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Skills available for Texas Algebra 2 standards

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College and Career Readiness Standards College and Career Readiness Standards
State of Texas Assessments of Academic Readiness (STAAR) State of Texas Assessments of Academic Readiness (STAAR)
Texas Essential Knowledge and Skills (TEKS): Algebra 2 Texas Essential Knowledge and Skills (TEKS): Algebra 2
Texas Essential Knowledge and Skills (TEKS): Mathematical processes Texas Essential Knowledge and Skills (TEKS): Mathematical processes
Texas College and Career Readiness Standards: Grades: 9-12 Texas College and Career Readiness Standards: Grades: 9-12

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2 Attributes of functions and their inverses

2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.
- A graph the functions f(x)=√x, f(x)=1/x, f(x)=x³, f(x)= ³√x, f(x)=b to the x power, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval;
- B graph and write the inverse of a function using notation such as f-¹ (x);
  - Find values of inverse functions from graphs (A2-P.10)
  - Find inverse functions and relations (A2-P.11)
- C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range; and
- D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

3 Systems of equations and inequalities

3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.
- A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic;
- B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;
  - Solve a system of equations in three variables using substitution (A2-E.12)
  - Solve a system of equations in three variables using elimination (A2-E.13)
- C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;
  - Solve a system of linear and quadratic equations: parabolas (A2-E.16)
- D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;
  - Solve a system of linear and quadratic equations by graphing: parabolas (A2-E.15)
- E formulate systems of at least two linear inequalities in two variables;
- F solve systems of two or more linear inequalities in two variables; and
- G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.
  - Is (x, y) a solution to the system of inequalities? (A2-F.1)

4 Quadratic and square root functions, equations, and inequalities

4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.
- A write the quadratic function given three specified points in the plane;
- B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;
- C determine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d;
  - Graph square root functions (A2-M.13)
- D transform a quadratic function f(x) = ax² + bx + c to the form f(x) = a(x - h)² + k to identify the different attributes of f(x);
  - Convert equations of parabolas from general to vertex form (A2-U.7)
  - Find properties of a parabola from equations in general form (A2-U.8)
- E formulate quadratic and square root equations using technology given a table of data;
- F solve quadratic and square root equations;
- G identify extraneous solutions of square root equations; and
- H solve quadratic inequalities.
  - Graph solutions to quadratic inequalities (A2-C.10)
  - Solve quadratic inequalities (A2-C.11)

5 Exponential and logarithmic functions and equations

5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.
- A determine the effects on the key attributes on the graphs of f(x) = b to the x power and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d;
  - Match exponential functions and graphs (A2-T.4)
- B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;
- C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;
- D solve exponential equations of the form y = ab to the x power where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions; and
- E determine the reasonableness of a solution to a logarithmic equation.

6 Cubic, cube root, absolute value and rational functions, equations, and inequalities

6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.
- A analyze the effect on the graphs of f(x) = x³ and f(x) = ³√x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- B solve cube root equations that have real roots;
- C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
  - Graph an absolute value function (A2-D.13)
  - Transformations of absolute value functions (A2-D.16)
- D formulate absolute value linear equations;
- E solve absolute value linear equations;
  - Solve absolute value equations (A2-B.4)
  - Graph solutions to absolute value equations (A2-B.5)
- F solve absolute value linear inequalities;
  - Solve absolute value inequalities (A2-C.6)
  - Graph solutions to absolute value inequalities (A2-C.7)
- G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- H formulate rational equations that model real-world situations;
- I solve rational equations that have real solutions;
  - Solve rational equations (A2-O.7)
- J determine the reasonableness of a solution to a rational equation;
  - Solve rational equations (A2-O.7)
- K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and
  - Rational functions: asymptotes and excluded values (A2-O.1)
- L formulate and solve equations involving inverse variation.
  - Write and solve inverse variation equations (A2-R.2)

7 Number and algebraic methods

7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.
- A add, subtract, and multiply complex numbers;
- B add, subtract, and multiply polynomials;
  - Add and subtract polynomials (A2-L.2)
  - Multiply polynomials (A2-L.3)
- C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;
  - Divide polynomials using long division (A2-L.4)
  - Divide polynomials using synthetic division (A2-L.5)
- D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;
  - Factor polynomials (A2-J.7)
- E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;
- F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;
  - Multiply and divide rational expressions (A2-O.5)
  - Add and subtract rational expressions (A2-O.6)
- G rewrite radical expressions that contain variables to equivalent forms;
  - Simplify radical expressions with variables I (A2-M.4)
  - Simplify radical expressions with variables II (A2-M.5)
- H solve equations involving rational exponents; and
- I write the domain and range of a function in interval notation, inequalities, and set notation.

8 Data

8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.
- A analyze data to select the appropriate model from among linear, quadratic, and exponential models;
  - Identify linear, quadratic, and exponential functions from graphs (A2)
  - Identify linear, quadratic, and exponential functions from tables (A2)
- B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data; and
- C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.
  - Interpret regression lines (A2-FF.7)
  - Analyze a regression line of a data set (A2-FF.8)

Texas

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2 Attributes of functions and their inverses

2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.

B graph and write the inverse of a function using notation such as f-¹ (x);

C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range; and

D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

3 Systems of equations and inequalities

3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.

A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic;

B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;

C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;

D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;

E formulate systems of at least two linear inequalities in two variables;

F solve systems of two or more linear inequalities in two variables; and

G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

4 Quadratic and square root functions, equations, and inequalities

4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.

A write the quadratic function given three specified points in the plane;

B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;

C determine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d;

D transform a quadratic function f(x) = ax² + bx + c to the form f(x) = a(x - h)² + k to identify the different attributes of f(x);

E formulate quadratic and square root equations using technology given a table of data;

F solve quadratic and square root equations;

G identify extraneous solutions of square root equations; and

H solve quadratic inequalities.

5 Exponential and logarithmic functions and equations

5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.

A determine the effects on the key attributes on the graphs of f(x) = b to the x power and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d;

B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;

C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;

D solve exponential equations of the form y = ab to the x power where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions; and

E determine the reasonableness of a solution to a logarithmic equation.

6 Cubic, cube root, absolute value and rational functions, equations, and inequalities

6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.

A analyze the effect on the graphs of f(x) = x³ and f(x) = ³√x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

B solve cube root equations that have real roots;

C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

D formulate absolute value linear equations;

E solve absolute value linear equations;

F solve absolute value linear inequalities;

G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

H formulate rational equations that model real-world situations;

I solve rational equations that have real solutions;

J determine the reasonableness of a solution to a rational equation;

K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and

L formulate and solve equations involving inverse variation.

7 Number and algebraic methods

7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.

A add, subtract, and multiply complex numbers;

B add, subtract, and multiply polynomials;

C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;

D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;

E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;

F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;

G rewrite radical expressions that contain variables to equivalent forms;

H solve equations involving rational exponents; and

I write the domain and range of a function in interval notation, inequalities, and set notation.

8 Data

8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.

A analyze data to select the appropriate model from among linear, quadratic, and exponential models;

B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data; and

C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.