# Precalculus Critical Areas

CRITICAL AREA #1: Extend work with complex numbers
Students continue their work with complex numbers. They perform arithmetic operations with complex numbers and represent them and the operations on the complex plane. Students investigate and identify the characteristics of the graphs of polar equations, using graphing tools. This includes classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.

The Domains and Clusters below relate to this Critical Area:

The Complex Number System

• Perform arithmetic operations with complex numbers.
• Represent complex numbers and their operations on the complex plane.
• Use complex numbers in polynomial identities and equations.

CRITICAL AREA #2: Expand understanding of logarithms and exponential functions
Students expand their understanding of functions to include logarithmic and trigonometric functions. They investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and solve equations and practical problems. This includes the role of e, natural and common logarithms, laws of exponents and logarithms, and the solutions of logarithmic and exponential equations. Students model periodic phenomena with trigonometric functions and prove trigonometric identities. Other trigonometric topics include reviewing unit circle trigonometry, proving trigonometric identities, solving trigonometric equations, and graphing trigonometric functions.

The Domain and Clusters below relate to this Critical Area:

Building Functions

• Build a function that models a relationship between two quantities.
• Build new functions from existing functions.
• Trigonometric Functions
• Extend the domain of trigonometric functions using the unit circle.
• Model periodic phenomena with trigonometric functions.
• Prove and apply trigonometric identities.
• Similarity, Right Triangles, and Trigonometry
• Apply trigonometry to general triangles.

CRITICAL AREA #3: Use characteristics of polynomial and rational functions to sketch graphs of those functions
Students investigate and identify the characteristics of polynomial and rational functions and use these to sketch the graphs of the functions. They determine zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, and maximum or minimum points. Students translate between the geometric description and equation of conic sections. They deepen their understanding of the Fundamental Theorem of Algebra.

The Domains and Clusters below relate to this Critical Area:

Interpreting Functions

• Analyze functions using different representations.

CRITICAL AREA #4: Perform operations with vectors
Students perform operations with vectors in the coordinate plane and solve practical problems using vectors. This includes the following topics: operations of addition, subtraction, scalar multiplication, and inner (dot) product; norm of a vector; unit vector; graphing; properties; simple proofs; complex numbers (as vectors); and perpendicular components.

The Domains and Clusters below relate to this Critical Area:

Vector and Matrix Quantities

• Represent and model with vector quantities.
• Perform operations on vectors.
• Perform operations on matrices and use matrices in applications.

STANDARDS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS

Circles

• Understand and apply theorems about circles.

Expressing Geometric Properties with Equations

• Translate between the geometric description and the equation for a conic section.

Geometric Measurement and Dimension

• Explain volume formulas and use them to solve problems.
• Visualize relationships between two-dimensional and three-dimensional objects.