Intersections Project

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


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


3/12

Amogh Sahu

Grade 11/Mr.Bishop

Due:18/01/2012

2) When A=3



x1=2.17

x2=2.57

x3=3.77

x4=4.54


SL=.4

SR=.77

Therefore,

SL- SR=0.37

and since D= SL- SR

D=0.37

3) When A=4


4/12

Amogh Sahu

Grade 11/Mr.Bishop

Due:18/01/2012



x1=2.22

x2=2.61

x3=3.64x4=4.28


SL=.39

SR=.64

Therefore,

SL- SR=0.25

and since D= SL- SR

D=0.25

4) When A=5



x1=2.28

x2=2.64

x3=3.56

x4=4.12


SL=.36

SR=.56

Therefore,

SL- SR=0.2

and since D= SL- SR

D=0.2


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


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)



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


7/12

Amogh Sahu

Grade 11/Mr.Bishop

Due:18/01/2012

x4=4.73


SL=.73

SR=1.73

Therefore,SL- SR=1

and since D= SL- SR

D=1

b) (2,1)



x1=1

x2=1.38

x3=3.62

x4=5


SL=.38

SR=1.38

Therefore,

SL- SR=1

and since D= SL- SR

D=1


8/12

Amogh Sahu

Grade 11/Mr.Bishop

Due:18/01/2012

c) (3,1)



x1=1.55

x2=2

x3=5

x4=6.45


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.


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



10/12

Amogh Sahu

Grade 11/Mr.Bishop

Due:18/01/2012


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



x1=1.13

x2=2

x3=5

x4=8.87


SL=.87

SR=3.87

Therefore,

SL- SR=3

and since D= SL- SR

D=3


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)


Therefore the value of D needs to recalculated. The Previous examples showed

x1=1.22

x2=1.45

x3=6.55

x4=7.78


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.


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

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Intersections Project