RATIONAL FUNCTIONS

1. Transformations of parent function
a. / b. / c.
d. / e. / f.
2. Vertical Asymptotes (V.A.)
a. / b. / c.
d. / e. / f.
g. / h. / i.
3. V.A. (singles & doubles), Roots
a. / b. / c.
d. / e. / f.
g.
4. V.A. (singles & doubles), Roots (singles & doubles), Donuts
a. / b. / c.
d. / e. / f.
g. / h.
5. V.A. (singles, doubles, & none), Roots (singles & doubles), Donuts
a. / b. / c.
d. / e. / f.
g. / h. / i.
j.

Useful Steps for Sketching Rational Functions

Steps

/ Example:
1. Factor the numerator and denominator. /
2. Donuts: Does anything in the factored function reduce? If so, there is a removable discontinuity (or “donut” or “hole”) where the canceled factors would have equaled zero. / The factor cancels. Thus, there is a removable discontinuity (donut) where , which is .
The function reduces to:

From this point forward, use the reduced form of the function.
(But don’t forget the donut at the end.)
3. Vertical Asymptotes: Set the denominator equal to zero. Solve for x. Wherever the denominator of a rational function equals zero, the function is undefined. /
(the vertical asymptote)
4. Roots: Set and solve for x. If the function is a single ratio, then you need only set the numerator equal to zero. /


5. Y-intercept: By definition, the y-intercept is the value of y, or f(x), when x=0. /
y-intercept is (0, 0)
6. End Behavior: End behavior is determined by the leading terms, including the coefficients, of the polynomials in the numerator and denominator.
Suppose a, b, h, and k are real numbers. /
Leading Terms:

End Behavior:
If and b>a, then .
If and a=b, then .
If and a>b, then .
7. Sign Line: Use the values for the roots and the vertical asymptotes to create a sign line. Find the sign of the function on each interval and mark +/- along the x-axis.
Remember that roots and vertical asymptotes of odd multiplicity have opposite signs to their left and right; roots and vertical asymptotes of even multiplicity have the same sign to both their left and right. /
8. Sketch.
Remember that the curve will approach but never cross the vertical asymptotes. Horizontal and oblique/slant asymptotes demonstrate end behavior only; thus, the function may cross them.
DON’T forget the donut! /

Create a mnemonic device to help you remember the steps for sketching a rational function.

F
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Rational Functions—Graphs for Worksheet



/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line: /

/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line:


/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line: /

/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line:

/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line: /

/ Donuts:
Vert. Asymp.:
Roots:
y-intercept:
End Behavior:
Sign line: