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Skills available for Texas Precalculus standards

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Texas Essential Knowledge and Skills (TEKS): 111.47 Statistics Texas Essential Knowledge and Skills (TEKS): 111.47 Statistics
Texas Essential Knowledge and Skills (TEKS): Mathematical processes Texas Essential Knowledge and Skills (TEKS): Mathematical processes
Texas Essential Knowledge and Skills (TEKS): Precalculus Texas Essential Knowledge and Skills (TEKS): Precalculus
Texas College and Career Readiness Standards: Grades: 9-12 Texas College and Career Readiness Standards: Grades: 9-12

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2 Functions

2 The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems.
- A use the composition of two functions to model and solve real-world problems;
- B demonstrate that function composition is not always commutative;
  - Composition of functions (PC-A.11)
- C represent a given function as a composite function of two or more functions;
- D describe symmetry of graphs of even and odd functions;
  - Symmetry and periodicity of trigonometric functions (PC-O.2)
- E determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations;
- F graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions;
- G graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems;
- H graph arcsin x and arccos x and describe the limitations on the domain;
- I determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing;
- J analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems;
- K analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes;
- L determine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities;
  - Rational functions: asymptotes and excluded values (PC-E.1)
  - Find and analyze points of discontinuity using graphs (C-I.4)
- M describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities;
- N analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems;
  - Exponential growth and decay: word problems (PC-F.16)
  - Compound interest: word problems (PC-F.17)
- O develop and use a sinusoidal function that models a situation in mathematical and real-world problems; and
- P determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems.
  - Find trigonometric ratios using right triangles (PC-M.5)
  - Find trigonometric ratios using reference angles (PC-M.8)

3 Relations and geometric reasoning

3 The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations.
- A graph a set of parametric equations;
- B convert parametric equations into rectangular relations and convert rectangular relations into parametric equations;
- C use parametric equations to model and solve mathematical and real-world problems;
- D graph points in the polar coordinate system and convert between rectangular coordinates and polar coordinates;
- E graph polar equations by plotting points and using technology;
  - Match polar equations and graphs (PC-T.5)
- F determine the conic section formed when a plane intersects a double-napped cone;
- G make connections between the locus definition of conic sections and their equations in rectangular coordinates;
  - Write equations of parabolas in vertex form (PC-P.2)
- H use the characteristics of an ellipse to write the equation of an ellipse with center (h, k); and
  - Write equations of ellipses in standard form (PC-P.9)
- I use the characteristics of a hyperbola to write the equation of a hyperbola with center (h, k).
  - Write equations of hyperbolas in standard form (PC-P.12)

4 Number and measure

4 The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems.
- A determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems;
  - Find trigonometric ratios using the unit circle (PC-M.6)
- B describe the relationship between degree and radian measure on the unit circle;
  - Convert between radians and degrees (PC-M.1)
- C represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position;
- D represent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity;
  - Radians and arc length (PC-M.2)
- E determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems;
- F use trigonometry in mathematical and real-world problems, including directional bearing;
- G use the Law of Sines in mathematical and real-world problems;
  - Law of Sines (PC-M.15)
  - Solve a triangle (PC-M.17)
- H use the Law of Cosines in mathematical and real-world problems;
  - Law of Cosines (PC-M.16)
  - Solve a triangle (PC-M.17)
- I use vectors to model situations involving magnitude and direction;
- J represent the addition of vectors and the multiplication of a vector by a scalar geometrically and symbolically; and
- K apply vector addition and multiplication of a vector by a scalar in mathematical and real-world problems.
  - Linear combinations of vectors (PC-U.14)

5 Algebraic reasoning

5 The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms.
- A evaluate finite sums and geometric series, when possible, written in sigma notation;
  - Identify arithmetic and geometric series (PC-W.9)
  - Find the sum of a finite geometric series (PC-W.12)
- B represent arithmetic sequences and geometric sequences using recursive formulas;
- C calculate the nth term and the nth partial sum of an arithmetic series in mathematical and real-world problems;
- D represent arithmetic series and geometric series using sigma notation;
  - Introduction to sigma notation (PC-W.10)
- E calculate the nth term of a geometric series, the nth partial sum of a geometric series, and sum of an infinite geometric series when it exists;
- F apply the Binomial Theorem for the expansion of (a + b)ⁿ in powers of a and b for a positive integer n, where a and b are any numbers;
- G use the properties of logarithms to evaluate or transform logarithmic expressions;
- H generate and solve logarithmic equations in mathematical and real-world problems;
  - Solve logarithmic equations with one logarithm (PC-F.11)
  - Solve logarithmic equations with multiple logarithms (PC-F.12)
- I generate and solve exponential equations in mathematical and real-world problems;
- J solve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems;
- K solve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems;
  - Graph solutions to higher-degree inequalities (PC-K.3)
  - Solve higher-degree inequalities (PC-K.4)
- L solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems;
- M use trigonometric identities such as reciprocal, quotient, Pythagorean, cofunctions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric expressions; and
- N generate and solve trigonometric equations in mathematical and real-world problems.
  - Solve trigonometric equations (PC-M.11)

Texas

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2 Functions

A use the composition of two functions to model and solve real-world problems;

B demonstrate that function composition is not always commutative;

C represent a given function as a composite function of two or more functions;

D describe symmetry of graphs of even and odd functions;

E determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations;

F graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions;

G graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems;

H graph arcsin x and arccos x and describe the limitations on the domain;

J analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems;

K analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes;

L determine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities;

M describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities;

N analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems;

O develop and use a sinusoidal function that models a situation in mathematical and real-world problems; and

P determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems.

3 Relations and geometric reasoning

3 The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations.

A graph a set of parametric equations;

B convert parametric equations into rectangular relations and convert rectangular relations into parametric equations;

C use parametric equations to model and solve mathematical and real-world problems;

D graph points in the polar coordinate system and convert between rectangular coordinates and polar coordinates;

E graph polar equations by plotting points and using technology;

F determine the conic section formed when a plane intersects a double-napped cone;

G make connections between the locus definition of conic sections and their equations in rectangular coordinates;

H use the characteristics of an ellipse to write the equation of an ellipse with center (h, k); and

I use the characteristics of a hyperbola to write the equation of a hyperbola with center (h, k).

4 Number and measure

4 The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems.

A determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems;

B describe the relationship between degree and radian measure on the unit circle;

C represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position;

D represent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity;

E determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems;

F use trigonometry in mathematical and real-world problems, including directional bearing;

G use the Law of Sines in mathematical and real-world problems;

H use the Law of Cosines in mathematical and real-world problems;

I use vectors to model situations involving magnitude and direction;

J represent the addition of vectors and the multiplication of a vector by a scalar geometrically and symbolically; and

K apply vector addition and multiplication of a vector by a scalar in mathematical and real-world problems.

5 Algebraic reasoning

5 The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms.

A evaluate finite sums and geometric series, when possible, written in sigma notation;

B represent arithmetic sequences and geometric sequences using recursive formulas;

C calculate the nth term and the nth partial sum of an arithmetic series in mathematical and real-world problems;

D represent arithmetic series and geometric series using sigma notation;

E calculate the nth term of a geometric series, the nth partial sum of a geometric series, and sum of an infinite geometric series when it exists;

F apply the Binomial Theorem for the expansion of (a + b)n in powers of a and b for a positive integer n, where a and b are any numbers;

G use the properties of logarithms to evaluate or transform logarithmic expressions;

H generate and solve logarithmic equations in mathematical and real-world problems;

I generate and solve exponential equations in mathematical and real-world problems;

J solve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems;

K solve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems;

L solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems;

M use trigonometric identities such as reciprocal, quotient, Pythagorean, cofunctions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric expressions; and

N generate and solve trigonometric equations in mathematical and real-world problems.

F apply the Binomial Theorem for the expansion of (a + b)ⁿ in powers of a and b for a positive integer n, where a and b are any numbers;