1. Section 2.6 Rational Functions Part 1

2. What you should learn How to find the domains of rational functions
How to find the horizontal and vertical asymptotes of graphs of rational functions
How to analyze and sketch graphs of rationale functions
How to sketch graphs of rational functions that have slant asymptotes
How to use rational functions to model real-life problems

3. Rational Function A rational function can be written in the form
Where N(x) and D(x) are polynomials.
D(x) is not the zero polynomial.

4. Find the domain of f(x)

5. Horizontal Asymptote at y = 0 when degree of numerator less than degree of the denominator.
Find the domain of g(x)
Vertical Asymptote at the zero of the denominator.

6. Asymptotes Vertical Horizontal at zeros of the denominator
N < D asymptote at y = 0
N = D asymptote at y = aN / aD
N > D no Horizontal asymptote

7. N = D asymptote at y = aN / aD

8. Guidelines for Analyzing the Graphs of Rational Functions
Find and plot the y-intercept by evaluating f(0).
Find the zeros of the numerator and plot the points.
Find the zeros of the denominator and sketch the vertical asymptotes.
Find and sketch the horizontal asymptote if it exists.

9. Guidelines for Analyzing the Graphs of Rational Functions
Test for symmetry
Plot at least one point between and one point beyond each x intercept and vertical asymptote.
Use smooth curves to complete the graph between and beyond the vertical asymptotes

10. Graph 1. y-intercept when x = 0 f(0) = undefined
x = 0 is a vertical asymptote

11. Graph 2. Find the zeros of the numerator 0 = x2 – 4 = (x + 2)(x – 2)

12. Graph 3. Find the zeros of the Denominator x = {0} 0 = x2
We already found this in step #1
x = 0 is a vertical asymptote

13. Graph 4. Find and sketch the horizontal asymptote
Since the degree of the numerator and the denominator are the same the horizontal asymptote will be
y = 1/1=1
y = 1

14. Graph 5. Test for Symmetry
Since f(x)= f(-x) we know this drawing will be symmetrical about the y-axis

15. 6. Plot some points
Graph
f(1)= f(-1) = -3
f(3)= f(-3) = 5/9

16. 7. Use smooth curves to complete the graph.

17. 2.6 Homework Part 1
Rational Functions
page 174
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