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# Properties of quartics

Extracts from this document...

Introduction

 Ecolint – DP 2009/10 Properties of Quartics Math HL – Portfolio Assignment
 Kathia Zimba6/23/2009

## Introduction

Quartic functions are functions that the highest exponent is 4. These types of functions are of the form 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  . The second derivative of this function  will gives the points inflection at  ; provided  .  FIND 2ND DERIVATIVE OF            when  and  At  and  Middle . 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 PQ  QR  RS, thus, leading to the ratio segments.

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

Conclusion   The quadratic equation will be used to obtain the roots which are equivalent to the missing points P and S.        or  The two roots found (above) are the x coordinates of the missing points and they happen to be irrational numbers.

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

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