Functions: Graphs, Properties and Analysis

What are the questions?

1. Develop skills for graphing and solving problems involving a variety of functions including polynomials, trigonometric functions, exponentials, and logarithms.
2. Distinguish between the function forms of linear, quadratic, polynomial, rational functions, and composite functions
3. Find roots/zeros/x-intercepts of quadratic functions and polynomial functions of various degrees.
4. Understand functional relationships as well as inverse relationships.
5. Use a variety of graphing techniques to sketch functions (shifts, inverses, reciprocals).
6. Apply the unit circle approach to understanding the fundamental properties of trigonometric functions and their applications, as well as graphs of these functions.
7. Define and derive properties of exponential and logarithmic functions.
8. Determine the end behavior of the graph of the function.
9. Analyze a wide variety of real-world problems using learned theoretical concepts of functions.
10. When do you use Lovelace's Favorite Method?
11. Using the real zeros of the numerator and denominator of the given equation for f(x), divide the x-axis into intervals and determine where the graph is above and below the x-axis by choosing a number in each interval and evaluating f(x) there.
12. Analyze the behavior of the graph of f(x) near each asymptote and indicate the behavior on the graph.
13. Locate any horizontal or oblique or vertical asymptotes.
14. Factor the numerator and denominator of a rational function and find its domain.
15. Find the x- and y-intercepts of the graph of the function.
16. Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept.
17. Use a graphing utility to graph the function.
18. Approximate the turning points of the graph. Round to four decimal places.
19. Use the graph of the function to determine where the function is increasing and where it is decreasing.