PreCalculus: Lesson Plan * 10/11/2006

Quickstart
1. Perform the indicated operation and write the result in standard notation.
( 4 - √-9)( 2 + √-9)
2. Write (3 + i)/i in standard form
3. Plot 6 - 5i and -3 + 2i in the complex plane.

Assignment
2.5 Group Quiz
In groups of 3 or 4, complete one problem from the following set p.187: 26, 28, 34, 36, 38

Lesson
2.6 Rational Functions and Asymptotes Objective: Students will know how to determine the domain and find asymptotes of rational functions

I Introduction to Rational Functions
a rational function is a function of the form ƒ(x) = N(x)/D(x), where N and D are both polynomials. The domain of ƒ is all x such that D(x) ≠ 0.


II. Horizontal and Vertical Asymptotes
A. Definition of asymptotes
1. The line x = a is a vertical asymptote of the graph of ƒ
if ƒ(x) approaches ±∞ as x approaches a, either from the right or left.
2.The line y = b is a horizontal asymptote of the graph of ƒ
if ƒ(x) approaches b as x approaches ±∞.
B. Asymptotes of Rational Functions Rules
Let ƒ be a rational function given by
ƒ(x) = N(x)/D(x)
= [a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀]/[b_mx^m + b_m-1x^m-1 + …+ b₁x + b₀].
1. The graph of ƒ has a vertical asymptote at x = a, if D(a) = 0 and N(a) ≠ 0.
2. The graph of ƒ has one horizontal asymptote or no horizontal asymptote, depending on the degree of N and D.
a. If n < m, y = 0
b. If n = m, y = a_n / b_m is the horizontal asymptote of the graph of ƒ.
c. n > m, then there is no horizontal asymptote of the graph of ƒ.


III Applications


2.7 Graphs of Rational Functions
Objective: Students will know how to sketch the graph of a rational function.

Guidelines for Graphing of Rational Functions [p. 199]
Let ƒ(x) = N(x)/D(x), where N(x) and D(x) are polynomials with no common factors.
1. Find and plot the y-intercept (if any) by evaluating ƒ(0).
2. Set the numerator equal to zero and solve the equation N(x) = 0. The real solutions represent the x-intercepts of the graph. Plot these intercepts.
3. Set the denominator equal to zero and solve the equation D(x) = 0. The real solutions represent the vertical asymptotes. Sketch these asymptotes using dashed vertical lines.
4. Find and sketch the horizontal asymptote of the graph using a dashed horizontal line.
5. Plot at least one point between and one point beyond each x-intercept and vertical asymptote.
6. Use smooth curves to complete the graph between and beyond the vertical asymptotes.

Homework
p.195 # 1 - 18 ALL
1 - 6, parts b and c only
7 - 12 are matching
13 - 18, parts a and b only