinates of points on a unit circle and
    1. coordinate geometry;
    2. cosine and sine values; and
    3. lengths of sides of special right triangles (30o-60o-90o and 45o-45o-90o). (AII/T.22)
  10. Find trigonometric function values of special angles and their related angles in both degrees and radians. (AII/T.23)
  11. Apply the properties of the unit circle without using a calculator. (AII/T.23)
  12. Use a conversion factor to convert from radians to degrees and vice versa without using a calculator. (AII/T.23)
  13. (Optional) Find the length of an arc, linear speed, and angular velocities.

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VII. Solving Triangles

Benchmark

Students solve real-world problems conducive to right triangle solutions.

Key Terms

Hero's formula

Law of Cosines

Law of SInes

Pythagorean Theorem

Essential Knowledge and Skills/Indicators

  1. Write a practical problem involving triangles. (AII/T.29)
  2. Solve practical problems involving triangles. (AII/T.29)
  3. Use the trigonometric functions, Pythagorean Theorem, Law of Sines, and Law of Cosines to solve practical problems. (AII/T.29)
  4. Identify a solution technique that could be used with a given problem. (AII/T.29)
  5. Find the area of triangles using Hero’s formula and the sine formula.

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VIII. Trigonometric Identities, Equations, and Inverses

Benchmark

Students apply algebraic techniques to establish and verify trigonometric relationships.

Key Terms

difference identities

infinite solution

reciprocal identities

double angle identities

 Pythagorean identities restricted domain
half-angle identities quotient identities sum identities
identities range trigonometric equation

Essential Knowledge and Skills/Indicators

  1. Use trigonometric identities to make algebraic substitutions to simplify and verify trigo-nometric identities. (AII/T.25)
    The basic trigonometric identities include:
    1. reciprocal identities;
    2. Pythagorean identities;
    3. sum and difference identities;
    4. double-angle identities; and
    5. half-angle identities.
  2. Solve trigonometric equations with restricted domains algebraically and by using a graph-ing utility. (AII/T.27)
  3. Solve trigonometric equations with infinite solutions algebraically and by using a graphing utility. (AII/T.27)
  4. Check for reasonableness of results, and verify algebraic solutions, using a graphing utility. (AII/T.27)
  5. Use a calculator to find the trigonometric function values of any angle in either degrees or radians. (AII/T.24)
  6. Define inverse trigonometric functions. (AII/T.24)
  7. Find angle measures by using the inverse trigonometric functions when the trigonometric function values are given. (AII/T.24)
  8. Find the domain and range of the inverse trigonometric functions. (AII/T.27)
  9. Use the restrictions on the domains of the inverse trigonometric functions in finding the values of the inverse trigonometric functions. (AII/T.27)
  10. (Optional) Use the half-angle identities to verify other trigonometric identities and solve equations.