If I am given 3 points on a graph, for instance, (0,0), (1,4), and (2,3), that form a parabola (the second point is the vertex) and want to find the slope at the vertex of that parabola, will a simple slope calculation (rise/run) of the secant line across the two end points (in the example, (0,0) and (2,3)) give the correct slope at the vertex or is it necessary to first find the equation of the parabola, calculate its derivative, and then substitute the vertex x-value (in the example, 1) into the derivative to find slope?
A (possible) big catch is that all of the x-values of the end points of the parabolas are "incremental." I just mean to say that given any 3 points, they are of the form (x-1, y0), (x, y), (x+1, y1) where (x, y) is the point at the vertex. I'm also only working with integers for x and y values.
In this particular example with points (0,0), (1,4) and (2,3), it works out that: the derivative of the parabola evaluated at the vertex x-value = rise/run of the secant line extending to the end points, (0,0) and (2,3).
The bottom-line question I have is: given 3 points that form a parabola (with the guidelines I set on what the x and y values can be, not sure if these guidelines actually have an effect), is it true that the slope at the vertex of the parabola is equal to the slope of the secant line extending across the parabola? Thank you!