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Math Investigation - Properties of Quartics

Extracts from this document...

Introduction

Investigation

Mathematical Investigation

Ritesh P. Kothari

10/25/2007

A quartic function with a “W” shape has two points of inflection, Q and R. In this investigation a line is drawn through Q and R to meet the quartic function again at P and S. The ratio PQ: QR: RS is to be investigated using specific examples to form a conjecture, and then examined formally to prove the findings.

IB Mathematics Portfolio Type I

Candidate name: Ritesh Kothari

Candidate Code: 001859-018

School name: Mahatma Gandhi International School

School Code: 001859


  1. Graph the functionimage06.pngimage06.pngf(x) = image37.pngimage37.png4 - 8image88.pngimage88.png 3 + 18image37.pngimage37.png2 - 12image37.pngimage37.png + 24

image07.png

The above graph is made using “Microsoft Excel 2007” and the displayed function is

F(x) = image08.pngimage08.png4 – 8image08.pngimage08.png3 + 18image08.pngimage08.png2 – 12image08.pngimage08.png + 24

Scale used: X: – 1.5 to 5.5

        Y: 6 to 66

  1. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersects the quartic function again, and calculate the ratio PQ:QR:RS.

Planning:

  • Find the points of inflection in the function by differentiating the function twice.
  • Substituting the X values into the function to get Y co-ordinates.
  • Get the equation of a line which passes through the inflection points, named Q and R respectively.
  • Find out two points P and S, where f(image08.pngimage08.png) intersects the line which passes through the inflection points (Q and R).
  • Finding out the roots of the function, ignoring the found points of inflection and use these roots to find other intersection points for the function.
  • To find the ratios we will need to find out the distances between the points using the properties of similar triangles.
  • Find the ratio between the points PQ, QR, and RS. Then simplify the points to get some result.
...read more.

Middle

image08.png3 + 18image08.pngimage08.png2 – 8image08.pngimage08.png – 3 = 0

We will get 4 roots from the above equation; we already know two roots which are Q and R respectively. Therefore, to find out the other two roots we will use Synthetic Division. We can thus divide the function by the point Q with the X-value of 1 to acquire a cubic function from quartic:

1   1        -8        18        -8        -3image00.png

                1        -7        11        3

     1         -7        11          3        0image01.png

The cubic is image08.pngimage08.png3 – 7image08.pngimage08.png2 + 11image08.pngimage08.png + 3 = 0

Another root of the function can be found by dividing point R with X-value of 3 to get a quadratic function from cubic:

3   1        -7        11        3image00.png

                3        -12        -3

     1         -4         -1         0image03.png

The quadratic is image08.pngimage08.png2 – 4image08.pngimage08.png – 1 = 0

Using the quadratic formula we can now find out the remaining two roots of the quartic.

image09.png

image10.png

image11.png

Roots of the function are therefore:

P: image08.pngimage08.png = image12.pngimage12.png

Q: image08.pngimage08.png = 1

R: image08.pngimage08.png = 3

S: image08.pngimage08.png = image13.pngimage13.png

The graph below shows 4 intersection points of the functions. The quartic function f(image08.pngimage08.png) and a linear function g(image08.pngimage08.png) have two inflection points Q and R, and the other intersection points are P and S respectively.

image14.png

To find out the relationships between PQ, QR and RS lines we must make triangles created by the intersection points. All the three triangles made would be similar because all the three angles of these triangles are equal. This is because all these triangles have been created by horizontal and a vertical line of a function with a constant slope of a line. As

...read more.

Conclusion

image82.pngimage82.png. We would not be able to find the ratio as point R will be missing in the function.

The last type of quartic does posses two distinct and real inflection points. From the definition we know that points of inflection occur whenimage83.pngimage83.png, and we also know that function is a straight line whenimage84.pngimage84.png. Points of inflection will be at X values whereimage85.pngimage85.png.

Given a general quarticimage78.pngimage78.png, and using previously arrived first and second derivatives, we arrive at this condition:

image86.png

image87.png

Any quartic which satisfies this equation will thus have at least one real inflection point.

In case of inflection points, the ratio PQ: QR: RS will be the same as in any “W” shaped quartic. This is because the general conjecture from question 5 applies to all quartics with two distinct inflection points.

Bibliography: -

Internet

  • The International Baccalaureate Organization,  Mathematics HL DP Agenda, the IBO, http://www.ibo.org/events/dipsjo108/documents/MathematicsHL.pdf
  • The Old Pro, “Graphing Polynomial Functions - Cubic and Quartic Equations”, The Old Pro, http://www.theoldpro.net/math/graphingpolynomials/
  • JSTOR, “Notes on Quartics”, JSTOR, http://links.jstor.org/sici?sici=0002-9890%28194903%2956%3A3%3C165%3ANOQC%3E2.0.CO%3B2-L&size=LARGE&origin=JSTOR-enlargePage
  • Planet Math, “Quartic Formula”, Planet Math, http://planetmath.org/encyclopedia/QuarticFormula.html

Books

  • Basset, Alfred Barnard, An Elementary Treatise on Cubic and Quartic Curves, Michigan, University of Michigan, 2005.

Done By: - Ritesh P. Kothari

IB Mathematics HL Portfolio – Investigation Type 1                                                             |

...read more.

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