Curve Sketching
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 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
Graph
x
y =
x^{2}  1
Solution:
xint at (0,0)
Same for yint
(x^{2}  1)(1) 
x(2x)
x^{2}  1
f '(x) =
=
(x^{2}  1)^{2}
(x^{2}  1)^{2}
Since the numerator is never zero, so there are no local extrema and
f(x) is
increasing when x > 1 or x < 1.
Now take the second derivative
The second derivative 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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