A List of Interesting features of a Graph
Below is a list of features of a graph that may assist in curve sketching:
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
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.
f '(x) = 3x2
- 6x - 9 = 3(x - 3)(x + 1).
We see that f is increasing on (-
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
and concave up on (1, ).
f(x) has an inflection point at
f has no horizontal asymptotes.
The graph of f is shown below:
x2 - 1
x-int at (0,0)
Same for y-int
(x2 - 1)(1) -
-x2 - 1
f '(x) =
(x2 - 1)2
(x2 - 1)2
Since the numerator is never zero, so there are no local extrema and
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
The horizontal asymptote is y = 0.
The graph is shown below.
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