Analyzing The Function H(x) = (2x - 6) / (x + 3) Determining Intercepts And Asymptotes

Introduction: Delving into the Realm of Rational Functions

In the captivating world of mathematics, functions serve as the fundamental building blocks that describe relationships between variables. Among the diverse family of functions, rational functions hold a special place, characterized by their expression as the ratio of two polynomials. Understanding the behavior of these functions is crucial in various fields, from physics and engineering to economics and computer science. In this article, we embark on an in-depth exploration of a particular rational function, $h(x) = (2x - 6) / (x + 3)$, dissecting its properties and unraveling the mysteries it holds. Our journey will involve analyzing its intercepts, asymptotes, and overall graphical representation, providing a comprehensive understanding of this fascinating mathematical entity.

Exploring the Function h(x) = (2x - 6) / (x + 3)

The function $h(x) = (2x - 6) / (x + 3)$ presents itself as a quintessential rational function, a ratio of two linear expressions. To truly grasp its essence, we must delve into its key characteristics. This function's behavior is governed by the interplay between its numerator, $2x - 6$, and its denominator, $x + 3$. The roots of these polynomials, along with their relative magnitudes, dictate the function's intercepts, asymptotes, and overall shape. By meticulously examining these elements, we can construct a vivid picture of the function's graph and its behavior across the coordinate plane.

Unveiling the Intercepts: Where the Graph Meets the Axes

Intercepts are the points where a function's graph intersects the coordinate axes, providing crucial anchors for sketching its trajectory. The $y$-intercept, the point where the graph crosses the $y$-axis, reveals the function's value when the input $x$ is zero. Conversely, the $x$-intercept, the point where the graph crosses the $x$-axis, signifies the input value(s) that make the function's output zero. Finding these intercepts is akin to pinpointing the function's ground level and launchpad, guiding our understanding of its overall behavior.

Determining the y-intercept

To find the $y$-intercept, we set $x = 0$ in the function's expression and evaluate the result. For our function, $h(x) = (2x - 6) / (x + 3)$, this translates to:

$$h(0) = (2(0) - 6) / (0 + 3) = -6 / 3 = -2$$

Therefore, the graph intersects the $y$-axis at the point $(0, -2)$, not at $(0, 3)$ as the statement suggests. This seemingly simple calculation unveils a crucial aspect of the function's behavior, placing it firmly below the $x$-axis at the point of intersection.

Finding the x-intercept

The $x$-intercepts, on the other hand, are the solutions to the equation $h(x) = 0$. For a rational function, this occurs when the numerator equals zero, provided the denominator is not simultaneously zero at the same value of $x$. In our case, we need to solve the equation:

$$2x - 6 = 0$$

Adding 6 to both sides and dividing by 2, we get:

$$x = 3$$

Thus, the graph intersects the $x$-axis at the point $(3, 0)$, signifying that the function's output is zero when the input is 3. This intercept marks a critical point where the function transitions from negative to positive values, or vice versa.

Asymptotes: Guiding the Function's Trajectory

Asymptotes are imaginary lines that a function's graph approaches but never quite touches, acting as guides that dictate its long-term behavior. Vertical asymptotes arise when the denominator of a rational function approaches zero, creating a point of discontinuity where the function's value shoots off towards infinity or negative infinity. Horizontal asymptotes, on the other hand, describe the function's behavior as $x$ approaches positive or negative infinity, indicating the value the function tends towards as it stretches out along the horizontal axis. Understanding asymptotes is like having a roadmap for the function's journey, revealing its ultimate destination and the boundaries it adheres to.

Unveiling Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero. For $h(x) = (2x - 6) / (x + 3)$, this happens when:

$$x + 3 = 0$$

Subtracting 3 from both sides, we find:

$$x = -3$$

Therefore, there is a vertical asymptote at $x = -3$. This vertical line acts as a barrier, preventing the function's graph from crossing it. As $x$ approaches -3 from the left, the function's value plummets towards negative infinity, while as $x$ approaches -3 from the right, the function's value soars towards positive infinity. This dramatic behavior near the vertical asymptote is a hallmark of rational functions.

Discovering Horizontal Asymptotes

Horizontal asymptotes dictate the function's long-term behavior as $x$ approaches infinity or negative infinity. To find the horizontal asymptote, we examine the degrees of the polynomials in the numerator and denominator. In our case, both the numerator ($2x - 6$) and the denominator ($x + 3$) are linear, meaning they have the same degree (1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is:

$$y = 2 / 1 = 2$$

This horizontal asymptote at $y = 2$ signifies that as $x$ becomes very large (positive or negative), the function's value approaches 2. The graph will get increasingly close to this horizontal line but never actually cross it, unless there is an intersection at a finite value of $x$.

Sketching the Graph: Visualizing the Function's Behavior

With the intercepts and asymptotes in hand, we can now sketch the graph of $h(x) = (2x - 6) / (x + 3)$. The vertical asymptote at $x = -3$ divides the graph into two distinct regions. The horizontal asymptote at $y = 2$ provides a ceiling and floor for the function's long-term behavior. The $y$-intercept at $(0, -2)$ and the $x$-intercept at $(3, 0)$ act as anchors, guiding the curve's trajectory.

Constructing the Graph Piece by Piece

To the left of the vertical asymptote ($x < -3$), the function approaches the horizontal asymptote from below, starting from negative infinity and gradually increasing towards $y = 2$. To the right of the vertical asymptote ($x > -3$), the function approaches the horizontal asymptote from above, starting from positive infinity and decreasing towards $y = 2$. The intercepts provide additional guidance, ensuring the graph passes through the points $(0, -2)$ and $(3, 0)$. By connecting these pieces, we obtain a complete picture of the function's graph, revealing its unique shape and behavior.

Conclusion: A Comprehensive Understanding of h(x)

In this exploration, we have meticulously analyzed the rational function $h(x) = (2x - 6) / (x + 3)$, uncovering its intercepts, asymptotes, and overall graphical representation. We determined that the graph intersects the $y$-axis at $(0, -2)$, not $(0, 3)$, as the initial statement suggested. We also identified a vertical asymptote at $x = -3$ and a horizontal asymptote at $y = 2$, which dictate the function's long-term behavior. By combining these insights, we can confidently sketch the graph of the function and understand its behavior across the coordinate plane.

This comprehensive analysis highlights the importance of understanding the individual components of a rational function – its intercepts, asymptotes, and polynomial degrees – in order to fully grasp its behavior. By mastering these concepts, we can unlock the secrets of rational functions and apply them to a wide range of real-world problems.

Summary of Findings

• The graph of $h(x) = (2x - 6) / (x + 3)$ intersects the $y$-axis at $(0, -2)$, not $(0, 3)$.
• The function has a vertical asymptote at $x = -3$.
• The function has a horizontal asymptote at $y = 2$.
• The graph intersects the $x$-axis at $(3, 0)$.

This detailed analysis provides a solid foundation for understanding the behavior of rational functions and their applications in various fields.