Exercise 37 Page 285

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

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

Let's tackle them one at a time!

End Behavior

Consider the given function. We can see above that the leading coefficient is $8$ 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.

$x$	$8x^3$	$y=8x^3$
$\text{-}1$	$8(\text{-}1{)}^3$	$\text{-}8$
$\text{-}0.5$	$8(\text{-}0.5{)}^3$	$\text{-}1$
$0$	$8(0{)}^3$	$0$
$0.5$	$8(0.5{)}^3$	$1$
$1$	$8(1{)}^3$	$8$

We found that $(\text{-}1,\text{-}8),$ $(\text{-}0.5,\text{-}1),$ $(0,0),$ $(0.5,1),$ and $(1,8)$ 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 there are no turning points.

Increasing/Decreasing Intervals

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

From the graph above, we see that the function increases from $\text{-}\infty$ to $\infty$.