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Start by finding the end behavior.
End Behavior: Down and up
Turning Points: (-0.3,-2.8) and (0.3,-3.2)
Decreasing: (-0.3,0.3)
Increasing: (-∞,-0.3)∪(0.3,∞)
To describe the shape of the graph of the given cubic function, we have to determine three things.
Let's tackle them one at a time!
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 | 3x3−x−3 | y=3x3−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).
Finally, we will determine the increasing and decreasing intervals. Let's consider the graph of the function one more time.