Common Core Standards
12th Grade · High school · Math
Official Common Core math standards for 12th Grade. These are the standards used in adopting states.
Standards
Click a standard for the full text and future study materials.
Building Functions
- HSF.BF.A.1 — Write a function that describes a relationship between two quantities.*
- HSF.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*
- HSF.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
- HSF.BF.B.4 — Find inverse functions.
- HSF.BF.B.5 — (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Interpreting Functions
- HSF.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- HSF.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- HSF.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
- HSF.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
- HSF.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*
- HSF.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
- HSF.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
- HSF.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- HSF.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Linear, Quadratic, & Exponential Models
- HSF.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.
- HSF.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- HSF.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
- HSF.LE.A.4 — For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
- HSF.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.
Trigonometric Functions
- HSF.TF.A.1 — Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
- HSF.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
- HSF.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 x, π + x, and 2π - x in terms of their values for x, where x is any real number.
- HSF.TF.A.4 — (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
- HSF.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
- HSF.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.
- HSF.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.*
- HSF.TF.C.8 — Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
- HSF.TF.C.9 — (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
The Complex Number System
- HSN.CN.A.1 — Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
- HSN.CN.A.2 — Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- HSN.CN.A.3 — (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- HSN.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.
- HSN.CN.B.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 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
- HSN.CN.B.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.
- HSN.CN.C.7 — Solve quadratic equations with real coefficients that have complex solutions.
- HSN.CN.C.8 — (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).
- HSN.CN.C.9 — (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Quantities
- HSN.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- HSN.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
- HSN.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
The Real Number System
- HSN.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
- HSN.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- HSN.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Vector & Matrix Quantities
- HSN.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).
- HSN.VM.A.2 — (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- HSN.VM.A.3 — (+) Solve problems involving velocity and other quantities that can be represented by vectors.
- HSN.VM.B.4 — (+) Add and subtract vectors.
- HSN.VM.B.5 — (+) Multiply a vector by a scalar.
- HSN.VM.C.6 — (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- HSN.VM.C.7 — (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- HSN.VM.C.8 — (+) Add, subtract, and multiply matrices of appropriate dimensions.
- HSN.VM.C.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.
- HSN.VM.C.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.
- HSN.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.
- HSN.VM.C.12 — (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
© Common Core State Standards Initiative. Used under CC BY 4.0. See thecorestandards.org for the official standards.
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