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Graph the rational function


Step 1: Find the domain of the rational function
Step 2: Write f(x) in lowest terms
Step 3: Locate the intercepts of the graph. The x-intercepts, if any, of in lowest terms satisfy the equation p(x)=0. The y-intercept, if there is one, is f(0)
Step 4: Test for symmetry. Replace x by −x in f(x). If f(−x)=f(x)there is symmetry with respect to the y-axis; if f(-x)=-f(x) there is symmetry with respect to the origin.
Step 5: Locate the vertical asymptotes. The vertical asymptotes, if any, of in lowest terms are found by identifying the real zeros of q(x). Each zero of the denominator gives rise to a vertical asymptote.
Step 6: Locate the horizontal or oblique asymptotes, if any, using the procedure given in page 2 of the hand-out. Determine the points, if any, at which the graph of f(x)(in lowest terms) intersects these asymptotes.
Step 7: Graph f(x) using graphing utility
Step 8: Use the results obtained in steps 1 through 7 to graph f(x) by hand


Finding Horizontal Asymtotes

Consider the Rational Function
in which the degree of the numerator is n and the degree of the denominator is m.
1. If n<m, then f(x) is a proper rational function, and the graph of f(x) will have the horizontal asymptote y=0
2. If n≥m, then f(x) is improper. Here long division is used.
a. If n=m the quotient obtained will be , and the line y= is the horizontal asymptote.
Note:an is the coefficient of the leading term of numerator and bn is the coefficient of the denominator.
b. If n=m+1 the quotient is the form ax+b , and the line y=ax+b is the horizontal asymptote.
c. If n>m+1 the quotient is a polynomial of degree 2 or higher, f(x) and has neither a horizontal asymptote nor an oblique asymptote.


1. Domain :_________________________________________
2. in lowest terms ____________________

3. x intercept(s): _______________

4. y intercept(s): _______________
a. Symmetry with respect to y axis (yes/no)?
b. Symmetry with respect to origin (yes/no)?

6. Vertical asymptote(s): ______________________________

7. Horizontal asymptote(s): ____________________________

8. Does the graph intersect the horizontal asymptote? If it does, at what points?

9. Graph f(x) in the graphing utility

10. Use all the results above and draw f(x) by hand