Grade: 11-12
Credit: One
Prerequisite: Advanced Algebra GT

Precalculus With Trigonometry includes all the topics of Trigonometry (3150) and an in-depth treatment of functions through the study of polynomials, transformations, rational, exponential, and logarithmic functions, inverses, polar equations, parametric equations, two-dimensional vectors, and selected topics in discrete mathematics. The course also includes the study of limits, continuity, maximum and minimum points and values, definition and properties of the derivative, rules of differentiation, equations of tangent lines to polynomial functions, infinite limits, and partial fractions. 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 Trigonometry or Mathematical Analysis. All Virginia Standards of Learning for Trigonometry and Mathematical Analysis are included in this course.

Benchmarks and indicators are organized by the following strands:

I.Functions, Relations, and Their Graphs

II.Trigonometry

III.Parametrics, Polars, and Vectors

IV.Preparation for Calculus

V.Selected Discrete Topics

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

1. Investigate and identify the characteristics of polynomial, radical, rational, and special functions (absolute value, step, and piecewise) in order to graph these functions, solve equations, and solve real-world problems. Sketch graphs with transformations (vertical and horizontal shift, reflection, stretching, and shrinking). Find compositions and inverses. Calculators will be used to find zeros and intersections.*
2. Solve equations and inequalities both algebraically and graphically.
3. Investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions, solve equations, and solve real-world problems. This will include the role of e, natural logarithms, common logarithms, laws of exponents, and solutions of logarithmic and exponential equations.*
4. Solve systems of equations including linear and nonlinear functions.
5. (optional) Decompose a rational expression into partial fractions.

Benchmark
Students apply algebraic techniques to analyze, describe, compare, and verify trigonometric relationships and solve real-world problems conducive to trigonometric ideas.

1. Change from radian to degree measure and vice versa, find angles that are coterminal with a given angle, and find the reference angle for a given angle.
2. Find the values of the six trigonometric functions of an angle in standard position. Circular functions, the unit circle, and the right triangle will be interrelated.*
3. Find exact and approximate answers for the six trigonometric functions of special angles.*
4. Evaluate inverse trigonometric relations and functions.*
5. Solve trigonometric equations graphically and algebraically.*
6. Find the length of an arc, linear speed, and angular velocities.
7. Solve right triangles and apply solutions to real-world problems.*
8. Apply the law of sines and the law of cosines to the solution of real-world problems involving triangles.*
9. Find the area of triangles using Hero’s formula and the sine formula.
10. Identify and use reciprocal identities, quotient identities, and Pythagorean identities to simplify expressions, verify other identities, and solve equations.*
11. Use the sum and difference identities and the half-angle and double angle identities to verify other trigonometric identities and solve equations.*
12. Find the domain, range, amplitude, period, phase shift, and vertical shift and sketch the graph of each of the six trigonometric functions given an equation in standard form (e.g., y = Asin(Bx + C) + D) by using transformations for at least one period.*
13. Write equations of trigonometric functions given the amplitude, period, phase shift, and vertical shift.
14. Graph inverse trigonometric functions and identify their domain and range.*
15. (optional) Graph compound trigonometric functions using addition of ordinates.

Benchmark
Students will analyze, describe, compare, and apply multiple representations to a variety of equations.

1. Eliminate the parameters in a given equation, sketch a curve defined by parametric equations, and use parametric equations to model and solve real-world problems.*
2. Investigate and identify characteristics of the graphs of polar equations using a graphing calculator. Change from polar to rectangular form and vice versa, classify polar equations, predict the effects of changes in parameters, find the maximum r-values, and determine the points of intersection of polar graphs.*
3. Operate with vectors in the coordinate plane, including addition, subtraction, scalar multiplication, inner (dot) product, normal of a vector, unit vector, components, and graphing properties, and solve practical problems using vectors. *
4. (optional) Apply DeMoivre’s theorem to convert expressions from complex form to polar and vice versa as well as to find powers and roots of complex numbers.
5. (optional) Perform operations with vectors in space, including the cross product, and solve practical problems.
6. Apply the techniques of rotation of axes in the coordinate plane to graph functions and conic sections.*

Benchmark
Students investigate fundamental principles of calculus through graphing technologies.

1. Solve problems involving infinite arithmetic and geometric sequences and series, including finding the sum of finite and infinite series that will lead to an intuitive understanding of the concept of limit.*
2. Find the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity. Graphical, numerical (tabular), and algebraic methods (including substitution and limit rules) will be used.*
3. Investigate and describe the end behavior and continuity of functions (including piecewise, absolute value, and step functions) with graphing calculators using the formal definition of continuity.*
4. Find the derivative of polynomial, rational, and radical functions using the definition of derivative and apply the derivative to problems concerning the equation of the tangent line at a point, increasing or decreasing intervals of a function, and maximum or minimum values of a function.*
5. (optional) Find the derivatives of polynomial, rational, and radical functions using the power, product, quotient, and chain rules.
6. (optional) Find the second derivative of a polynomial function and use it to determine the intervals of concavity and inflection points of the function.

Benchmark
Students solve real-world problems involving discrete data.

1. Apply the method of mathematical induction to prove formulas and statements.*
2. (optional) Express a discrete data relation implicitly and recursively and apply this method to real-world data.
3. (optional) Find a best-fit curve for real-world discrete data using linear, quadratic, polynomial, exponential, power, and logarithmic functions.

Assessment
There is no Virginia Standards of Learning Test for Precalculus or Trigonometry.

Last update: August 21, 1998