US MATH PRECALCULUS

A. Numbers and Operations

A.01. Understand basic arithmetic operations with integers.
A.02. Apply the order of operations to evaluate integer expressions.
A.03. Convert between fractions, decimals, and percentages.
A.04. Perform addition and subtraction of rational numbers.
A.05. Perform multiplication and division of rational numbers.
A.06. Estimate and approximate values of irrational numbers.
A.07. Explore real-world applications of rational and irrational numbers.

B. Function Concepts

C. Function Transformations

D. Quadratic Functions

E. Polynomial Expressions and Equations

E.01. Understand polynomial vocabulary (degree, terms, coefficients).
E.02. Perform polynomial long division.
E.03. Perform polynomial synthetic division.
E.04. Evaluate polynomials using synthetic division.
E.05. Factor polynomials using various techniques, including factoring sums and differences of cubes and quadratic patterns.
E.06. Solve polynomial equations using factoring and other techniques.
E.07. Apply Pascal’s Triangle to expand binomial expressions.
E.08. Apply the Binomial Theorem to expand binomial expressions.
E.09. Understand and apply the Remainder Theorem.
E.10. Understand and apply the Factor Theorem.

F. Polynomial Functions

F.01. Find the roots (zeros) of factored polynomials.
F.02. Write a polynomial given its roots.
F.03. Apply the Rational Root Theorem.
F.04. Apply the Complex Conjugate Theorem.
F.05. Apply Descartes’ Rule of Signs.
F.06. Understand the Fundamental Theorem of Algebra.
F.07. Analyze the end behavior of polynomial functions.
F.08. Analyze the relationship between the zeros and the graph of a polynomial function.
F.09. Determine the domain and range of polynomial functions.
F.10. Identify even and odd polynomial functions.
F.11. Sketch the graph of a polynomial function given its properties.

G. Rational Functions

H. Exponential Functions

I. Logarithms

J. Logarithmic Functions

K. Exponential and Logarithmic Equations

L. Radical Expressions and Functions

M. Rational Exponents

N. Systems of Equations

O. Systems of Inequalities

P. Nonlinear Inequalities

Q. Matrices

Q.01. Understand matrix vocabulary (dimension, element, row, column).
Q.02. Understand matrix operation rules.
Q.03. Perform addition, subtraction, and scalar multiplication of matrices.
Q.04. Perform matrix multiplication.
Q.05. Simplify matrix expressions.
Q.06. Understand properties of matrices (associative, commutative, distributive).
Q.07. Solve matrix equations.
Q.08. Find the determinant of a matrix.
Q.09. Determine if a matrix is invertible.
Q.10. Find the inverse of a matrix (2×2 and 3×3).
Q.11. Solve matrix equations using inverses.
Q.12. Identify and apply transformation matrices.

R. Trigonometry

R.01. Convert between radians and degrees.
R.02. Understand radians and arc length.
R.03. Understand quadrants and their properties.
R.04. Identify coterminal and reference angles.
R.05. Find trigonometric ratios using right triangles.
R.06. Find trigonometric ratios using the unit circle.
R.07. Find trigonometric ratios of special angles.
R.08. Find trigonometric ratios using reference angles.
R.09. Understand and apply inverse trigonometric functions.
R.10. Solve trigonometric equations.
R.11. Solve right triangles.
R.12. Apply the Law of Sines.
R.13. Apply the Law of Cosines.
R.14. Solve triangles (AAS, ASA, SAS, SSS).
R.15. Find the area of a triangle using sine and Heron’s formula.

S. Trigonometric Functions

T. Trigonometric Identities

U. Conic Sections

U.01. Identify and analyze parabolas (vertex, focus, directrix).
U.02. Write equations of parabolas in vertex form.
U.03. Graph parabolas.
U.04. Identify and analyze circles (center, radius).
U.05. Write equations of circles in standard form.
U.06. Graph circles.
U.07. Identify and analyze ellipses (center, vertices, foci, major and minor axes).
U.08. Determine the eccentricity of an ellipse.
U.09. Write equations of ellipses in standard form.
U.10. Identify and analyze hyperbolas (center, vertices, foci, asymptotes).
U.11. Determine the eccentricity of a hyperbola.
U.12. Write equations of hyperbolas in standard form.
U.13. Convert equations of conic sections from general to standard form.

V. Complex Numbers

W. Complex Plane

X. Polar Form

Y. Two-Dimensional Vectors

Y.01. Find the magnitude of a vector.
Y.02. Find the component form of a vector.
Y.03. Find the direction angle of a vector.
Y.04. Find the component form of a vector from its magnitude and direction angle.
Y.05. Graph resultant vectors using the triangle and parallelogram methods.
Y.06. Add and subtract vectors.
Y.07. Find the magnitude and direction of a vector sum.
Y.08. Multiply vectors by scalars.
Y.09. Find unit vectors.
Y.10. Understand linear combinations of vectors.

Z. Three-Dimensional Vectors

AA. Sequences

BB. Series

BB.01. Identify arithmetic and geometric series.
BB.02. Understand and apply sigma notation.
BB.03. Find the sum of an arithmetic series.
BB.04. Find the sum of a finite geometric series.
BB.05. Understand partial sums.
BB.06. Find partial sums of arithmetic and geometric series.
BB.07. Determine if a geometric series is convergent or divergent.
BB.08. Find the sum of an infinite geometric series.
BB.09. Write a repeating decimal as a fraction using geometric series.

Course Content

A. Numbers and Operations

A.01. Understand basic arithmetic operations with integers.

A.02. Apply the order of operations to evaluate integer expressions.

A.03. Convert between fractions, decimals, and percentages.

A.04. Perform addition and subtraction of rational numbers.

A.05. Perform multiplication and division of rational numbers.

A.06. Estimate and approximate values of irrational numbers.

A.07. Explore real-world applications of rational and irrational numbers.

B. Function Concepts

B.01. Define and identify functions.

B.02. Determine the domain and range of a function.

B.03. Evaluate functions given an input value.

B.04. Interpret function values from graphs and tables.

B.05. Create tables of values for functions.

B.06. Interpret functions within real-world scenarios.

B.07. Perform arithmetic operations with functions (addition, subtraction, multiplication, division).

B.08. Understand and apply the composition of functions.

C. Function Transformations

C.01. Identify and describe basic function transformations (translations, reflections, dilations).

C.02. Apply transformation rules to graphs of functions.

C.03. Predict the graph of a transformed function given the original function and transformation rules.

C.04. Write the equation of a transformed function given its graph.

C.05. Combine multiple transformations to create complex function transformations.

C.06. Analyze and interpret the effects of transformations on key function features (e.g., intercepts, asymptotes).

D. Quadratic Functions

D.01. Graph quadratic functions by plotting points or using vertex form.

D.02. Identify the vertex, axis of symmetry, and intercepts of a quadratic function.

D.03. Determine the maximum or minimum value of a quadratic function.

D.04. Solve quadratic equations using square roots, factoring, completing the square, and the quadratic formula.

D.05. Interpret and solve quadratic word problems.

D.06. Analyze the discriminant to determine the number and type of solutions.

D.07. Apply quadratic functions to model real-world situations.

E. Polynomial Expressions and Equations

E.01. Understand polynomial vocabulary (degree, terms, coefficients).

E.02. Perform polynomial long division.

E.03. Perform polynomial synthetic division.

E.04. Evaluate polynomials using synthetic division.

E.05. Factor polynomials using various techniques, including factoring sums and differences of cubes and quadratic patterns.

E.06. Solve polynomial equations using factoring and other techniques.

E.07. Apply Pascal’s Triangle to expand binomial expressions.

E.08. Apply the Binomial Theorem to expand binomial expressions.

E.09. Understand and apply the Remainder Theorem.

E.10. Understand and apply the Factor Theorem.

F. Polynomial Functions

F.01. Find the roots (zeros) of factored polynomials.

F.02. Write a polynomial given its roots.

F.03. Apply the Rational Root Theorem.

F.04. Apply the Complex Conjugate Theorem.

F.05. Apply Descartes’ Rule of Signs.

F.06. Understand the Fundamental Theorem of Algebra.

F.07. Analyze the end behavior of polynomial functions.

F.08. Analyze the relationship between the zeros and the graph of a polynomial function.

F.09. Determine the domain and range of polynomial functions.

F.10. Identify even and odd polynomial functions.

F.11. Sketch the graph of a polynomial function given its properties.

G. Rational Functions

G.01. Identify asymptotes and excluded values of rational functions.

G.02. Solve rational equations.

G.03. Determine if two rational functions are inverses of each other.

G.04. Analyze the behavior of rational functions near asymptotes.

G.05. Sketch the graph of a rational function, including asymptotes and intercepts.

H. Exponential Functions

H.01. Identify exponential functions.

H.02. Determine the domain and range of exponential functions.

H.03. Graph exponential functions.

H.04. Compare linear and exponential growth and decay.

H.05. Analyze and interpret real-world exponential models (e.g., population growth, compound interest).

H.06. Solve basic exponential equations.

I. Logarithms

I.01. Convert between exponential and logarithmic forms.

I.02. Evaluate logarithms using the definition and properties.

I.03. Understand and apply the change of base formula.

I.04. Understand and apply the product, quotient, and power properties of logarithms.

I.05. Solve logarithmic equations using properties.

I.06. Use logarithms to solve exponential equations.

J. Logarithmic Functions

J.01. Determine the domain and range of logarithmic functions.

J.02. Graph logarithmic functions.

J.03. Analyze and interpret real-world logarithmic models.