Course Outline

 

Course Description:

Precalculus is a prerequisite for calculus and for most courses in science and engineering. It provides an essential background for many other college-level mathematics courses. The topics covered in this course include the real number system; algebraic, exponential, logarithmic, trigonometric, and inverse functions; polynomials; and systems of equations. Graphing calculators are utilized to further illustrate many concepts.

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A precalculus algebra and trigonometry course to prepare students majoring in mathematics, engineering, or physical science for courses in calculus and higher-level mathematics. Topics included are polynomial, rational exponential, logarithmic, trigonometric, and inverse trigonometric functions, and their graphs; trigonometric identities and trigonometric equations; appropriate applications of trigonometry; and analytic geometry.

 

Course Outline:

1. Functions

1. Define a function.

2. Determine if the given relationship is a function.

3. Determine if a function is one-to-one.

4. Use functional notation and find function values.

5. Use the vertical line test to determine if a graph is a function.

6. Use the horizontal line test to determine if a graph is one-to-one.

7. Determine if an equation is a function.

8. Graph a variety of equations and functions.

9. Find the distance between two points.

10.Find the equation of a circle.

11.Find the domain and range of a function.

12.Get information about a function from its graph.

13.Evaluate and graph piecewise functions.

14.Recognize and evaluate step functions.

15.Find the average rate of change of a function.

16.Find a function’s difference quotient.

17.Determine various characteristics of graphs such as: increasing, decreasing, constant, concave up, & concave down intervals, & relative maxima/minima.

18.Perform function combinations.

19.Form composite functions.

20.Find the domain of a composite function.

21.Decompose a function.

22.Apply combinations and compositions to practical problems.

23.Define transformations of functions.

24.Use shifts, reflections, compressions, and/or stretches to graph functions.

25.Define an inverse function.

26.Find the inverse function.

27.Use inverse functions to find the range of a function.

28.Apply inverse functions in the real world.

B. Polynomial and Rational Functions

1. Understand and perform operations on complex numbers.

2. Graph quadratic functions.

3. Solve problems modeled by quadratic functions.

4. Learn properties of the graphs of polynomial functions.

5. Determine the end behavior & zeros of polynomial functions.

6. Identify the relationship between degrees, real zeros, and turning points.

7. Graph polynomial functions.

8. Use the Division Algorithm.

9. Use the Remainder and Factor Theorems.

10.Use the Rational Zeros Test.

11.Find the possible # of positive & negative zeros of polynomials.

12.Find the bounds on the real zeros of polynomials.

13.Use the Conjugate Pairs Theorem to find zeros of polynomials.

14.Define a rational function.

15.Define vertical and horizontal asymptotes.

16.Graph translations of f(x) = 1/x.

17.Find vertical, horizontal, & oblique asymptotes (if any).

18.Graph rational functions.

19.Graph rational functions with oblique asymptotes.

C. Exponential and Logarithmic Functions

1. Define an exponential function.

2. Graph exponential functions.

3. Develop formulas for compound interest.

4. Understand the number e.

5. Graph the natural exponential function.

6. Model using exponential functions.

7. Define logarithmic functions.

8. Evaluate logarithms.

9. Find the domains of logarithmic functions.

10.Graph logarithmic functions.

11.Use logarithms to solve exponential equations.

12.Learn the rules of logarithms.

13.Change the base of a logarithm.

14.Apply logarithms to growth and decay.

15.Solve exponential equations.

16.Solve applied problems involving exponential equations.

17.Solve logarithmic equations.

18.Use the logistic growth model.

19.Use logarithmic and exponential inequalities.

D. Trigonometric Functions

1. Learn the vocabulary associated with angles.

2. Use degree and radian measure.

3. Convert between degree and radian measure.

4. Find complements and supplements.

5. Find the length of an arc of a circle.

6. Compute linear and angular speed.

7. Find the area of a sector.

8. Review properties of the unit circle.

9. Define trigonometric functions of real numbers.

10.Find exact trig function values using a terminal point on the unit circle.

11.Define trigonometric functions of angles.

12.Find trig. function values of an angle in standard position.

13.Approximate trigonometric function values using a calculator.

14.Determine the signs of the trig. functions in each quadrant.

15.Find a reference angle.

16.Use basic trigonometric identities.

17.Discuss properties of the sine and cosine functions.

18.Graph the trigonometric functions.

19.Find the amplitude, period, & phase shift of trig functions.

20.Discuss properties of the tangent function.

21.Graph the inverse sine, cosine, & tangent functions.

22.Evaluate inverse trigonometric functions.

23.Find the exact values of composite functions involving inverse trig functions.

E. Analytic Trigonometry

1. Use fundamental trigonometric identities to evaluate trigonometric functions.

2. Simplify a complicated trigonometric expression.

3. Prove that a given equation is not an identity.

4. Verify a trigonometric identity.

5. Solve trigonometric equations of the form, etc.

6. Solve trigonometric equations involving multiple angles.

7. Solve trig. equations by using the zero-product property.

8. Solve trig equations that contain more than one trigonometric function.

9. Solve trigonometric equations by squaring both sides.

10.Use the sum and difference formulas for trig functions.

11.Use cofunction identities.

12.Use double-angle formulas.

13.Use power-reducing formulas.

14.Use half-angle formulas.

15.Solve trigonometric equations involving multiple angles and half-angles.

16.Derive product-to-sum formulas & sum-to-product formulas.

17.Verify trigonometric identities involving multiple angles.

 

F. Applications of Trigonometric Functions

1. Express the trigonometric functions using a right triangle.

2. Evaluate trigonometric functions of angles in a right triangle.

3. Solve right triangles.

4. Use right-triangle trigonometry in applications.

5. Use vocabulary and conventions for solving triangles.

6. Derive the Law of Sines.

7. Solve AAS and ASA triangles by using the Law of Sines.

8. Solve for possible triangles in the ambiguous SSA case.

9. Find the area of an SAS triangle.

10.Derive the Law of Cosines.

11.Use the Law of Cosines to solve SAS & SSS triangles.

12.Use Heron’s formula to find the area of a triangle.