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Steps for Graphing Rational Functions
Suppose r is a rational function.
1. Find the domain of r.
2. Reduce r(x) to lowest terms, if applicable.
3. Find the x- and y -intercepts of the graph of y = r(x), if they exist.
4. Determine the location of any vertical asymptotes or holes in the graph, if they exist.
Analyze the behavior of r on either side of the vertical asymptotes, if applicable.
5. Analyze the end behavior of r. Use long division, as needed.
6. Use a sign diagram and plot additional points, as needed, to sketch the graph of y = r(x). 248 Rational Functions 3x
x2 − 4
Solution. We follow the six step procedure outlined above.
Example 4.2.1. Sketch a detailed graph of f (x) = 1. As usual, we set the denominator equal to zero to get x2 − 4 = 0. We ﬁnd x = ±2, so our
domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞).
2. To reduce f (x) to lowest terms, we factor the numerator and denominator which yields
f (x) = (x−2)(x+2) . There are no common factors which means f (x) is already in lowest
3. To ﬁnd the x-intercepts of the graph of y = f (x), we set y = f (x) = 0. Solving (x−2)(x+2) = 0
results in x = 0. Since x = 0 is in our domain, (0, 0) is the x-intercept. To ﬁnd the y -intercept,
we set x = 0 and ﬁnd y = f (0) = 0, so that (0, 0) is our y -intercept as well.3 4. The two numbers excluded from the domain of f are x = −2 and x = 2. Since f (x) didn’t
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