Alaska Precalculus Math Standards

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9-12.N Number and Quantity

• 9-12.N-CN The Complex Number System
  • Perform arithmetic operations with complex numbers.
    • 9-12.N-CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
  • Represent complex numbers and their operations on the complex plane.
    • 9-12.N-CN.4 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.
    • 9-12.N-CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 – √3i)³ = 8 because (1 – √3i) has modulus 2 and argument 120°.
    • 9-12.N-CN.6 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.
• 9-12.N-VM Vector and Matrix Quantities
  • Represent and model with vector quantities.
    • 9-12.N-VM.1 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).
    • 9-12.N-VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
    • 9-12.N-VM.3 Solve problems involving velocity and other quantities that can be represented by vectors.
  • Perform operations on vectors.
    • 9-12.N-VM.4 Add and subtract vectors.
      • 9-12.N-VM.4.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.
      • 9-12.N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
      • 9-12.N-VM.4.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.
    • 9-12.N-VM.5 Multiply a vector by a scalar.
      • 9-12.N-VM.5.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).
      • 9-12.N-VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
  • Perform operations on matrices and use matrices in applications.
    • 9-12.N-VM.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
    • 9-12.N-VM.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
    • 9-12.N-VM.8 Add, subtract, and multiply matrices of appropriate dimensions.
    • 9-12.N-VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
    • 9-12.N-VM.10 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.
    • 9-12.N-VM.11 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.
    • 9-12.N-VM.12 Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

9-12.A Algebra

• 9-12.A-REI Reasoning with Equations and Inequalities
  • Solve systems of equations.
    • 9-12.A-REI.8 Represent a system of linear equations as a single matrix equation in a vector variable.
    • 9-12.A-REI.9 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).

9-12.F Functions

• 9-12.F-IF Interpreting Functions
  • Analyze functions using different representations.
    • 9-12.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
      • 9-12.F-IF.7.d Graph rational functions, identifying zeros and discontinuities (asymptotes/holes) using technology, and algebraic methods when suitable factorizations are available, and showing end behavior.
• 9-12.F-BF Building Functions
  • Build a function that models a relationship between two quantities.
    • 9-12.F-BF.1 Write a function that describes a relationship between two quantities.
      • 9-12.F-BF.1.c Compose functions.
  • Build new functions from existing functions.
    • 9-12.F-BF.4 Find inverse functions.
      • 9-12.F-BF.4.b Verify by composition that one function is the inverse of another.
      • 9-12.F-BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
      • 9-12.F-BF.4.d Produce an invertible function from a non-invertible function by restricting the domain.
    • 9-12.F-BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
• 9-12.F-TF Trigonometric Functions
  • Extend the domain of trigonometric functions using the unit circle.
    • 9-12.F-TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
    • 9-12.F-TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  • Model periodic phenomena with trigonometric functions.
    • 9-12.F-TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
    • 9-12.F-TF.7 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.
  • Prove and apply trigonometric identities.
    • 9-12.F-TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

9-12.G Geometry

• 9-12.G-GPE Expressing Geometric Properties with Equations
  • Translate between the geometric description and the equation for a conic section.
    • 9-12.G-GPE.3 Derive the equations of ellipses and hyperbolas given foci and directrices.
• 9-12.G-GMD Geometric Measurement and Dimension
  • Explain volume formulas and use them to solve problems.
    • 9-12.G-GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

9-12.S Statistics and Probability

• 9-12.S-MD Using Probability to Make Decisions
  • Calculate expected values and use them to solve problems.
    • 9-12.S-MD.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.
    • 9-12.S-MD.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
    • 9-12.S-MD.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.
    • 9-12.S-MD.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
  • Use probability to evaluate outcomes of decisions.
    • 9-12.S-MD.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
      • 9-12.S-MD.5.a Find the expected payoff for a game of chance.
      • 9-12.S-MD.5.b Evaluate and compare strategies on the basis of expected values.