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Properties of Quartics

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Introduction

Table of Contents Introduction In this investigation quartic functions will be explored, in particular quartic functions with a unique "W" shape. These quartic functions will have two points of inflection, which will be referred to as Point Q and Point R. When Line QR is drawn it intercepts the quartic function twice more; those interceptions will be referred to as Point P and Point S. The ratio of PQ:QR:RS will be investigated and any findings will be formally proven. This investigation will include the given function f(x) = x4 - 8x3 + 18x2 - 12x +24 and the chosen function f(x) = x4 + 3x3 -5x . These functions will be depicted with graphs. Method 1. Graph the function f(x) = x4 - 8x3 + 18x2 - 12x +24. 2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersect the quartic function again, and calculate the ratio PQ:QR:RS. Find Q and R Find First Derivative: f(x) = x4 - 8x3 + 18x2 - 12x +24 f'(x) ...read more.

Middle

2) d = 1.24 PQ:QR:RS = 1.24:2:1.24 3. Simplify this ratio so that PQ = 1 and comment upon your results. Divide by PQ (1.24/1.24):(2/1.24):(1.24/1.24) 1:1.618:1 When PQ is set to equal 1, QR becomes 1.618. This number is known as "Phi" (?) or the "Divine Proportion", as it is a number found in ratios in most things in nature, such as in spirals in Nautilus Shells or in the construction of the Human Body. 4. Choose another quartic function with a "W" shape and investigate the same ratios. Choose a Function f(x) = x4 + 3x3 -5x Find Q and R Find First Derivative: f(x) = x4 + 3x3 -5x f'(x) = 4x3 +9x2 -5 Find Second Derivative f"(x) = 12x2 + 18x Set to Zero 0 = 12x2 + 18x Apply Quadratic Equation x = -1.5, 0 Concavity Checks f(-2) = 12 positive A switch from positive to negative indicates an inflection point. f(-1) = -6 negative A switch from negative to positive indicates an inflection point. f(4) = 30 positive Find Y Point Q f(-1.5) ...read more.

Conclusion

= x4 - 8x3 + 18x2 - 12x +24 and chosen equation f(x) = x4 + 3x3 -5x , quartic equations will always be proportionate in terms of ?. This is proven by how a "W" shape of a quartic will only appear if the proportions are exactly 1:1.681:1. If the ratio is not exactly this proportion, then the "w" shape will no longer be present. However, for the Divine Proportion to be present in a quatric equation, two distinct inflection points must be present, and it does not have to result in a "w" shape quatric function. 6. Extend this investigation to other quartic functions that are not strictly of a "W" shape. For quartic functions not strictly of a "W" shape then the proportion of 1:1.681:1 does not exist. For example, take the quartic equation f(x) = x4 + 4x. As can be seen by the graph there is no inflection point from which to derive a ratio. For other types of quartic equations, like those with one inflection point, the proportion could not be found as there is no other point for which to compare to. ?? ?? ?? ?? 1 ...read more.

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