We started by plotting a parabola, y = x^2, (using a table of Values) Next we plotted 2 points on the parabola and connected them.
Our task was to find the slope and y-intercept of the line segment by looking at the coordinates of the end points?
In this paper we will be explaining if and how we can do this.

First, we plotted two random points on the parabola, connected them, and found the slope using the slope formula, m = \frac{y_1-y_2}{x_1-x_2}.
We thought that there might be an easier way to find the slope by just looking at the points. So we started to experiment, found a couple, and tried them out on some other points to see if it was a true statement.
We eventually found the equation m = q - p,which seemed to work on all the points we picked out at random. After finding that, we found the y-intercept by plugging the slope in to this equation, . We thought that there would be an easier way to find the y-intercept also. We experimented and found some equations, tested them out on quit a few points and eventually found the right one.
We found that there was one equation that worked. It was y-intercept = (x_1)(x_2). We then plotted two points randomly on the parabola (that we haven’t used yet), and tested it one last time. We then had to figure out how exactly this equation works for this graph.
Then we took the slope equation and the points (Q, Q^2) as point one and (P, P^2) as point two. Plugging those points into the slope equation we get,m = \frac{Q^2 - P^2}{Q - P}. And as we already know, a^2 - b^2 = (a + b)(a - b). So we reduced the equation and got m \frac{(Q + P)(Q - P)}{(Q - P)}. Because \frac{(Q - P)}{(Q - P)} divide out, you’re left with (Q + P), which is what we got for the easier way to find slope.
Now we have to do the same kind of thing to find how we got the y intercept. We will take the slope intercept equation and the same random two points (connect as a line segment) and find how we got that easier way to the y intercept. First we have chosen two points (-3, 9) and (7, 49) which we are going to find the slope first with m = Q - P.  Which would be m = 7 - 3 \quad or \quad m = 4. Now we are going to plug the solution into the slope-intercept equation, to find the y-intercept.  We now have 9 = 4(-3) + b. Simplifying, we have 21 = b.  Now we know how we have found the slope and y-intercept, in an easier way. We have found that we can in fact find the slope and y-intercept by just looking at the coordinates of the points.

In summary,
Given a line segment connnected by two points on the parabola y = x^2, the slope of the segment and it's y-intercept can be found by the following:
If (P, P^2) and (Q, Q^2),
then m = Q - P and b = PQ.

Investigate using our GeoGebra applet and try it for yourself.

Thank you for your time.