Idaho Precalculus Math Standards

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N Number and quantity

• N.CN The Complex Number System
  • N.CN.A Perform arithmetic operations with complex numbers.
    • N.CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find absolute value and quotients of complex numbers.
  • N.CN.B Represent complex numbers and their operations on the complex plane.
    • N.CN.B.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.
    • N.CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
    • N.CN.B.6 (+) Calculate the distance between numbers in the complex plane as the absolute value of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
• N.VM Vector and Matrix Quantities
  • N.VM.A Represent and model with vector quantities.
    • N.VM.A.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).
    • N.VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
    • N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
  • N.VM.B Perform operations on vectors.
    • N.VM.B.4 (+) Add and subtract vectors.
      • N.VM.B.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.
      • N.VM.B.4.b (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
      • N.VM.B.4.c (+) Demonstrate understanding of 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 componentwise.
    • N.VM.B.5 (+) Multiply a vector by a scalar.
      • N.VM.B.5.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).
      • N.VM.B.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).
  • N.VM.C Perform operations on matrices and use matrices in applications.
    • N.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
    • N.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
    • N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
    • N.VM.C.9 (+) Demonstrate understanding that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
    • N.VM.C.10 (+) Demonstrate understanding 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.
    • N.VM.C.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.
    • N.VM.C.12 (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

A Algebra

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

F Functions

• F.IF Interpreting Functions
  • F.IF.C Analyze functions using different representations.
    • F.IF.C.7 Graph functions expressed symbolically and show key features of the graphs, by hand in simple cases and using technology for more complicated cases.
      • F.IF.C.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
      • F.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
      • F.IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
      • F.IF.C.7.d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
      • F.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
• F.BF Building Functions
  • F.BF.A Build a function that models a relationship between two quantities.
    • F.BF.A.1 Write a function that describes a relationship between two quantities. Functions could include linear, exponential, quadratic, simple rational, radical, logarithmic, and trigonometric.
      • F.BF.A.1.b Combine standard function types using arithmetic operations.
  • F.BF.B Build new functions from existing functions.
    • F.BF.B.4 Find inverse functions algebraically and graphically.
      • F.BF.B.4.b (+) Verify by composition that one function is the inverse of another.
      • F.BF.B.4.c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
      • F.BF.B.4.d (+) Produce an invertible function from a non-invertible function by restricting the domain.
    • F.BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
• F.TF Trigonometric Functions
  • F.TF.A Extend the domain of trigonometric functions using the unit circle.
    • F.TF.A.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, cosine, and tangent for π − x, π + x, and 2π − x in terms of their values for x, where x is any real number.
    • F.TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  • F.TF.B Model periodic phenomena with trigonometric functions.
    • F.TF.B.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.
    • F.TF.B.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.
  • F.TF.C Prove and apply trigonometric identities.
    • F.TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G Geometry

• G.GPE Expressing Geometric Properties with Equations
  • G.GPE.A Translate between the geometric description and the equation for a conic section.
    • G.GPE.A.3 (+) 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.
      • G.GPE.A.3.a (+) Use equations and graphs of conic sections to model real-world problems.
• G.GMD Geometric Measurement and Dimension
  • G.GMD.A Explain volume formulas and use them to solve problems.
    • G.GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

S Statistics and probability

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