Subjects
Shows
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows [...]
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret [...]
Prove the Pythagorean identity (sin A)^2 + (cos A)^2 = 1 and use it to find sin A, cos A, [...]
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, [...]
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the [...]
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, [...]
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on [...]
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only [...]
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of [...]
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of [...]
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect [...]
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior [...]
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic [...]
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body [...]
(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) [...]
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between [...]
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) [...]
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write [...]
(+) Verify by composition that one function is the inverse of another.
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
(+) Produce an invertible function from a non-invertible function by restricting the domain.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Distinguish between situations that can be modeled with linear functions and with exponential functions.*
Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over [...]
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*
Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric [...]
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) [...]
For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers [...]
Interpret the parameters in a linear or exponential function in terms of a context.*
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted [...]
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) [...]
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of [...]
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element [...]
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of [...]
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci [...]
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of [...]
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, [...]
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified [...]
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for [...]
Graph linear and quadratic functions and show intercepts, maxima, and minima.*
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*
Subjects
Shows
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows [...]
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret [...]
Prove the Pythagorean identity (sin A)^2 + (cos A)^2 = 1 and use it to find sin A, cos A, [...]
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, [...]
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the [...]
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, [...]
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on [...]
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only [...]
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of [...]
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of [...]
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect [...]
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior [...]
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic [...]
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body [...]
(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) [...]
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between [...]
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) [...]
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write [...]
(+) Verify by composition that one function is the inverse of another.
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
(+) Produce an invertible function from a non-invertible function by restricting the domain.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Distinguish between situations that can be modeled with linear functions and with exponential functions.*
Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over [...]
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*
Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric [...]
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) [...]
For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers [...]
Interpret the parameters in a linear or exponential function in terms of a context.*
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted [...]
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) [...]
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of [...]
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element [...]
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of [...]
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci [...]
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of [...]
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, [...]
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified [...]
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for [...]
Graph linear and quadratic functions and show intercepts, maxima, and minima.*
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*
