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Properties of quartics

Extracts from this document...

Introduction

Ecolint – DP 2009/10

Properties of Quartics

Math HL – Portfolio Assignment

Kathia Zimba

6/23/2009


Introduction

Quartic functions are functions that the highest exponent is 4. These types of functions are of the formimage08.png

The graphs of a Quartic functions usually exert two shapes; “W” shape or “M” shape. For this investigation, an analysis of a “W” shaped function is to be carried out to explore the properties of the function. The points of inflection of the Quartic function, will be looked at very closely so that the ratio between the distances of the points if intersection when the Quartic graph is cut by a straight line is found.

Analysis

Let’s take image22.pngimage22.png. The second derivative of this function image45.pngimage45.png will gives the points inflection at image53.pngimage53.png; provided image63.pngimage63.png.image09.pngimage00.png

FIND 2ND DERIVATIVE OF image10.pngimage10.png

image22.png

image27.png

image32.pngimage32.png

image33.pngimage33.png

image34.pngimage34.png

image35.pngimage35.png when image36.pngimage36.png and image37.pngimage37.png

At image38.pngimage38.png and image39.pngimage39.png

...read more.

Middle

image42.png. When this straight line is drawn, the line meets the Quartic again at another two points P and S creating three identical segments.

It is important the coordinates of these two points, so that the ratio PQ:QR:RS is determined. The next step for completing this investigation is to calculate the equation of the straight line. From the equation of the straight line, the coordinates of point P and S will be obtained, hence, the distance between PQimage49.pngimage49.pngQRimage49.pngimage49.pngRS, thus, leading to the ratio segments.

So, the graph of image50.pngimage50.png with the points of inflection would look like this.


image51.pngimage00.pngimage05.pngimage06.pngimage04.png

image07.png

image05.png


As mentioned before a line is to be drawn, passing through the points of inflection (Q and R). After this is done the equation of the line can be calculated.

...read more.

Conclusion

image03.png

image73.pngimage73.png

The quadratic equation will be used to obtain the roots which are equivalent to the missing points P and S.

image11.pngimage11.png

image12.pngimage12.png

image13.pngimage13.png

image14.pngimage14.png or image15.pngimage15.png

The two roots found (above) are the x coordinates of the missing points and they happen to be irrational numbers.

 P since image16.pngimage16.pngimage17.pngimage17.png

image18.pngimage18.pngimage19.pngimage19.png

image20.pngimage20.png

image21.pngimage21.png

image23.pngimage23.png

S since image24.pngimage24.pngimage18.pngimage18.png S image25.pngimage25.png

image26.pngimage26.png

image20.pngimage20.png

image28.pngimage28.png

image29.pngimage29.png

        Once found the points P and S, the new graph would look like the one below.

The goal of this investigation is with the points of inflections and the extension of the line passing through them –image31.pngimage31.png  – the ratio between the points would be found. The distance formula has its roots from the Pythagoras’ theorem. Hence, if a triangle is drawn between the points linking them, a triangle should be visible (see the graph below). Again, the y coordinates can be ignored as to they are not exact. Thus, when calculating the ratio the x coordinates are used.image30.png

...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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