This exercise is designed to illustrate how numerical information from a function and its derivatives can be used to get a very good sense of how the function looks. While it is a good idea to use your graphing calculator to check your final answers, it would be missing the point to use it earlier. Consider the function f(x)=(x-1)/(x^2-5x+6). Use appropriate calculus notation and reasoning to answer the following questions.

1. What happens to f(x) when x goes to ? What happens when x goes to -? What happens when x = 0?

2. Does the graph have any vertical asymptotes? If so, what are they? If not, why not?

3. Where are the zeros (x-intercepts) of this function?

4. On what intervals is this function increasing? On what intervals is it decreasing?

5. Where are the local maxima and minima?

6. It is a fact that f''(x)=(2x^3-6x^2-6x+22)/((x^2-5x+6)^3). Where is f concave upward? Where is f concave downward? (You may use your calculator to solve the equation formed by setting the numerator equal to 0.)

7. Where are the inflection points?

8. Using this information, sketch a graph of this function on a separate piece of paper.