There is a pattern between a parabolas intersection points of the lines y=x and y=2x. There are going to be four points for every situation of intersecting of lines. For each of the points that are found, label them x1,x2,x3,x4. People use parabolas for real life situations like money and profits or economics. This a project that I have given to my students, it's tedious but enjoyable.
You can manipulate the parabolas equation, the only “restriction” is that it has to intersect with y=x and y=2x. The intersecting points that are found are going to be very important, with them you will need to find x2-x1 and x4-x3 and for the answers you get name them SL and SR. Then find the absolute value of SL-SR, (absolute value means that the answer will always be positive) name the answer as “D”. I started playing around with parabolas equations in quadrant 1. I would manipulate the value of “a” starting at 1 and just started getting creative but not to complex to where the answers would have been extreme.
When you start playing around with the equation for the parabola, you might start to notice a pattern if you look closely. When I put “3x^2” my final answer was “⅓” and if i did 5 i got “⅕” and when I did 1 I came to the answer of “1/1” or 1. My hypothesis is that the answer will basically be the inverse of your a value for you parabola equation. The vertex of the parabola being up or down does not effect the outcome, as long as the parabola has four intersections then it should work with the two lines y=x and y=2x. An easy example would be the line f(x)=(x^2-6x)+11 intersecting lines y=x and y=2x. I entered the parabola and the two intersecting lines and the four values of the intersections that were given to me by the calculator were x1=1.763932023, x2=2.381966011, x3=4.618033989, and x4=6.236067978. Once the four values from the x-axis are found then find the answers for x2-x1 and x4-x3 (enter the exact number do not round because then the final answer will be to far off from accurate). Another name for x2-x1 is SL and another name for x4-x3 is SR, then find the absolute value for SL-SR and the equation for that is .618033988-1.618033988, so then the final answer is "1" and for this problem the value of "a" was 1, based on the data above and the few more tries my conjecture will have been proven correct.
I conjecture that the value of a has everything to do with the outcome of the answers. Keep the roots and vertex inside quadrant 1, manipulate the value of a, and get results that much the pattern where a is the inverse of the final answer.