Grade: 11-12
Credit: one-half
Prerequisite: Trigonometry

Mathematical Analysis provides a thorough treatment of functions through the study of polynomials, transformations, and rational, logarithmic, exponential, and inverse functions. Topics also include an intuitive approach to limits, continuity, maximum and minimum points, and values. A transformational approach to graphing is used with families of related graphs. Numerical, graphical, and algebraic solutions should be considered for all problems where appropriate. Graphing utilities, especially graphing calculators, are integral to the course.

Indicators marked with an asterisk * relate directly to the Virginia Standards of Learning for Mathematical Analysis.

Benchmarks and indicators are organized by the following strands:

1. Mechanics
2. Functions and Relations
3. ISequences and Series
4. Preparation for Calculus

Benchmark
Students analyze and compare polynomial equations.

1. Perform operations on functions, find composite functions, find the inverse of a function (if it exists), and determine the domain and range of the function.*
2. Determine the roots (zeros) and bounds of polynomial equations and apply the fundamental theorem of algebra and the remainder and factor theorems.*
3. Identify all possible rational roots and determine all roots of a polynomial equation.
4. Perform operations with vectors in the coordinate plan and solve practical problems using vectors.*

Benchmark
Students apply a transformational approach with graphing calculators to analyze, describe, and compare a variety of functions.

1. Identify symmetrical graphs, use symmetry to complete a graph, and identify odd and even functions. Sketching will include a transformational approach (vertical and horizontal shift, reflection, stretching, and shrinking).*
2. Solve equations and inequalities both algebraically and graphically.
3. Graph rational functions including the determination of horizontal, vertical, and slant asymptotes.*
4. Graph polynomial, absolute value, and radical functions.*
5. Graph piecewise, step, and greatest integer functions.*
6. Investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and to solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and logarithms, and the solution of logarithmic and exponential equations.*
7. Solve systems of equations including linear and nonlinear functions.
8. Graph parametric equations, write functions and their inverses in parametric form, and graph the relations with graphing calculators.*
9. Investigate and identify graphs of polar equations using graphing utilities.*
10. Apply the techniques of translation and rotation of axes in the coordinate plane to graphing functions and conic sections.*

Benchmark
Students investigate patterns of numbers and sums to develop an intuitive approach to limits.

1. Investigate and apply the properties of arithmetic and geometric sequences and series to solve problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include å and a_n.*
2. Investigate numbers and sums and their behavior as they approach a finite number or infinity. The emphasis is on an intuitive approach to limits.
3. Expand binomials having positive integral exponents through the use of the binomial theorem, combinations, and Pascal’s triangle.*
4. Apply the method of mathematical induction to prove formulas or statements.*

Benchmark
Students investigate the fundamental principles of calculus using graphing technologies.

1. Explain and illustrate the limit of an algebraic function through graphs.*
2. Investigate and describe the continuity and end behavior of functions including piecewise, step, and absolute value functions using graphing calculators.*
3. Explain and illustrate the derivative of a polynomial function as the slope of the line tangent to the graph of a function at a given point.
4. Find the increasing and decreasing intervals and the relative extrema of the graph of a polynomial function and determine if each is a minimum or maximum with graphing calculators.*

Assessment
There is no Virginia Standards of Learning Test for Mathematical Analysis.

Last update: August 21, 1998