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

. 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

Apply Quadratic Equation

;

x = 1, 3

Concavity Checks

f(0) =36 positive

A switch from positive to negative indicates an inflection point.

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

Apply Quadratic Equation

;

x = 1, 3

Concavity Checks

f(0) =36 positive

A switch from positive to negative indicates an inflection point.