Overview

• Find the factors (linear components) of a quadratic given in standard form symbolically, graphically, or from a table.
• Determine if a function given in table form is quadratic by looking at the change in the change.
• Determine the domain and range in the context of a given quadratic situation.
• Express quadratic functions in vertex form, factored form, and standard form.
• Apply transformations to quadratic functions and represent symbolically.
• Solve quadratic equations and inequalities graphically, with a table, and symbolically.
• Recognize and define the imaginary number i.
• Determine if a given situation can be modeled by a linear function, a quadratic function, or neither. If it is linear or quadratic, write a function to model it.

A.PA.08.02 For basic functions, e.g., simple quadratics, direct and indirect variation, and population growth, describe how changes in one variable affect the others.

A.PA.08.03 Recognize basic functions in problem context, e.g., area of a circle is ?r², volume of a sphere is ⁴/₃?r ³, and represent them using tables, graphs, and formulas.

A.RP.08.04 Use the vertical line test to determine if a graph represents a function in one variable.

A.RP.08.05 Relate quadratic functions in factored form and vertex form to their graphs, and vice versa. In particular, note that solutions of a quadratic equation are the x-intercepts of the corresponding quadratic function.

A.RP.08.06 Graph factorable quadratic functions, finding where the graph intersects the x-axis and the coordinates of the vertex; use words "parabola" and "roots"; include functions in vertex form and those with leading coefficient -1, e.g., y = x² - 36, y = (x - 2)² - 9; y = - x ²; y = - (x - 3)².

A.FO.08.07 Recognize and apply the common formulas: (a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b² ; (a + b)(a - b) = a² - b²; represent geometrically.

A.FO.08.08 Factor simple quadratic expressions with integer coefficients, e.g.,
x² + 6x + 9, x² + 2x - 3, and x² - 4; solve simple quadratic equations, e.g., x² = 16 or x² = 5 (by taking square roots); x² - x - 6 = 0, x² - 2x = 15 (by factoring); verify solutions by evaluation.

A3.3.1 Write the symbolic form and sketch the graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.)

A3.3.2 Identify the elements of a parabola (vertex, axis of symmetry, and direction of opening) given its symbolic form or its graph and relate the elements to the coefficient(s) of the symbolic form of the function.

A3.3.3 Convert quadratic functions from standard to vertex form by completing the square.

A3.3.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

A3.3.5 Express quadratic functions in vertex form to identify their maxima and minima and in factored form to identify their zeros.

A1.1.3 Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities.

A1.2.1 Write and solve equations and inequalities with one or two variables to represent mathematical or applied situations.

A1.2.2 Associate a given equation with a function whose zeros are the solutions of the equation.

A1.2.3 Solve linear and quadratic and inequalities, including systems of up to three linear equations with three unknowns. Justify steps in the solutions, and apply the quadratic formula appropriately.

A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution.

A2.1.1 Determine whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.

A2.1.2 Read, interpret, and use function notation and evaluate a function at a value in its domain.

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.

A2.1.6 Identify the zeros of a function and the intervals where the values of a function are positive or negative, and describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations.

A2.1.7 Identify and interpret the key features of a function from its graph or its formula(s).

A2.2.1 Combine functions by addition, subtraction, multiplication and division.

A2.2.2 Apply given transformations to parent functions and represent symbolically.

A2.2.3 Determine whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.

A2.3.2 Describe the tabular pattern associated with functions having constant rate of change (linear) or variable rates of change.

A2.4.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers.

A2.4.3 Using the adapted general symbolic form draw reasonable conclusions about the situation being modeled.