Graphing Rational Functions
A List of Interesting features of a Graph
Below is a list of features of a graph that may assist in curve
Domain and Range
Intervals of Increase and Decrease
Most graphs contain only some of these eleven features, so
to sketch a graph we find as many interesting features as possible and use
these features to sketch the graph.
y = x3 - 3x2 - 9x
We find the x intercepts by factoring out the x and putting into
the quadratic formula.
(-1.8,0), (0,0), (4.9,0).
Note that the y intercept is also
The domain is R (all real numbers) since this is a polynomial.
The function is continuous since it is a polynomial.
There are no vertical asymptotes since we have a polynomial.
The function is differentiable everywhere.
f '(x) = 3x2 - 6x - 9 = 3(x - 3)(x + 1).
We see that f is increasing on (-,-1) and on
is decreasing on (-1,3).
By the first derivative test, f has a relative maximum at
a relative minimum at (3,-27).
f ''(x) = 6x -
so that f is concave down on (-,1)
concave up on (1,).
f(x) has an inflection point at (1,-11).
f has no horizontal asymptotes.
The graph of f is shown below:
x2 - 1
The x-intercept is at (0,0)
Same for the y-int
(x2 - 1)(1) -
-x2 - 1
f '(x) =
(x2 - 1)2
(x2 - 1)2
f '(x) = 0 has no solution since the numerator is always
negative, so there are no local extrema. Since the denominator is always
nonnegative, f(x) is decreasing for all x
not equal to -1 or 1
where the function is undefined.
(x2 - 1)2(-2x) - (-x2 - 1)[(2x)(2)(x2
f ''(x) =
(x2 - 1)4
(2x)(x2 - 1)[-(x2 - 1) + 2(x2 + 1)]
(x2 - 1)4
2x(x2 + 3)
(x2 - 1)3
is 0 when x = 0 and is positive when
x is between - 1 and
0 or x is greater than
1. This is where f(x) is concave up. It is concave down elsewhere except
at 0 and 1.
f(x) has a vertical asymptote at x = 1 and
The horizontal asymptote is y = 0.
The graph is shown below.
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