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You are having a conference call with the CEO of a paper company. You have interpreted the number of trees cut down versus profit as the function P(x) = –x4 + x3 + 7x2 − x − 6. Describe to the CEO what the graph looks like and, in general, how to sketch the graph without using technology. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0).

2 Answers

4 votes

Answer:

First, lets find the zeros.

-x4 + x3 + 7x2 - x - 6 = 0

Using synthetic division, the left side factors to

(x - 3)(-x3 - 2x2 + x + 2) = 0

Then by group factoring the second factor,

(x - 3)[-x3 + x - 2x2 + 2] = 0

(x - 3)[-x(x2 - 1) - 2(x2 - 1)] = 0

(x - 3)(x2 - 1)(-x - 2) = 0

(x - 3)(x - 1)(x + 1)(-x - 2) = 0

Our zeros are then

x = -2

x = -1

x = 1

x = 3

The company will break even when 1 tree is cut down or 3 trees are cut down.

Since we have a negative 4th degree function, the graph will start off increasing, and then end off decreasing. This is a symmetrical function. Using this fact, we know can tell the shape the graph. It will have 2 maximums and 1 minimum. There is a maximum in the interval (-2, -1), a minimum in the interval (-1, 1), and another maximum in the interval (1, 3). This is in the general sense of the graph.

However, the number of trees must be positive. So the company makes a minimum profit when they don't cut down any trees. The company makes a maximum profit when 2 trees are cut down. When no trees are cut down, the company loses profit by 6.

As for the end behaviors, x cannot be negative because you cannot cut down negative number of trees. Therefore, the graph starts at x=0.

The profits start to increase when at most 2 trees are cut down.

After 2 trees are cut down, the profits decrease.

To get a better visual, you can draw the graph on a graphing calculator.

Explanation:

User Zaratruta
by
5.4k points
2 votes
Since we have a negative 4th degree function, the graph will start off increasing, and then end off decreasing. This is a symmetrical function. Using this fact, we know can tell the shape the graph. It will have 2 maximums and 1 minimum. There is a maximum in the interval (-2, -1), a minimum in the interval (-1, 1), and another maximum in the interval (1, 3). This is in the general sense of the graph.

However, the number of trees must be positive. So the company makes a minimum profit when they don't cut down any trees. The company makes a maximum profit when 2 trees are cut down. When no trees are cut down, the company loses profit by 6.

As for the end behaviors, x cannot be negative because you cannot cut down negative number of trees. Therefore, the graph starts at x=0.
User Idobr
by
6.0k points