Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Create equations that describe numbers or relationships
A-CED.1
Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
Solve equations and inequalities in one variable
A-REI.4
Solve quadratic equations in one variable.
A-REI.4.a
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
A-REI.4.b
Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Solve systems of equations
A-REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
Interpret the structure of expressions
A-SSE.1
Interpret expressions that represent a quantity in terms of its context
A-SSE.1.a
Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.1.b
Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
A-SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
Write expressions in equivalent forms to solve problems
A-SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.3.a
Factor a quadratic expression to reveal the zeros of the function it defines.
A-SSE.3.b
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A-SSE.3.c
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t 1.01212t to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.
Build a function that models a relationship between two quantities
F-BF.1
Write a function that describes a relationship between two quantities
F-BF.1.a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.1.b
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Build new functions from existing functions
F-BF.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.
F-BF.4
Find inverse functions.
F-BF.4.a
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠1.
Interpret functions that arise in applications in terms of the context
F-IF.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.
F-IF.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.
F-IF.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.
Analyze functions using different representations
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.
F-IF.7.a
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.7.b
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.8.a
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F-IF.8.b
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F-IF.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.
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.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.
Interpret expressions for functions in terms of the situation they model
F-LE.6
Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.
Prove and apply trigonometric identities
F-TF.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.
Understand and apply theorems about circles
G-C.1
Prove that all circles are similar.
G-C.2
Identify and describe relationships among inscribed angles, radii, and chords.
G-C.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Find arc lengths and areas of sectors of circles
G-C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians.
Prove geometric theorems
G-CO.10
Prove theorems about triangles.
G-CO.11
Prove theorems about parallelograms.
G-CO.9
Prove theorems about lines and angles.
Explain volume formulas and use them to solve problems
G-GMD.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
G-GMD.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G-GMD.5
Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k2, and k3, respectively; determine length, area and volume measures using scale factors.
G-GMD.6
Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.
Translate between the geometric description and the equation for a conic section
G-GPE.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G-GPE.2
Derive the equation of a parabola given a focus and directrix.
Use coordinates to prove simple geometric theorems algebraically
G-GPE.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G-GPE.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Understand similarity in terms of similarity transformations
G-SRT.1
Verify experimentally the properties of dilations given by a center and a scale factor:
G-SRT.1.a
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G-SRT.1.b
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
G-SRT.4
Prove theorems about triangles.
G-SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
G-SRT.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.7
Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-SRT.8.1
Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°).
Perform arithmetic operations with complex numbers.
N-CN.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.
N-CN.2
Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Use complex numbers in polynomial identities and equations.
N-CN.7
Solve quadratic equations with real coefficients that have complex solutions.
Extend the properties of exponents to rational exponents.
N-RN.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.
N-RN.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
N-RN.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.