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
sketching:

xintercepts

yintercepts

Domain and Range

Continuity

Vertical Asymptotes

Differentiability

Intervals of Increase and Decrease

Relative Extrema

Concavity

Inflection Points

Horizontal Asymptotes
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.
Examples
Example 1
Graph
y = x^{3}  3x^{2}  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
(0,0).

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.

We find
f '(x) = 3x^{2}  6x  9 = 3(x  3)(x + 1).
We see that f is increasing on (,1) and on
(3,). f
is decreasing on (1,3).

By the first derivative test, f has a relative maximum at
(1,5) and
a relative minimum at (3,27).

f ''(x) = 6x 
6
so that f is concave down on (,1)
and
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:
Example 2
Graph
x
y =
x^{2}  1
Solution:
The xintercept is at (0,0)
Same for the yint
(x^{2}  1)(1) 
x(2x)
x^{2}  1
f '(x) =
=
(x^{2}  1)^{2}
(x^{2}  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.
(x^{2}  1)^{2}(2x)  (x^{2}  1)[(2x)(2)(x^{2}
 1)]
f ''(x) =
(x^{2}  1)^{4}
(2x)(x^{2}  1)[(x^{2}  1) + 2(x^{2} + 1)]
=
(x^{2}  1)^{4}
2x(x^{2} + 3)
=
(x^{2}  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
1.
The horizontal asymptote is y = 0.
The graph is shown below.
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