# Properties of Quartics

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

. 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) = 4x3 - 24x2 + 36x - 12

Find Second Derivative

f"(x) = 12x2 - 48x + 36

Set to Zero

0 = 12x2 - 48x + 36

;

x = 1, 3

Concavity Checks

f(0) =36 positive

A switch from positive to negative indicates an inflection point.

f(2) = -12 negative

A switch from negative to positive indicates an inflection point.

f(4) = 36 positive

Find Y

Point Q f(1) = 23

Point R f(3) = 34 - 8(3)3 + 18(3)2 - 12(3) + 24 = 15

Inflection points are:

Q (1, 23) R (3, 15)

Line QR

Equation of a Line Passing through Two Points:

y - y1 = (y2 - y2)/(x2 - x1) * (x - x1)

Plug in Known Values

y - 15 = (23-15)/(1-3) * (x-3)

y ...