BQ# 7: Unit V

     The origins of the the difference quotient comes from a graph and using an old equation from early this year.

Here is the graph f(x) and the line that's barely touching it at x is the tangent line. So the coordinates for the graph is ( x, f(x) ). 

This is another graph that has a secant line going through it. It still has the original point as the first one but it has another point in it. Since we moved a little bit from the original to the new point, then it's a change in the graph. That change can be written as delta x or as i put it h. So the new coordinates for this graph is ( x, f(x) ), ( x+h, f(x+h) ).

Highlighted in blue is the two coordinates from the graph. Highlighted in green is the slope formula that will help find the difference quotient. The one in pink has everything plugged in and in the denominator we see that the xs' cancel so there's only an  h left in the denominator. The last one in purple is the difference quotient and that is how we get the equation.