To describe the shape of the graph of the given cubic function, we have to determine three things.

The end behavior of the function.
The turning points of the function.
The decreasing and increasing intervals of the function.

Let's tackle them one at a time!

End Behavior

Consider the given function.

y = 3 x^{3} - x - 3

We can see above that the leading coefficient is

3

and the degree is

3 .

With this information, we know that the end behavior of the cubic polynomial is down and up.

Turning Points

Since the degree of the polynomial is $3,$ it will have at most two turning points. Let's use a table of values to find some points on the function.

$x$	$3 x^{3} - x - 3$	$y = 3 x^{3} - x - 3$
$- 1$	$3 (- 1)^{3} - (- 1) - 3$	$- 5$
$- 0.5$	$3 (- 0.5)^{3} - (- 0.5) - 3$	$- 2.875$
$0$	$3 (0)^{3} - (0) - 3$	$- 3$
$0.5$	$3 (0.5)^{3} - (0.5) - 3$	$- 3.125$
$1$	$3 (1)^{3} - (1) - 3$	$- 1$

We found that $(- 1, - 5),$ $(- 0.5, - 2.875),$ $(0, - 3),$ $(0.5, - 3.125),$ and $(1, - 1)$ are points on the graph of the function. We can plot and connect these points with a smooth curve. Remember, the end behavior is down and up.

Looking at the graph, we can see two turning points. Although we cannot state an exact answer, we can approximate that the turning points are located at $(- 0.3, - 2.8)$ and $(0.3, - 3.2) .$

Increasing/Decreasing Intervals

Finally, we will determine the increasing and decreasing intervals. Let's consider the graph of the function one more time.

Blue cubic function labeled y = 3x^3 - x - 3. Green arrows show the increasing direction of the function, while red arrows show the decreasing direction. The turning points are represented with red points.

The function decreases between the two turning points. It also increases before the first turning point and after the second one.

Decreasing Interval : Increasing Interval : (- 0.3, 0.3) (- \infty, - 0.3) \cup (0.3, \infty)

Exercise

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