The change in water vapor in a cloud is modeled by a polynomial function, C(x). Describe how to find the x-intercepts of C(x) and how to construct a rough graph of C(x) so that the meteorologist can predict when there will be no change in the water vapor. You may create a sample polynomial of degree 3 or higher to use in your explanations. In your lab, a substance's temperature has been observed to follow the function T(x) = (x + 5)3 + 7. The turning point of the graph is where the substance changes from a liquid to a solid. Using complete sentences in your written answer, explain to your fellow scientists how to find the turning point of this function. Hint: The turning point of the graph is similar to the vertex of a quadratic function.

Two students in your class, Hunter and Maggie, are disputing a function. Hunter says that for the function, between x = −2 and x = 2, the average rate of change is 0. Maggie says that for the function, between x = −2 and x = 2, the graph goes up through a turning point, and then back down. Explain how Hunter and Maggie can both be correct, using complete sentences.

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).