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 inte...
... middle of paper ...
... the value of D will be 0, as it is similar to the cubic graph.
Bibliography:
Websites:
http://en.wikibooks.org/wiki/Theory_of_Equations
(Date accessed – 11/05/09)
http://id.mind.net/~zona/mmts/intersections/intersectionLineAndParabolla1/intersectionLineParabolla1.html
(Date accessed – 09/05/09)
http://www.analyzemath.com/Calculators/Parabola_Line.html
(Date accessed – 13/05/09)
http://www.mathforum.org/library/drmath/view/65384.html
(Date accessed – 14/05/09)
http://en.wikipedia.org/wiki/Quadratic_equations
(Date accessed – 10/05/09)
http://www.math.vanderbilt.edu/~schectex/courses/cubic/
(Date accessed – 16/05/09)
http://en.wikipedia.org/wiki/Cubic_equation
(Date accessed – 17/05/09)
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