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Program of Study for High School Mathematics
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.
Some of the Virginia Standards of Learning (SOL) for Mathematical Analysis are addressed in this course. The number in parenthesis following each Essential Knowledge and Skill refers to the related SOL objective. Required FCPS Program of Studies (POS) indicators that are not addressed in the Virginia SOL are in italics type.
- Lial. Pre-Calculus with Limits. Prentice Hall. 2001.
- Hungerford. Pre-Calculus: A Graphing Approach. Holt, Rinehart, & Winston. 2002.
Benchmark
Students apply a transformational approach with graphing calculators to analyze, describe, and compare a variety of functions.
Key Terms
| absolute value |
asymptote |
binomial theorem |
| domain & range |
even function |
factor theorem |
| greatest integer |
inverse function |
intercepts |
| odd function |
Pascal's Triangle |
piecewise function |
| polynomail function |
rational function |
remainder theorem |
| root |
step function |
symmetry |
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zeros of a function |
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Indicators
- Identify a polynomial function, given an equation or graph. (MA.1)
- Identify rational functions, given an equation or graph. (MA.1)
- Sketch the graph of a polynomial function. (MA.1)
- Sketch the graph of a rational function. (MA.1)
- Investigate and verify characteristics of a polynomial or rational function, using a graphing calculator. (MA.1)
- Find the composition of functions. (MA.2)
- Find the inverse of a function algebraically and graphically. (MA.2)
- Determine the domain and range of the composite functions. (MA.2)
- Determine the domain and range of the inverse of a function. (MA.2)
- Identify absolute value, step, and piece-wise-defined functions. (MA.3)
- Use transformations to sketch absolute value, step, and rational functions. (MA.3)
- Verify the accuracy of sketches of functions, using a graphing utility. (MA.2)
- Identify all possible rational roots and determine all roots of a polynomial equation.
- Solve equations and inequalities both algebraically and graphically.
- Solve systems of equations including linear and nonlinear functions.
- Expand binomials having positive integral exponents. (MA.4)
- Use the Binomial Theorem, the formula for combinations, and Pascal’s Triangle to expand binomials. (MA.4)
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II. Exponential & Logarithmic Functions
Benchmark
Students investigate and identify the characteristics of exponential and logarithmic functions to solve practical problems using graphing utilities.
| common logarithm |
e |
exponential function |
| horizontal asymptote |
logarithmic function |
natural logarithm |
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vertical asymptote |
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Indicators
- Identify exponential functions from an equation or a graph. (MA.9)
- Identify logarithmic functions from an equation or a graph. (MA.9)
- Define e, and know its approximate value. (MA.9)
- Write logarithmic equations in exponential form and vice versa. (MA.9)
- Identify common and natural logarithms. (MA.9)
- Use laws of exponents and logarithms to solve equations and simplify expressions. (MA.9)
- Model practical problems, using exponential and logarithmic functions. (MA.9)
- Graph exponential and logarithmic functions, using a graphing utility, and identify asymptotes, intercepts, domain, and range. (MA.9)
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III. Preparing for Calculus
Benchmark
Students develop an intuitive approach to limits and investigate the derivative of polynomial functions.
Key Terms
| continuity |
discontinuity |
derivative |
| end behavior |
extrema |
finite |
| infinite |
limit |
relative maximum |
| relative minimum |
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tangent line |
Indicators
- Verify intuitive reasoning about the limit of a function, using a graphing utility. (MA.7)
- Find the limit of a function algebraically, and verify with a graphing utility. (MA.7)
- Find the limit of a function numerically, and verify with a graphing utility. (MA.7)
- Describe continuity of a function. (MA.3)
- Investigate the continuity of absolute value, step, rational, and piece-wise-defined functions. (MA.3)
- Identify zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, points of discontinuity, end behavior, and maximum and minimum points, given a graph of a function. (MA.1)
- 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.
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IV. Parametric Equations (Optional)
Benchmark
Students investigate and identify the graphs of parametric equations, using graphing utilities, to model and solve problems.
Key Terms
| motion |
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parametric equation |
Indicators
- Graph parametric equations, using a graphing utility. (MA.12)
- Use parametric equations to model motion over time. (MA.12)
- Determine solutions to parametric equations, using a graphing utility. (MA.12)
- Compare and contrast traditional solution methods with parametric methods. (MA.12)
Craig Herring, Mathematics Specialist
Craig.Herring@fcps.edu
Alan Leis Instructional Center
7423 Camp Alger Ave.
Falls Church, VA 22042
703-208-7738
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