Download - Intersections Project
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7/29/2019 Intersections Project
1/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
Intersections ProjectQuestion 1:The parabola is graphed, along with the lines and . Theirintersections were found, and they are illustrated below:
In this case, the points D,C,B,A(in order) will be referred to as x1,x2,x3 and x4 and the intersection
between the line and the parabola. The Values of the four different x-values are given below(as the
x-coordinates of the points A,B,C,D) :
- x1=1.76
- x2=2.38
- x3=4.62
- x4=6.24
The values will then be subtracted by each other in the format:
1) x2- x1= 0.62
2) x4- x3 = 1.62
Expression 1) can now be referred to as SL and Expression 2) will be referred to as SR.
SL- SR=1
D= SL- SR
D=1
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7/29/2019 Intersections Project
2/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
Question 2:
In this question, the value of the parabola will remain fixed except for the change to the first
coefficient, referred to in the standard formula as A. Four Values of A will be tested, and a
conjecture about the value of D with respect to the changes in the value in A will be found, with D
being defined as the expression SL- SR found in the previous question .
1) When A=2
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=2
x2=2.5
x3=4
x4=5
From this the values of SL and SR can be derived as follows
SL=0.5
SR=1Therefore,
SL- SR=0.5
and since D= SL- SR
D=0.5
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7/29/2019 Intersections Project
3/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
2) When A=3
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=2.17
x2=2.57
x3=3.77
x4=4.54
From this the values of SL and SR can be derived as follows
SL=.4
SR=.77
Therefore,
SL- SR=0.37
and since D= SL- SR
D=0.37
3) When A=4
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7/29/2019 Intersections Project
4/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=2.22
x2=2.61
x3=3.64x4=4.28
From this the values of SL and SR can be derived as follows
SL=.39
SR=.64
Therefore,
SL- SR=0.25
and since D= SL- SR
D=0.25
4) When A=5
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=2.28
x2=2.64
x3=3.56
x4=4.12
From this the values of SL and SR can be derived as follows
SL=.36
SR=.56
Therefore,
SL- SR=0.2
and since D= SL- SR
D=0.2
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7/29/2019 Intersections Project
5/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
From the values that have been found above, it is now possible to create a conjecture based
on data, about the value of D relative to a change in the coefficient A in the equation Ax^2+Bx+C.
The data combining these different values is given below:
A X1 X2 X3 X4 SL SR D
1 1.76 2.38 4.62 6.24 0.62 1.62 1
2 2 2.5 4 5 0.5 1 0.5
3 2.17 2.57 3.77 4.54 0.4 0.77 0.37
4 2.22 2.61 3.64 4.28 0.39 0.64 0.25
5 2.28 2.64 3.56 4.12 0.36 0.56 0.2
The relationship between A and D can further be investigated by graphing the values of A relative to
the values of D, and comparing the graph to a certain parent function. This is done below:-
This looks a lot like a reciprocal function, specifically that of (1/x), which is shown below.
Considering this similarity, as well as the fact that the data reflects this relationship even when
investigated arithmetically, by substituting the specific A-Values into (1/x), it is safe to assume that
the conjecture is true for all real values of A when the intersecting lines and the vertex is held
constant.
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6
D relative to A
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7/29/2019 Intersections Project
6/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
General Conjecture:
Considering that this conjecture applies to the examples given above, it is necessary to test this
conjecture by changing the other factors involved in the polynomial.
Question 3:
Here, the variable that will be changed will be the vertex of the parabolic function to see the effect
of the change on relationship between D and A. Four new Vertices will be tried out, and their
values are given below:
a) (2,2)
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=1.27
x2=2x3=3
when D can be defined as SL-SR when two parallel lines
intersect with a parabola, and A as the first coefficient in
the standard form given that the vertex andthe number of intersecting lines is held constant
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7/29/2019 Intersections Project
7/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
x4=4.73
From this the values of SL and SR can be derived as follows
SL=.73
SR=1.73
Therefore,SL- SR=1
and since D= SL- SR
D=1
b) (2,1)
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=1
x2=1.38
x3=3.62
x4=5
From this the values of SL and SR can be derived as follows
SL=.38
SR=1.38
Therefore,
SL- SR=1
and since D= SL- SR
D=1
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7/29/2019 Intersections Project
8/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
c) (3,1)
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=1.55
x2=2
x3=5
x4=6.45
From this the values of SL and SR can be derived as follows
SL=.45
SR=1.45
Therefore,
SL- SR=1
and since D= SL- SR
D=1
Conclusion:
The relationship between D and A does not change. According to the conjecture mentioned
previously, the value of A did not change in all of the previous examples, and neither did the value of
D. The vertex was the only factor that changed. Therefore, it is wise to conclude that the value of the
vertex does not impact the value of D relative to A.
Proof:
The conjecture described above can be proved by turning all the constants into variables. The
standard form of the quadratic equation can be equated to x and 2x, and the process from which D
was derived previously can be repeated in variable form to see if that relationship holds.
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7/29/2019 Intersections Project
9/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
Solving both equations,
SL-SR=
Question 4:
Considering that every factor but the intersecting lines have been changed, it is necessary to
investigate the effect of changing intersecting lines on the value of D relative to A. There will be
three different pairs of intersecting lines, and the conjecture will be modified according to the
results of the conjecture. The pairs are given below:
1) Y=x, Y=3x
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
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7/29/2019 Intersections Project
10/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
Therefore the value of D needs to recalculated.
x1=1.3
x2=2
x3=5
x4=7.7From this the values of SL and SR can be derived as follows
SL=.7
SR=2.7
Therefore,
SL- SR=2
and since D= SL- SR
D=2
2) y=x, Y=4x
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated.
x1=1.13
x2=2
x3=5
x4=8.87
From this the values of SL and SR can be derived as follows
SL=.87
SR=3.87
Therefore,
SL- SR=3
and since D= SL- SR
D=3
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7/29/2019 Intersections Project
11/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
3)Y=6x, Y=4x(with a change in the value of A)
The Values of x1,x2,x3 and x4,as given previously, have changed as the parabola has changed.
Therefore the value of D needs to recalculated. The Previous examples showed
x1=1.22
x2=1.45
x3=6.55
x4=7.78
From this the values of SL and SR can be derived as follows
SL=.23
SR=1.23
Therefore,
SL- SR=1
and since D= SL- SR
D=1
The Relationship:
A X1 X2 X3 X4 SL SR D Y2-Y1 Diff.Coefficient/A
1 1.3 2 5 7.7 0.7 2.7 2 22x 2
1 1.13 2 5 8.87 0.87 3.87 3 3x 3
2 1.22 1.45 6.55 7.78 .23 1.23 1 2x 1
Considering the trend given above, the relationship between the change in the intersecting lines and
the value of A, a conjecture can be derived which explains this relationship.
when D can be defined as SL-SR when two lines intersect with aparabola, and A as the first coefficient in the standard form
given, and that M2 and M1 are assumed as the
coefficients of the two intersecting lines.
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7/29/2019 Intersections Project
12/12
Amogh Sahu
Grade 11/Mr.Bishop
Due:18/01/2012
Proof:
x
Question 5:
The conjecture will now be tested for a cubic function to see whether the values change when the
function changes to become a cubic. The