Turning Points and Asymptotes of a Rational Function

Introduction

Understanding turning points and asymptotes is crucial for analyzing the behavior of rational functions. This article focuses on a specific rational function F(x) to illustrate the concepts of turning points, differentiating the function, and finding asymptotes. The goal is to uncover the local extrema and the reflection property associated with a particular point on the graph of the function.

Background and Definitions

Rational functions are ratios of two polynomials. A turning point of a function occurs at points where the derivative is zero or undefined and the function changes its direction, indicating a local minimum or maximum. Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes are determined by the degrees of the numerator and denominator.

Analyzing the Function F(x)

The given function is defined as F(x), which is differentiable over the domain D (-∞, -2/3) U (-2/3, ∞). To find the turning points, we need to differentiate F(x) and set the derivative to zero.

Differentiating F(x)

After differentiating, we have:

F'(x) 2 - 12/3x^2

Setting F'(x) 0, we get:

2 - 12/3x^2 0

Multiplying all terms by 3 to simplify:

6 - 12x^2 0

Multiplying all terms by -1/6 to isolate x^2:

x^2 6/12 1/2

Therefore, the solutions are:

x ±sqrt(1/2) ±sqrt(6)/3

These points, approximately 0.15 and -1.49, are the turning points of F(x).

Calculating the Function Values at Turning Points

Evaluating the function F(x) at these turning points:

F(0.149830) ≈ 2.93265

F(-1.489163) ≈ -3.59941

Reflection Property of Asymptotes

The reflection property of a rational function is an interesting feature. The reflection point can be found as the intersection of the vertical and horizontal asymptotes. For the function given:

Vertical Asymptote

The vertical asymptote is determined by setting the denominator to zero:

3x^2 0

x -2/3

Horizontal Asymptote

The horizontal asymptote can be found by examining the behavior of the function as x approaches infinity. By dividing the numerator and denominator by the highest power of x, we get:

lim_{x → ±∞} F(x) 0

This implies a horizontal asymptote at y 0.

Oblique Asymptote

An oblique asymptote can be found when the degree of the numerator is one more than the degree of the denominator. Here, the oblique asymptote is:

G(x) 2x - 1

The reflection point is the intersection of the vertical and oblique asymptotes, which can be calculated:

x -2/3, y -1/3

The reflection point is (-2/3, -1/3).

Verification of Reflection Property

To verify the reflection property, we choose a point on the function, say (0, 3). Reflecting this point through the reflection point (-2/3, -1/3) should land on another point on the graph:

x_0 0, y_0 3

The displacements are:

x_0 - x 2/3, y_0 - y 3 1/3 10/3

Reversing the directions, we get:

x_1 -2/3 - 2/3 -4/3, y_1 -1/3 - 10/3 -11/3

The point (-4/3, -11/3) should be on the graph of F(x).

Substituting x -4/3 into F(x):

F(-4/3) 2 - 4/3(1) 4/3(-4/3)^2

2 - 4/3 4/3(16/9)

2 - 4/3 64/27

2 - 4/3 21.67/27

2 - 4/3 21.67/27

2 - 4/3 0.76 -11/3

The point (-4/3, -11/3) is indeed on the graph of F(x), verifying the reflection property.

Conclusion

Understanding turning points and asymptotes is crucial for analyzing the behavior of rational functions. The given function F(x) illustrates how to find these points and verify their properties. The reflection point, in particular, is an interesting feature that highlights the symmetry in the graph of the function.