Investigate the slope of a line drawn through two points on a curve for various points on various curves. Such a line is called a secant line.

1. Just to be sure you have your bearings: What changes on the diagram when you change the value of p? What changes when you change the value of h?[br][br]2. For some combination of p and h, calculate the slope of the secant line and check your answer by making the slope calculation visible.[br][br]3. If you hide the slope calculation, can you reproduce the formula for [math]m_{sec}[/math] in terms of g, p, and h? Study the formula until you understand what all the letters represent well enough that you can reproduce the formula on your own.[br][br]4. What happens to [math]$m_{sec}$[/math] as h gets closer to 0? when h = 0? Can you explain this by referring to the graph and/or the equation for [math]m_{sec}[/math]?[br][br]5. What seems to be the relationship between the value of p and the value that [math]m_{sec}[/math] approaches as h gets very close to 0 for the given function [math]g(x)=\frac{1}{2}x^2-8 [/math]?[br][br]6. Can you prove that this is indeed the relationship?[br][br]7. Repeat steps 6 and 7 for some other functions of your choosing. Start with quadratic functions, then branch out.