Analyzing The Graph Of H(x) = (4x² - 100) / (8x - 20) A Comprehensive Guide

Jul 12, 2025 by ADMIN 76 views

Unveiling the Secrets of Rational Functions: Analyzing the Graph of h(x) = (4x² - 100) / (8x - 20)

Navigating the world of rational functions can be a fascinating journey, especially when we delve into understanding their graphical representations. Today, we'll dissect a specific rational function, h(x) = (4x² - 100) / (8x - 20), to reveal its hidden characteristics and accurately describe its graph. This involves a meticulous examination of its asymptotes, potential discontinuities, and overall behavior. Our primary goal is to select the correct statement that accurately portrays the graph of this function from the options provided. Let's embark on this mathematical exploration together, ensuring a comprehensive and insightful understanding.

Deciphering the Function: A Step-by-Step Analysis

Before we jump into the graphical aspects, let's simplify and analyze the function h(x) = (4x² - 100) / (8x - 20). This preliminary step is crucial as it unveils the function's core structure and any potential simplifications that might be present. By simplifying the function, we can identify key features such as holes, vertical asymptotes, and the function's overall behavior as x approaches infinity or negative infinity.

1. Factoring for Clarity

The first step is to factor both the numerator and the denominator of the rational function. This will help us identify common factors that can be canceled out, leading to a simplified form of the function. Factoring is a fundamental technique in algebra, allowing us to rewrite expressions in a more manageable and insightful way. It's like taking apart a complex machine to understand its individual components and how they interact.

Numerator: The numerator, 4x² - 100, is a difference of squares. We can factor out a 4 first, giving us 4(x² - 25). Then, we apply the difference of squares factorization: 4(x - 5)(x + 5).
Denominator: The denominator, 8x - 20, can be factored by taking out the greatest common factor, which is 4. This gives us 4(2x - 5).

2. The Simplified Form

After factoring, our function looks like this:

h(x) = [4(x - 5)(x + 5)] / [4(2x - 5)]

Now, we can cancel out the common factor of 4, simplifying the function to:

h(x) = [(x - 5)(x + 5)] / (2x - 5)

This simplified form is much easier to work with and reveals important information about the function's behavior.

3. Identifying Potential Discontinuities

Discontinuities in rational functions occur where the denominator equals zero. These points are crucial for understanding the function's graph, as they can lead to vertical asymptotes or holes. To find these points, we set the denominator equal to zero and solve for x.

2x - 5 = 0
2x = 5
x = 5/2

This tells us that there is a potential discontinuity at x = 5/2. However, we need to investigate further to determine whether it's a vertical asymptote or a hole.

4. Unmasking Vertical Asymptotes and Holes

A vertical asymptote occurs when the denominator of the simplified function equals zero, and the factor does not cancel out with a factor in the numerator. A hole, on the other hand, occurs when a factor in the denominator cancels out with a factor in the numerator.

In our simplified function, h(x) = [(x - 5)(x + 5)] / (2x - 5), the factor (2x - 5) in the denominator does not cancel out with any factor in the numerator. Therefore, we have a vertical asymptote at x = 5/2.

It's essential to distinguish between these two types of discontinuities, as they have different graphical representations. Vertical asymptotes are represented by vertical lines that the function approaches but never crosses, while holes are represented by open circles on the graph.

Asymptotes: The Guiding Lines of the Graph

Asymptotes are invisible lines that the graph of a function approaches as x or y approaches infinity. They act as guides, shaping the overall behavior of the function. Understanding asymptotes is crucial for accurately sketching the graph of a rational function. There are three main types of asymptotes: vertical, horizontal, and oblique (or slant). We've already identified the vertical asymptote; now, let's explore the possibility of horizontal or oblique asymptotes.

Horizontal Asymptotes: The Long-Term Trend

Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. To determine if a horizontal asymptote exists, we compare the degrees of the numerator and denominator in the original function.

Original Function: h(x) = (4x² - 100) / (8x - 20)
Degree of Numerator: 2
Degree of Denominator: 1

Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, there is a possibility of an oblique asymptote.

Oblique Asymptotes: A Slanted Guide

An oblique asymptote (also called a slant asymptote) occurs when the degree of the numerator is exactly one greater than the degree of the denominator. In our case, this condition is met, so we know an oblique asymptote exists. To find the equation of the oblique asymptote, we perform polynomial long division.

Polynomial Long Division

Dividing (4x² - 100) by (8x - 20):

        (1/2)x + (5/4)
    ____________________
8x - 20 | 4x² + 0x - 100
        - (4x² - 10x)
        ____________________
              10x - 100
              - (10x - 25)
              ____________________
                    -75

The quotient we obtain is (1/2)x + (5/4), and the remainder is -75. The oblique asymptote is given by the quotient, ignoring the remainder. Therefore, the equation of the oblique asymptote is:

y = (1/2)x + (5/4)

This line will act as a guide for the function's behavior as x approaches positive or negative infinity.

The Correct Statement: Putting It All Together

Now that we've thoroughly analyzed the function, we can confidently select the correct statement describing its graph. Let's recap our findings:

Vertical Asymptote: x = 5/2
Horizontal Asymptote: None
Oblique Asymptote: y = (1/2)x + (5/4)

Based on this analysis, the correct statement is:

B. The graph has an oblique asymptote.

This statement accurately captures a key characteristic of the function's graph. The presence of an oblique asymptote significantly influences the function's long-term behavior, making it a crucial feature to identify.

Conclusion: Mastering Rational Function Analysis

Analyzing rational functions requires a systematic approach, combining algebraic manipulation with graphical interpretation. By factoring, simplifying, identifying discontinuities, and determining asymptotes, we can gain a comprehensive understanding of a rational function's behavior and accurately describe its graph. This process not only helps in selecting the correct statement but also builds a strong foundation for further mathematical explorations. Mastering these techniques unlocks a deeper appreciation for the intricate world of functions and their graphical representations. Remember, practice is key to solidifying these concepts and building confidence in your mathematical abilities. So, keep exploring, keep analyzing, and keep uncovering the beauty hidden within the world of mathematics!