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Parabolic Investigation Essay

2369 words - 9 pages

In this portfolio task I have investigated the patterns in the intersection of parabolas and various lines. I have formed a conjecture to find the value of D of the parabolas, which are intersected by 2 lines, of varying slopes and shown the proof of its validity. I have used the TI-84 graphic display calculator, the software Geoegebra and Microsoft Excel to do my calculations.
I have even investigated the values of D, for polynomials of higher powers and tried to come up with a general solution for all equations. I have been able to do this portfolio from the knowledge learnt from classroom discussions and through various other resources.

Question 1
“Consider the parabola y = (x−3)2 + 2 = x2−6x+11 and the lines y=x and y=2x.
(a) Using technology find the four intersections illustrated on the right.
(b) Label the x-values of these intersections as they appear from left to right on the x-axis as x1 , x2 , x3 , and x4 .
(c) Find the values of x2 – x1 and x4 – x3 and name them respectively SL and SR .
(d) Finally, calculate D = | SL − SR|”

SOLUTIONS:
Graphical Method -
Using the software Geogebra, the following graph for f(x), g(x) and h(x) is obtained –

From the graph, we can get to know that the line g(x) = x , intersects the parabola f(x) at points P (2.38,2.38) and O (4.62,4.62) , and the line h(x) = 2x intersects f(x) at points N (1.76, 3.53) and M (6.24, 12.47).
The points X1, X2, X3, and X4 have been plotted on the graph.
SL = X2 − X1
= 2.38 − 1.76
= 0.62

SR = X4 − X3
= 6.24 – 4.62
= 1.62

D = |SL − SR|
= |0.62 – 1.62|
= |-1|
= 1

THEORITICAL METHOD -
f(x) = (x−3)2 + 2
= x2 – 6x + 11
g(x) = x
Since, f(x) is intersected by g(x), the points of intersection will have same x-coordinates and y-coordinates.
Therefore, equating the y-values of both the equations,
x2 – 6x + 11 = x
x2 – 7x + 11 = 0
Using the quadratic formula to find the roots of the equation,
=
=
=
=
= 4.618 or 2.382
Therefore, X2 = 2.382 and X3 = 4.618

h(x) = 2x
Since, f(x) is also intersected by h(x), the points of intersection will have same x-coordinates and y-coordinates.
Therefore, equating the y-values of both the equations,

Using the quadratic formula to find the roots of the equation,
=
=
=
=
= 6.236 or 1.764
Therefore, X1 = 1.764 and X4 = 6.236

SL = X2 − X1
= 2.382 – 1.764
= 0.618
SR = X4 – X3
= 6.236 – 4.618
= 1.618
D = |SL − SR|
= |0.618 – 1.618|
= |-1|
= 1
Hence, The graphical and theoretical methods give the same answers.

Question 2
“For different 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 value of D for these parabolas.”
SOLUTION:
Let’s...

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