Standards - Mathematics

Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals using rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Define the imaginary number i such that $i^2 = -1$.

Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

Use the properties of exponents to transform expressions for exponential functions.

Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.

Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation.

Select an appropriate method to solve a quadratic equation in one variable.

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. Explain how the quadratic formula is derived from this form.

Solve quadratic equations by inspection (such as $x^2 = 49$), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

Select an appropriate method to solve a system of two linear equations in two variables.

Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.

Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.

Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.

Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. Note: The graph of a relation often forms a curve (which could be a line).

Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: 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 $x$.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions.

Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation $y = f(x)$.

Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Use function composition to combine different types of standard functions to write and evaluate functions.

Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.

Explain why the x-coordinates of the points where the graphs of the equations $y = f(x)$ and $y = g(x)$ intersect are the solutions of the equation $f(x) = g(x)$.

Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Note: Include cases where $f(x)$ is a linear, quadratic, exponential, or absolute value function and $g(x)$ is constant or linear.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.

Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.

Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.

Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions.