Skill Descriptor: Apply understanding of the absolute value of real numbers.

Global Competencies: CTPS

Achievement Indicators:

• Relate the distance of two real numbers in the form ± 𝑎, 𝑎 ∈ 𝑅 to the absolute value of a (|a|)
• Determine the absolute values of positive and negative real numbers
• Solve numerical expressions by determining absolute value
• Order the absolute values of real numbers in given sets

Skill Descriptor: Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands.

Global Competencies: CTPS

Achievement Indicators:

• Order radical expressions with numerical radicands in given sets
• Convert between entire radicals and mixed radicals
• Simplify radical expressions with numerical or variable radicands (maximum index of 2)
• Rationalize the denominator of rational expressions with monomial or binomial denominators
• Relate the conjugate to the difference of squares by demonstrating products as rationalized
• Identify the values of the variable for which given radical expressions are defined, using equalities and inequalities

Skill Descriptor: Solve radical equations.

Global Competencies: CTPS, ICE

Achievement Indicators:

• Determine any restrictions on values for the variable in radical equations
• Determine the roots of radical equations algebraically and verify using technology
• Verify the values determined in solving radical equations
• Demonstrate roots that are extraneous by solving radical equations algebraically
• Model situations using radical equations

Skill Descriptor: Determine equivalent forms of rational expressions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Demonstrate the non-permissible value(s) for given rational expressions
• Identify all non-permissible values, including those not found in given simplified expressions, and explain why they still exist
• Simplify rational expressions
• Identify and correct errors in simplifications of rational expressions

Skill Descriptor: Perform operations with rational expressions and express in simplified form.

Global Competencies: CTPS

Achievement Indicators:

• Determine the non-permissible values when performing operations on rational expressions
• Determine the sum or difference of rational expressions with the same and with different denominators
• Determine the product or quotient with rational expressions
• Determine the simplified form of expressions that involve two or more operations

Skill Descriptor: Solve rational equations.

Global Competencies: CM, CTPS

Achievement Indicators:

• Determine the non-permissible values for the variable in rational equations
• Determine the solution to rational equations algebraically
• Explain why values obtained in solving rational equations may not be solutions of the equations
• Model situations using rational equations

Skill Descriptor: Apply understanding of angles from 0° to 360° in standard position.

Global Competencies: CTPS

Achievement Indicators:

• Sketch angles in standard position, given the measure of the angles
• Determine the reference angle for angles in standard position
• Determine the angles from 0° to 360° that have the same reference angle as given angles
• Determine the quadrant in which given angles in standard position terminate
• Draw angles in standard position given any point P (𝑥, 𝑦) on the terminal arm of the angles

Skill Descriptor: Solve problems for angles from 0° to 360° in standard position using the three primary trigonometric ratios.

Global Competencies: CTPS

Achievement Indicators:

• Determine the distance from the origin to a point 𝑃 (𝑥, 𝑦) on the terminal arm of angles
• Determine the value of sin 𝜃, cos 𝜃, tan 𝜃, given any point 𝑃 (𝑥, 𝑦) on the terminal arm of angle 𝜃
• Determine the values of sin 𝜃, cos 𝜃, and tan 𝜃 when they exist and without the use of technology, where 𝜃 = 0°, 90°, 180°, 270° 𝑜𝑟 360°
• Determine the exact value of the sine, cosine, or tangent of given angles with reference angles 30°, 45° 𝑜𝑟 60°
• Find the sign of given trigonometric ratios for given angles without the use of technology
• Solve equations of the form 𝑠𝑖𝑛 𝜃 = 𝑎, 𝑐𝑜𝑠 𝜃 = 𝑎, tan 𝜃 = 𝑎 for all values of 𝜃, where 𝑎 is a real number
• Represent trigonometric problems by sketching diagrams
• Model situations using trigonometric ratios

Skill Descriptor: Factor polynomial expressions.

Global Competencies: CTPS

Achievement Indicators:

• Factor given polynomial expressions that require the identification of common factors
• Determine whether given binomials are factors for given polynomial expressions
• Factor given quadratic polynomial expressions, including in the forms:

𝑎𝑥² +𝑏𝑥+𝑐,𝑎≠0

𝑎²𝑥² −𝑏²𝑦²,𝑎≠0,𝑏≠0

• Factor given polynomial expressions that have a quadratic pattern, including:

𝑎(𝑓(𝑥))² +𝑏(𝑓(𝑥))+𝑐,𝑎≠0

𝑎² (𝑓(𝑥))² –𝑏² (𝑔(𝑦))² ,𝑎≠0,𝑏≠0

Skill Descriptor: Graph and analyze absolute value functions.

Global Competencies: CTPS

Achievement Indicators:

• Create tables of values for y = |𝑓 (𝑥)|, given tables of values for 𝑦 = 𝑓 (𝑥)
• Generalize a rule for writing absolute value functions as piecewise functions
• Sketch the graph of y = | 𝑓 (𝑥)| showing the process and stating the intercepts, domain, and range
• Model problems that involve an absolute value function

Skill Descriptor: Solve absolute value equations.

Global Competencies: CM, CTPS

Achievement Indicators:

• Solve absolute value equations graphically
• Solve equations with a single absolute value algebraically and verify the solutions
• Explain why the absolute value equation |𝑓 (𝑥)| < 0 has no solution
• Determine and correct errors in solutions to absolute value equations

Skill Descriptor: Analyze quadratic functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain why functions given in the form 𝑦 = 𝑎(𝑥 − 𝑝)² + 𝑞 are quadratic functions
• Compare graphs of sets of functions to generalize a rule about the effect of the parameters 𝑎, 𝑝, and 𝑞
• Determine the coordinates of the vertex for quadratic functions of the form 𝑦 = 𝑎(𝑥−𝑝)² +𝑞
• State the vertex for quadratic functions of the form 𝑦 = 𝑎(𝑥 − 𝑝)² + 𝑞 as coordinate point (𝑝, 𝑞)
• Sketch the graph of 𝑦 = 𝑎(𝑥 − 𝑝)² + 𝑞, and state the vertex, domain and range, direction of opening, axis of symmetry and 𝑥- and 𝑦- intercepts
• Explain how the values of 𝑎 and 𝑞 may be used to determine whether a quadratic function has zero, one or two 𝑥-intercepts
• Write quadratic functions in the form 𝑦 = 𝑎(𝑥 − 𝑝)² + 𝑞 for given graphs or given sets of characteristics of graphs

Skill Descriptor: Analyze quadratic functions to identify characteristics of the corresponding graph to solve problems.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain the process of completing the square and its application
• Demonstrate the relationship between completing the square and the quadratic formula by deriving the formula
• Write quadratic functions given in the form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐, in the form 𝑦 = 𝑎(𝑥−𝑝)² +𝑞
• Identify and explain errors in examples of completing the square
• Sketch the graph and determine the characteristics of quadratic functions, given in the form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐, by changing to vertex form 𝑦 = 𝑎(𝑥−𝑝)² +𝑞
• Verify that a quadratic function in the form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 represents the same function as a given quadratic function in the form 𝑦 = 𝑎(𝑥−𝑝)² +𝑞
• Model given situations with quadratic functions, explaining any assumptions made, to solve problems
• Solve problems by analyzing given quadratic functions

Skill Descriptor: Solve problems that involve quadratic equations.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain the relationship among the roots of a quadratic equation, the zeros of the corresponding quadratic function and the x- intercepts of the graph of the quadratic function by using examples
• Solve quadratic equations of the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 by using strategies that include: determining square roots; factoring; completing the square; applying the quadratic formula; graphing its corresponding function
• Select methods for solving quadratic equations and justify the choices
• Explain how the discriminant may be used to determine whether a quadratic equation has two real and distinct roots; two real and equal roots; no real roots
• Relate the number of zeros to the graph of corresponding quadratic functions
• Identify and explain errors in solutions to quadratic equations
• Model problems with quadratic equations

Skill Descriptor: Solve problems that involve systems of linear-quadratic and quadratic equations in two variables.

Global Competencies: CM, CTPS, ICE

Achievement Indicators:

• Model situations using systems of linear-quadratic or quadratic-quadratic equations
• Determine the solutions of systems of linear-quadratic or quadratic equations graphically with technology
• Determine and verify the solutions of systems of linear-quadratic or quadratic equations algebraically
• Explain the contextual meaning of the point(s) of intersection of systems of linear-quadratic or quadratic equations within problems
• Explain why systems of linear-quadratic or quadratic equations may have zero, one, two or an infinite number of solutions

Skill Descriptor: Solve problems that involve linear and quadratic inequalities in two variables.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain how test points can be used to determine the solution region that satisfies an inequality
• Determine when a solid or dashed line should be used in solutions for inequalities
• Sketch the graphs of quadratic inequalities
• Model problems that involve quadratic inequalities

Skill Descriptor: Solve problems that involve quadratic inequalities in one variable.

Global Competencies: CTPS

Achievement Indicators:

• Determine the solutions and show the process of quadratic inequalities in one variable
• Model quadratic inequalities in one variable
• Interpret the solutions within the context of problems that involve quadratic inequalities in one variable