Essay about Parabolic Investigation

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A Parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape. This is u
In this task I will investigate the patterns in the intersections of parabolas and the lines y = x and y = 2x. Forming a conjecture that holds true for the vertex of the parabola being in the first quadrant and then change it so it holds true for the vertex is in any quadrant. Then I will prove my conjectures for other lines like y = 3x and 4x and so on and I will also change the degree of the polynomials and their values to prove the conjuncture to be true for values greater than 3.

Using the dynamic graphing
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‘Find values of D for other parabolas of the form y = ax2 + bx + c , a>0, with vertices in quadrant 1, intersected by the lines y = x and y = 2x. Consider various values of a, beginning with a = 1. Make a conjecture about the values of D for these parabolas’

Ans 2. To find a conjecture for D we need to find the factor that affects the values of D (a,b,c).
Case I – Change the values of ‘a’, ‘b’ & ‘c’ in the equation ax2+bx+c.

a b c x1 x2 x3 x4 SL SR D
1 -6 11 1.76 2.38 4.62 6.24 0.62 1.62 1
2 -2.50 1.4 0.37 0.62 1.13 1.88 0.25 0.75 0.50
3 -3.25 1.1 0.24 0.34 1.08 1.51 0.10 0.43 0.33
4 -3 0.75 0.17 0.25 0.75 1.08 0.08 0.33 0.25
5 -4 1 0.20 0.28 0.78 1 0.08 0.28 0.20
8 -6.5 1.5 0.22 0.29 0.65 0.84 0.07 0.19 0.12
10 -8 2 0.28 0.40 0.50 0.72 0.12 0.22 0.10

From the above table we can observe that when there is a change in the values of ‘a’, ‘b’, ‘c’ there is a change in the value of

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