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Skills available for West Virginia Precalculus standards

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College- and Career-Readiness Standards: Advanced Mathematical Modeling College- and Career-Readiness Standards: Advanced Mathematical Modeling
College- and Career-Readiness Standards: Mathematics I College- and Career-Readiness Standards: Mathematics I
College- and Career-Readiness Standards: Mathematics II College- and Career-Readiness Standards: Mathematics II
College- and Career-Readiness Standards: Mathematics III LA College- and Career-Readiness Standards: Mathematics III LA
College- and Career-Readiness Standards: Mathematics III STEM College- and Career-Readiness Standards: Mathematics III STEM
College- and Career-Readiness Standards: Mathematics III Technical Readiness College- and Career-Readiness Standards: Mathematics III Technical Readiness
College- and Career-Readiness Standards: Mathematics IV - Trigonometry/Pre-Calculus College- and Career-Readiness Standards: Mathematics IV - Trigonometry/Pre-Calculus
College- and Career-Readiness Standards: Mathematics IV Technical Readiness College- and Career-Readiness Standards: Mathematics IV Technical Readiness
College- and Career-Readiness Standards: STEM Readiness College- and Career-Readiness Standards: STEM Readiness
College- and Career-Readiness Standards: Transition Mathematics for Seniors College- and Career-Readiness Standards: Transition Mathematics for Seniors
West Virginia Next Generation Content Standards and Objectives (Common Core): High School Mathematics IV West Virginia Next Generation Content Standards and Objectives (Common Core): High School Mathematics IV
West Virginia Next Generation Content Standards and Objectives (Common Core): High School Mathematics STEM Readiness West Virginia Next Generation Content Standards and Objectives (Common Core): High School Mathematics STEM Readiness

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11-12.M.4HS.CVM Building Relationships among Complex Numbers, Vectors, and Matrices

11-12. Perform arithmetic operations with complex numbers.
- 11-12.M.4HS.CVM.1 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
11-12. Represent complex numbers and their operations on the complex plane.
- 11-12.M.4HS.CVM.2 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- 11-12.M.4HS.CVM.3 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
- 11-12.M.4HS.CVM.4 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
  - Midpoints in the complex plane (PC-S.7)
  - Distance in the complex plane (PC-S.8)
11-12. Represent and model with vector quantities.
- 11-12.M.4HS.CVM.5 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
- 11-12.M.4HS.CVM.6 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- 11-12.M.4HS.CVM.7 Solve problems involving velocity and other quantities that can be represented by vectors.
11-12. Perform operations on vectors.
- 11-12.M.4HS.CVM.8 Add and subtract vectors.
  - 11-12.M.4HS.CVM.8.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
  - 11-12.M.4HS.CVM.8.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
    - Find the magnitude and direction of a vector sum (PC-U.9)
  - 11-12.M.4HS.CVM.8.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
- 11-12.M.4HS.CVM.9 Multiply a vector by a scalar.
  - 11-12.M.4HS.CVM.9.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).
    - Multiply a vector by a scalar (PC-U.10)
    - Scalar multiples of three-dimensional vectors (PC-V.4)
  - 11-12.M.4HS.CVM.9.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
    - Find the magnitude of a vector scalar multiple (PC-U.11)
    - Determine the direction of a vector scalar multiple (PC-U.12)
11-12. Perform operations on matrices and use matrices in applications.
- 11-12.M.4HS.CVM.10 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- 11-12.M.4HS.CVM.11 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- 11-12.M.4HS.CVM.12 Add, subtract, and multiply matrices of appropriate dimensions.
- 11-12.M.4HS.CVM.13 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- 11-12.M.4HS.CVM.14 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- 11-12.M.4HS.CVM.15 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- 11-12.M.4HS.CVM.16 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
11-12. Solve systems of equations
- 11-12.M.4HS.CVM.17 Represent a system of linear equations as a single matrix equation in a vector variable.
- 11-12.M.4HS.CVM.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

11-12.M.4HS.ASF Analysis and Synthesis of Functions

11-12. Analyze functions using different representations
- 11-12.M.4HS.ASF.1 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  - 11-12.M.4HS.ASF.1.a Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
  - 11-12.M.4HS.ASF.1.b utilize an informal notion of limit to analyze asymptotes and continuity in rational functions.
    - Rational functions: asymptotes and excluded values (A1-HH.1)
    - Rational functions: asymptotes and excluded values (A2-O.1)
11-12. Build a function that models a relationship between two quantities
- 11-12.M.4HS.ASF.2 write a function that describes a relationship between two quantities, including composition of functions.
11-12. Build new functions from existing functions
- 11-12.M.4HS.ASF.3 Find inverse functions.
  - 11-12.M.4HS.ASF.3.a Verify by composition that one function is the inverse of another.
  - 11-12.M.4HS.ASF.3.b Read values of an inverse function from a graph or a table, given that the function has an inverse.
  - 11-12.M.4HS.ASF.3.c Produce an invertible function from a non-invertible function by restricting the domain.
- 11-12.M.4HS.ASF.4 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

11-12.M.4HS.TF Trigonometric and Inverse Trigonometric Functions of Real Numbers

11-12. Extend the domain of trigonometric functions using the unit circle
- 11-12.M.4HS.TF.1 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi–x, pi+x, and 2pi–x in terms of their values for x, where x is any real number.
- 11-12.M.4HS.TF.2 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  - Symmetry and periodicity of trigonometric functions (A2-BB.2)
  - Symmetry and periodicity of trigonometric functions (PC-O.2)
11-12. Model periodic phenomena with trigonometric functions
- 11-12.M.4HS.TF.3 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- 11-12.M.4HS.TF.4 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
- 11-12.M.4HS.TF.5 solve more general trigonometric equations.
  - Solve trigonometric equations I (A2-Z.14)
  - Solve trigonometric equations II (A2-Z.15)
11-12. Prove and apply trigonometric identities
- 11-12.M.4HS.TF.6 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
11-12. Apply transformations of function to trigonometric functions.
- 11-12.M.4HS.TF.7 graph trigonometric functions showing key features, including phase shift.

11-12.M.4HS.AG Derivations in Analytic Geometry

11-12. Translate between the geometric description and the equation for a conic section
- 11-12.M.4HS.AG.1 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
11-12. Explain volume formulas and use them to solve problems
- 11-12.M.4HS.AG.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

11-12.M.4HS.MP Modeling with Probability

11-12. Calculate expected values and use them to solve problems
- 11-12.M.4HS.MP.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
- 11-12.M.4HS.MP.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
  - Expected values of random variables (A2-EE.4)
  - Expected values of random variables (PC-Y.4)
- 11-12.M.4HS.MP.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
- 11-12.M.4HS.MP.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
11-12. Use probability to evaluate outcomes of decisions
- 11-12.M.4HS.MP.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
  - 11-12.M.4HS.MP.5.a Find the expected payoff for a game of chance.
    - Expected values for a game of chance (A2-EE.8)
    - Expected values for a game of chance (PC-Y.8)
  - 11-12.M.4HS.MP.5.b Evaluate and compare strategies on the basis of expected values.
    - Choose the better bet (A2-EE.9)
    - Choose the better bet (PC-Y.9)

11-12.M.4HS.SL Series and Informal Limits

11-12. Use sigma notations to evaluate finite sums.
- 11-12.M.4HS.SL.1 develop sigma notation and use it to write series in equivalent form.
  - Introduction to sigma notation (PC-W.10)
- 11-12.M.4HS.SL.2 apply the method of mathematical induction to prove summation formulas.
11-12. Extend geometric series to infinite geometric series.
- 11-12.M.4HS.SL.3 develop intuitively that the sum of an infinite series of positive numbers can converge and derive the formula for the sum of an infinite geometric series.
  - Convergent and divergent geometric series (PC-W.17)
  - Find the value of an infinite geometric series (PC-W.18)
- 11-12.M.4HS.SL.4 apply infinite geometric series models. For example, find the area bounded by a Koch curve.
  - Write a repeating decimal as a fraction (PC-W.19)

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11-12.M.4HS.CVM Building Relationships among Complex Numbers, Vectors, and Matrices

11-12. Perform arithmetic operations with complex numbers.

11-12.M.4HS.CVM.1 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

11-12. Represent complex numbers and their operations on the complex plane.

11-12.M.4HS.CVM.2 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

11-12.M.4HS.CVM.3 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

11-12.M.4HS.CVM.4 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

11-12. Represent and model with vector quantities.

11-12.M.4HS.CVM.5 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

11-12.M.4HS.CVM.6 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

11-12.M.4HS.CVM.7 Solve problems involving velocity and other quantities that can be represented by vectors.

11-12. Perform operations on vectors.

11-12.M.4HS.CVM.8 Add and subtract vectors.

11-12.M.4HS.CVM.8.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

11-12.M.4HS.CVM.8.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

11-12.M.4HS.CVM.9 Multiply a vector by a scalar.

11-12.M.4HS.CVM.9.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

11-12.M.4HS.CVM.9.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

11-12. Perform operations on matrices and use matrices in applications.

11-12.M.4HS.CVM.10 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

11-12.M.4HS.CVM.11 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

11-12.M.4HS.CVM.12 Add, subtract, and multiply matrices of appropriate dimensions.

11-12.M.4HS.CVM.13 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

11-12.M.4HS.CVM.14 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

11-12.M.4HS.CVM.15 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

11-12.M.4HS.CVM.16 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

11-12. Solve systems of equations

11-12.M.4HS.CVM.17 Represent a system of linear equations as a single matrix equation in a vector variable.

11-12.M.4HS.CVM.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

11-12.M.4HS.ASF Analysis and Synthesis of Functions

11-12. Analyze functions using different representations

11-12.M.4HS.ASF.1 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

11-12.M.4HS.ASF.1.a Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

11-12.M.4HS.ASF.1.b utilize an informal notion of limit to analyze asymptotes and continuity in rational functions.

11-12. Build a function that models a relationship between two quantities

11-12.M.4HS.ASF.2 write a function that describes a relationship between two quantities, including composition of functions.

11-12. Build new functions from existing functions

11-12.M.4HS.ASF.3 Find inverse functions.

11-12.M.4HS.ASF.3.a Verify by composition that one function is the inverse of another.

11-12.M.4HS.ASF.3.b Read values of an inverse function from a graph or a table, given that the function has an inverse.

11-12.M.4HS.ASF.3.c Produce an invertible function from a non-invertible function by restricting the domain.

11-12.M.4HS.ASF.4 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

11-12.M.4HS.TF Trigonometric and Inverse Trigonometric Functions of Real Numbers

11-12. Extend the domain of trigonometric functions using the unit circle

11-12.M.4HS.TF.1 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi–x, pi+x, and 2pi–x in terms of their values for x, where x is any real number.

11-12.M.4HS.TF.2 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

11-12. Model periodic phenomena with trigonometric functions

11-12.M.4HS.TF.3 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

11-12.M.4HS.TF.4 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

11-12.M.4HS.TF.5 solve more general trigonometric equations.

11-12. Prove and apply trigonometric identities

11-12.M.4HS.TF.6 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

11-12. Apply transformations of function to trigonometric functions.

11-12.M.4HS.TF.7 graph trigonometric functions showing key features, including phase shift.

11-12.M.4HS.AG Derivations in Analytic Geometry

11-12. Translate between the geometric description and the equation for a conic section

11-12.M.4HS.AG.1 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

11-12. Explain volume formulas and use them to solve problems

11-12.M.4HS.AG.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

11-12.M.4HS.MP Modeling with Probability

11-12. Calculate expected values and use them to solve problems

11-12.M.4HS.MP.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

11-12.M.4HS.MP.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

11-12.M.4HS.MP.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

11-12.M.4HS.MP.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

11-12. Use probability to evaluate outcomes of decisions

11-12.M.4HS.MP.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

11-12.M.4HS.MP.5.a Find the expected payoff for a game of chance.

11-12.M.4HS.MP.5.b Evaluate and compare strategies on the basis of expected values.

11-12.M.4HS.SL Series and Informal Limits

11-12. Use sigma notations to evaluate finite sums.

11-12.M.4HS.SL.1 develop sigma notation and use it to write series in equivalent form.

11-12.M.4HS.SL.2 apply the method of mathematical induction to prove summation formulas.

11-12. Extend geometric series to infinite geometric series.

11-12.M.4HS.SL.3 develop intuitively that the sum of an infinite series of positive numbers can converge and derive the formula for the sum of an infinite geometric series.

11-12.M.4HS.SL.4 apply infinite geometric series models. For example, find the area bounded by a Koch curve.

11-12.M.4HS.CVM.9.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).