Investigating Secant Lines
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?
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.
3. If you hide the slope calculation, can you reproduce the formula for 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.
4. What happens to as h gets closer to 0? when h = 0? Can you explain this by referring to the graph and/or the equation for ?
5. What seems to be the relationship between the value of p and the value that approaches as h gets very close to 0 for the given function ?
6. Can you prove that this is indeed the relationship?
7. Repeat steps 6 and 7 for some other functions of your choosing. Start with quadratic functions, then branch out.