1. Rational Functions and Their Graphs
Section 3.6 Part 2
Rational Functions and Their Graphs
Objectives:
Graph rational functions.
Identify slant asymptotes.
Solve applied problems involving rational functions.

2. Strategy for Graphing a Rational Function
Suppose that f(x) = p(x) / q(x),
where p(x) and q(x) are polynomial functions with no common factors.
1. Determine whether the graph of f has symmetry.
f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
Find the y-intercept (if there is one) by evaluating f (0).
Find the x-intercepts (if there are any) by solving the equation p(x) = 0.
Find any vertical asymptote(s) by solving the equation q (x) = 0.
Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function.
Plot at least one point between and beyond each x-intercept and vertical asymptote.
Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

3. Sketch the graph of:
Symmetry
y-intercept
x –intercept
Vertical asymptote
Horizontal asymptote
Test points

4. the graph has no symmetry The y-intercept is (0,-3/10)
Solution:
the graph has no symmetry
The y-intercept is (0,-3/10)
The x-intercept is (3/2, 0)
The vertical asymptote is x = -2
The horizontal asymptote is y = 2/5
Test points include (-3, 9/5), (0, -3/10), (2, 1/20)

5. Practice #1
Graph:

6. Practice #1 (cont.)

7. Practice #2
Graph:

8. Practice #2 (cont.)

9. Text Example Find the slant asymptote of
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x If it cannot be factored, use long division. Ignore any remainders when determining the slant asymptote.
The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph is shown. -2 -1 4 5 6 7 8 3 2 1 -3
Vertical asymptote:
x = 3
Slant asymptote:
y = x - 1

10. Practice #4
Find the slant asymptote of

11. Real Life Application
The average cost per unit for a company to produce x units is the sum of its fixed and variable costs divided by the number of units produced. The average cost function is modeled below:
A company is planning to manufacture wheelchairs that are light,
Fast, and beautiful. Fixed monthly cost sill be $500,00, and it will
Cost $400 to produce each radically innovative chair.
(a) Write the average cost function of producing x wheelchairs.

12. (b) Find and interpret:
(c) What is the horizontal asymptote for the average cost function? Describe what this represents for the company.
(d) Why would this model create difficulties for small businesses?