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Ib math HL portfolio parabola investigation

Extracts from this document...

Introduction

Math HL Portfolio

Parabola Investigation

image01.jpg

INTRODUCTION

In this portfolio I am going to investigate the patterns formed when a parabola,

y=ax2+bx+c intersects with the lines y=x and y=2x. Further I would be broadening the scope of this investigation to other lines and other orders of polynomials and observing patterns in their respective intersections.  

The first part of my portfolio would include parabolas with their turning points located in the 1st quadrant of a graph. I would be investigating the patterns formed when the two lines y=x and y=2x intersect with parabolas that have different coefficients of x2 but lie in the first quadrant and form a conjecture. Then I would prove it and test its limitations.

The second part would include testing and modifying my conjecture for the turning point of the parabola situated in any quadrant and testing for all real co-efficient of x2.

The third part would include modifications to the conjecture if the intersecting lines are changed and hence finding its limitations.

In the final part of my portfolio I will also try and make a conjecture for polynomials of higher order and derive any patterns or observations plausible.

PARABOLAS IN THE FIRST QUADRANT INTERSECTING WITH y=x AND y=2x.

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by:

image02.png

...read more.

Middle

X1

X2

X3

X4

SL

SR

D= | SL- SR |

 D = 1/a

1

1.763932

2.381966

4.618034

6.236068

0.618034

1.618034

1

TRUE

2

2

2.5

4

5

0.5

1

0.5

TRUE

3

2.131483

2.565741

3.767592

4.535184

0.434258

0.767592

0.33333

TRUE

4

2.219224

2.609612

3.640388

4.280776

0.390388

0.640388

0.25

TRUE

0.5

1.535898

2.267949

5.732051

8.464102

0.732051

2.732051

2

TRUE

2.5

2.07335

2.536675

3.863325

4.72665

0.463325

0.863325

0.4

TRUE

20.5

1.882709

2.441355

4.265752

5.31505

0.558646

1.049298

0.490652

TRUE

100.5

2.147934

2.573967

3.742261

4.484522

0.426033

0.742261

0.316228

TRUE

-1

-0.414214

-0.302776

3.302776

2.414214

0.111438

-0.88856

1

FALSE

-2

0.5

0.633975

2.366025

3

0.133975

0.633975

0.5

FALSE

-3

0.81954

1

2.333333

2.847127

0.18046

0.513794

0.33333

FALSE

-4

1

1.190983

2.309017

2.75

0.190983

0.440983

0.25

FALSE

-0.5

-1.645751

-2.44949

2.44949

3.645751

-0.80374

1.196261

2

FALSE

-100.5

1.038615

0.855132

2.82864

2.328929

-0.18348

-0.49971

0.316228

FALSE

Table 3: Showing the testing for all real values of a.

Hence I can modify my conjecture1:

D=  image82.pngimage82.png , a image51.png R.

  • Where D is defined as |(x2 - x1) – (x4 – x3)| where x1 and x4 are the points of intersection of lines y=2x and x2 and x3 are the points of intersection of lines y=x with the parabola y=a(x-h)2 + k
  • h and k > 0 ( Vertex in the first quadrant)
  • a, h and k  image51.png R

(The proof for this modified conjecture is the same as the proof given above on page 7 where we got D = image82.pngimage82.png )

PARABOLAS IN OTHER QUADRANTS INTERSECTING WITH LINES y=x and y=2x.

I will now see if my conjecture holds true for parabolas having their vertex in other quadrants.

image84.png

Graph 15: Value of a =-4 and turning point is in the 2nd quadrant.

image85.png

Graph 16: Value of a= 2 and turning point is in the 3rd quadrant.

image86.png

Graph 17: Value of a= 3 and turning point is in the 4th quadrant.

a (quadrant #)

X1

X2

X3

X4

SL

SR

D=|SL -SR|

D=1/|a|

a=-4 (2nd quad)

-3.395644

-3

-1.25

-1.10436

0.395644

0.145644

0.25

TRUE

a=2 (3rd quad)

-2.366025

-3

-0.5

-0.63398

-0.63398

-0.13398

0.5

TRUE

a=3 (4th quad)

1.389683

1.52519

4.808143

5.276984

0.135507

0.468841

0.33333

TRUE

Table 4: Showing the testing of my conjecture for turning points in different quadrants.

Hence I can modify my conjecture1:

D=  image82.pngimage82.png ,  a image51.png R.

  • Where D is defined as |(x2 - x1) – (x4 – x3)| where x1 and x4 are the points of intersection of lines y=2x and x2 and x3 are the points of intersection of lines y=x with the parabola y=a(x-h)2 + k
  • a, h and k  image51.png R

(The proof for this modified conjecture is the same as the proof given above on page 7 where we got D = image82.pngimage82.pngand we know that the value of D only depends on the coefficient of x2 not the values of b and c. Hence the ONLY condition for my conjecture is that the parabola MUST have four intersection points with the two lines.)

ANY PARABOLA INTERSECTING WITH ANY 2 LINES

In the previous part of my portfolio I investigated the patterns formed in the intersection of two lines y=x and y=2x with any parabola and found Dimage87.pngimage87.png.

 Now I will further expand my portfolio by investigating patterns formed in with any line.

Let us assume that:

  • P, N are the x coefficients of straight lines y= Px + p and y=Nx + n.
  • P, N image11.pngimage11.pngimage12.pngimage12.png
  • P < N

The basic quadratic equation y = image89.pngimage89.png would intersect the two lines at four distinct points and I have found these four points, labeling the intersections with y=Px+p as X2 and X3 and the intersections with y=Nx+n as X1 and X4. Using these four points I will again find the value of D = | (X2 – X1) – (X4 – X3) |.

image90.png

image91.png

image92.png

image93.png

image94.png

image95.pngimage95.png + n

image96.png

image97.png

image98.png

image99.png

...read more.

Conclusion

image45.pngimage45.pngwhere ‘a’ is the coefficient of xn and b is the coefficient of xn-1.  

Now that we know that the sum of the roots of any polynomial is image33.pngimage33.pngwe can find the value of D for any polynomial.

We already know that value of D for a parabola intersecting with two lines and for a cubic curve intersecting with two lines.

For a polynomial higher than that of degree 3, the coefficients of Xn and xn-1 always remain ‘a’ and ‘b’ respectively. The two lines y=Px+p and y=Nx+n do not have an effect on the value of ‘a’ and ‘b.

Thus the value of D is

image23.pngimage23.png=| (sum of the roots of the first line) – (sum of the roots of the second line) |

We know that (sum of the roots of first line) = image25.pngimage25.png and (sum of the roots of the second line)  =  image25.pngimage25.png

image26.pngimage26.png- (image28.pngimage28.png|

= image29.pngimage29.png+ image30.pngimage30.png

=0

Thus D = 0 for any polynomial higher than order 2.

CONCLUSION

I can conclude my portfolio by saying that there was a pattern found with the intersection of 2 lines and a polynomial. The value of D which is defined as the modulus of the sum of the roots of the first intersecting line minus the sum of the roots of the second intersecting line was found for each polynomial.

For Polynomial of degree 2D= |image10.pngimage10.png, where P and N image11.pngimage11.pngimage12.pngimage12.png and are the x coefficients of any 2 straight lines and a image13.pngimage13.png and is the coefficient of x2 in a parabola.

For a Polynomial of degree 3 or higher→ D = 0.

These are the patterns that I found and proved in my portfolio. This portfolio has further scope and the intersections of a cubic curve with a parabola can also give interesting patterns.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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