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The steps to graph rational functions, including arranging the function, finding the domain, factoring, identifying holes and vertical asymptotes, and determining horizontal and slant asymptotes, x- and y-intercepts, and test points.
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a. Factor the denominator of the function, completely. b. Find the real zeros of the denominator by setting the factors equal to zero and solving. c. Write “Domain = {x|x 6 = }”
a. Set each factor from the denominator of the reduced function equal to zero and solve. b. Write each equation in the form x =. c. Draw each vertical asymptote as a dotted line on the graph.
Refer to the reduced function. You will need to know the leading terms from the numerator and the denominator. That is, F (x) =
anxn^ +... bmxm^ +...
where the degree of the numerator is n and the degree of the denominator is m.
i. If n < m, then the x-axis (given by y = 0) is the horizontal asymptote of the graph. ii. If n = m, then the line y =
an bm is the horizontal asymptote of the graph. iii. If n > m, then the graph has no horizontal asymptote.
The graph will only have a slant asymptote if n = m + 1. To find the slant asymptote, use long division. The slant asymptote will be the line given by y = quotient.
quotient
divisor
dividend
a. Set the factors of the numerator from the reduced function equal to zero and solve. b. Write the x-intercepts in the form ( , 0). c. Plot these points.
a. Substitute zero everywhere x appears in the reduced function and simplify. (Hint: This is just the ratio of the constants from the numerator and denominator.) b. Write the y-intercept in the form (0, ). c. Plot this point.
(NOTE: If you end up with division by zero during Step 8, then there is no y-intercept.)
Make a T-chart with a few more points to the right and left of each vertical asymptote.
x f (x)
♦ Refer to the vertical asymptotes you found in Step 5. When adding more points to your graph, remember that one vertical asymptote creates two branches of the graph, two vertical asymptotes creates three branches, and so on. So, you must find additional points that lie on either side of each vertical asymptote.
♦ Avoid simple arithmetic mistakes when you substitute x values into the function. Use a calcula- tor, if necessary, and pay attention to signs.
♦ Sketch your curves so that they follow the asymptotes, without adding any extra curlicues to the graph.
♦ The curves will never cross any vertical asymptote but they may cross a horizontal asymptote.