Functions Worksheet - Math 121: Extra Practice For Test 2 Page 4
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Rational Functions
Recall that a rational function is a function of the form
P (x)
R(x) =
Q(x)
where P (x) and Q(x) are polynomials and Q(x) is not the zero polynomial.
(a) What is the domain of a rational function?
(b) What do the horizontal/slant asymptote tell you?
(c) What does a vertical asymptote tell you?
(d) When does the graph of a rational function have horizontal/slant/vertical asymptotes?
(e) What can you conclude about the graph of the rational function R(x) if as x
, R(x)
3.
(f) What can you conclude about the graph of the rational function R(x) if as x
3, R(x)
.
(g) When does a rational function have horizontal/slant asymptote?
(h) When does R(x) have vertical asymptotes?
(i) When does R(x) have hole(s) in the graph?
(j) Can R(x) have both a horizontal and a slant asymptote? Can it have more than one horizon-
tal/slant asymptotes?
(k) Can R(x) have more than one vertical asymptotes?
(l) Can the graph of R(x) cross any of the asymptotes?
13. Find all the vertical, horizontal and slant asymptotes, and any holes in the graphs of the given rational
functions:
2
2
2x
+ 7x + 3
x
4
(a) f (x) =
(d) f (x) =
x + 1
x + 2
2
2
10000x
+ 3x
1
2x
+ 1000000 + x
(e) f (x) =
(b) f (x) =
3
2
x
3x
x
4
x + 2
3x
+ 3
(c) f (x) =
(f) R(x) =
2
4
x
4
7x + 2x
TO SKETCH THE GRAPH OF A RATIONAL FUNCTION:
(a) Find the domain.
(b) Find the x-intercepts (if the numerator and the denominator do not have common factors then
the x-intercepts of the rational function are the zeros of the numerator.)
(c) Find the y intercept.
(d) Find the asymptotes and the holes, if any.
(e) Check if the graph of the function crosses the horizontal/slant asymptote.
(f) Plot the zeros and the points where the function is undefined on a number line. Pick a number
in each interval to determine if the graph of the rational function is above or below the x axis.
(g) Plot the asymptotes, the intercepts and use the information from the previous steps to obtain
the graph of R(x).
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