Graphing The Rational Function F(x) = (2x^2 - 4x - 30) / (x^2 - 2x - 8) A Step-by-Step Guide

Introduction: Understanding Rational Functions

In this comprehensive guide, we will delve into the process of graphing the rational function f(x) = (2x^2 - 4x - 30) / (x^2 - 2x - 8). Graphing rational functions is a fundamental skill in algebra and calculus, requiring a thorough understanding of their key characteristics. Rational functions, defined as the ratio of two polynomials, often exhibit interesting behaviors such as vertical and horizontal asymptotes, holes, and varying rates of change. To accurately sketch the graph of a rational function, we need to systematically analyze these features. This involves several steps, including factoring the numerator and denominator, identifying intercepts, determining asymptotes, and analyzing the function's behavior in different intervals. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems and gain a deeper appreciation for the beauty and power of mathematical functions. This article aims to provide a step-by-step approach, making the process clear and accessible even for those new to the topic. We will break down the function, identify its critical points, and then use this information to construct an accurate and informative graph. So, let's embark on this mathematical journey and unlock the secrets hidden within this rational function!

Step 1: Factoring the Numerator and Denominator

The first crucial step in analyzing any rational function is to factor both the numerator and the denominator. Factoring helps us identify common factors, which can lead to holes in the graph, and it also allows us to determine the roots (x-intercepts) and vertical asymptotes. For our function, f(x) = (2x^2 - 4x - 30) / (x^2 - 2x - 8), we begin by factoring the numerator, 2x^2 - 4x - 30. We can factor out a 2, giving us 2(x^2 - 2x - 15). Now, we need to factor the quadratic x^2 - 2x - 15. We are looking for two numbers that multiply to -15 and add to -2. These numbers are -5 and 3. Therefore, the factored form of the numerator is 2(x - 5)(x + 3). Next, we factor the denominator, x^2 - 2x - 8. We need two numbers that multiply to -8 and add to -2. These numbers are -4 and 2. Thus, the factored form of the denominator is (x - 4)(x + 2). Putting it all together, our factored function is f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)) . This factored form is essential for identifying key features of the graph, such as the zeros of the function, vertical asymptotes, and any potential holes. By breaking down the numerator and denominator into their simplest factors, we gain a clearer understanding of the function's behavior and its graphical representation. The factored form is the foundation upon which we will build our analysis and ultimately sketch the graph.

Step 2: Identifying Intercepts

Intercepts are the points where the graph of the function intersects the x-axis (x-intercepts) and the y-axis (y-intercept). These points provide valuable information about the function's behavior and its location on the coordinate plane. To find the x-intercepts, we set f(x) = 0 and solve for x. In other words, we need to find the values of x that make the numerator equal to zero, since a fraction is zero only if its numerator is zero. Using the factored form of our function, f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)), we set the numerator 2(x - 5)(x + 3) = 0. This gives us two solutions: x = 5 and x = -3. Therefore, the x-intercepts are the points (5, 0) and (-3, 0). These points are crucial anchors for our graph, indicating where the function crosses the x-axis. To find the y-intercept, we set x = 0 and evaluate f(0). Substituting x = 0 into the original function, we get f(0) = (2(0)^2 - 4(0) - 30) / ((0)^2 - 2(0) - 8) = -30 / -8 = 15 / 4. So, the y-intercept is the point (0, 15/4), or (0, 3.75). This point shows where the function intersects the y-axis, providing another key reference point for our graph. By identifying both the x and y-intercepts, we gain a better understanding of how the function is positioned in the coordinate plane and how it behaves near these points. Intercepts are essential landmarks that guide us in sketching an accurate graph of the rational function.

Step 3: Determining Asymptotes

Asymptotes are lines that the graph of a function approaches but never quite touches or crosses. They are crucial for understanding the end behavior and overall shape of a rational function. There are three main types of asymptotes: vertical, horizontal, and oblique (or slant). Vertical asymptotes occur where the denominator of the rational function is equal to zero, but the numerator is not. These indicate values of x where the function becomes undefined, causing the graph to shoot off towards positive or negative infinity. From our factored form, f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)), we see that the denominator (x - 4)(x + 2) is zero when x = 4 and x = -2. Since the numerator is not zero at these points, we have vertical asymptotes at x = 4 and x = -2. These vertical lines act as barriers that the graph will approach but never cross. Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, we compare the degrees of the polynomials in the numerator and denominator. In our case, both the numerator and denominator are quadratic (degree 2). 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 line indicates the value that the function approaches as x becomes very large or very small. Oblique (slant) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In our function, the degrees are equal, so there is no oblique asymptote. By identifying the vertical and horizontal asymptotes, we gain a framework for understanding the function's long-term behavior and the boundaries within which the graph will exist. Asymptotes are essential guides in sketching an accurate representation of the rational function.

Step 4: Identifying Holes

Holes in the graph of a rational function occur when a factor is present in both the numerator and the denominator. These common factors effectively cancel each other out, but they create a point of discontinuity in the graph. At this point, the function is undefined, resulting in a "hole" in the graph. To identify holes, we look for common factors in the factored form of the function. In our case, f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)), we can see that there are no common factors between the numerator and the denominator. This means that there are no holes in the graph of this function. If there were a common factor, such as (x - a), we would set (x - a) = 0 to find the x-coordinate of the hole, which would be x = a. Then, we would substitute this value of x into the simplified function (after canceling the common factor) to find the y-coordinate of the hole. The hole would then be represented as the point (a, f(a)). However, since our function has no common factors, we can confidently say that there are no holes in its graph. This simplifies our analysis and makes the process of sketching the graph more straightforward. The absence of holes means that we only need to focus on the intercepts and asymptotes to accurately represent the function's behavior. Understanding how to identify holes is crucial for a complete analysis of rational functions, but in this particular case, we can proceed without considering them.

Step 5: Analyzing Intervals and Test Points

Analyzing intervals and selecting test points is a crucial step in graphing rational functions. This process helps us determine the function's behavior between key points such as intercepts and vertical asymptotes. By knowing whether the function is positive or negative in each interval, we can accurately sketch the graph. To begin, we identify the critical points on the x-axis, which are the x-intercepts and the vertical asymptotes. From our previous analysis, we know that the x-intercepts are x = 5 and x = -3, and the vertical asymptotes are x = 4 and x = -2. These points divide the x-axis into several intervals: (-∞, -3), (-3, -2), (-2, 4), (4, 5), and (5, ∞). Next, we choose a test point within each interval. For example, we can choose x = -4 for the interval (-∞, -3), x = -2.5 for (-3, -2), x = 0 for (-2, 4), x = 4.5 for (4, 5), and x = 6 for (5, ∞). We then evaluate the function f(x) at each test point. Using the factored form, f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)), we can determine the sign of f(x) in each interval. For x = -4, f(-4) = 2(-4 - 5)(-4 + 3) / ((-4 - 4)(-4 + 2)) = 2(-9)(-1) / (-8)(-2) = 18 / 16 > 0, so the function is positive in (-∞, -3). For x = -2.5, f(-2.5) = 2(-2.5 - 5)(-2.5 + 3) / ((-2.5 - 4)(-2.5 + 2)) = 2(-7.5)(0.5) / (-6.5)(-0.5) = -7.5 / 3.25 < 0, so the function is negative in (-3, -2). For x = 0, f(0) = 2(0 - 5)(0 + 3) / ((0 - 4)(0 + 2)) = 2(-5)(3) / (-4)(2) = -30 / -8 > 0, so the function is positive in (-2, 4). For x = 4.5, f(4.5) = 2(4.5 - 5)(4.5 + 3) / ((4.5 - 4)(4.5 + 2)) = 2(-0.5)(7.5) / (0.5)(6.5) = -7.5 / 3.25 < 0, so the function is negative in (4, 5). For x = 6, f(6) = 2(6 - 5)(6 + 3) / ((6 - 4)(6 + 2)) = 2(1)(9) / (2)(8) = 18 / 16 > 0, so the function is positive in (5, ∞). This analysis tells us where the graph is above or below the x-axis in each interval. This information, combined with our knowledge of intercepts and asymptotes, allows us to sketch the graph with confidence. By systematically analyzing intervals and test points, we gain a clear understanding of the function's behavior across its domain.

Step 6: Sketching the Graph

Sketching the graph is the final step, where we bring together all the information we've gathered to create a visual representation of the function f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)). We start by plotting the intercepts, which are (5, 0), (-3, 0), and (0, 15/4). These points serve as anchors for our graph. Next, we draw the vertical asymptotes at x = 4 and x = -2 as dashed vertical lines. These lines indicate where the function approaches infinity or negative infinity. Then, we draw the horizontal asymptote at y = 2 as a dashed horizontal line. This line indicates the value the function approaches as x goes to positive or negative infinity. Now, we use the information from our interval analysis. In the interval (-∞, -3), the function is positive, so the graph is above the x-axis. It approaches the horizontal asymptote y = 2 as x goes to negative infinity and approaches the vertical asymptote x = -2 from above. In the interval (-3, -2), the function is negative, so the graph is below the x-axis. It starts from the x-intercept (-3, 0) and approaches the vertical asymptote x = -2 from below. In the interval (-2, 4), the function is positive, so the graph is above the x-axis. It approaches the vertical asymptote x = -2 from above, crosses the y-axis at (0, 15/4), and approaches the vertical asymptote x = 4 from above. In the interval (4, 5), the function is negative, so the graph is below the x-axis. It approaches the vertical asymptote x = 4 from below and reaches the x-intercept (5, 0). In the interval (5, ∞), the function is positive, so the graph is above the x-axis. It starts from the x-intercept (5, 0) and approaches the horizontal asymptote y = 2 as x goes to infinity. By connecting these pieces of information, we can sketch a smooth curve that represents the function. The graph should approach the asymptotes but never cross them, except for the horizontal asymptote, which the graph may cross in the middle. The graph should also pass through the intercepts we plotted earlier. The final sketch gives us a comprehensive visual understanding of the function's behavior. It shows the intercepts, asymptotes, and the overall shape of the graph, providing a complete picture of the rational function.

Conclusion: Mastering Graphing Rational Functions

In conclusion, graphing the rational function f(x) = (2x^2 - 4x - 30) / (x^2 - 2x - 8) involves a systematic approach that combines algebraic manipulation and graphical analysis. We began by factoring the numerator and denominator to identify key features such as intercepts and asymptotes. Factoring the function into f(x) = 2(x - 5)(x + 3) / ((x - 4)(x + 2)) allowed us to easily find the zeros and potential vertical asymptotes. Identifying intercepts, both x and y, provided crucial points for anchoring the graph. The x-intercepts were found by setting the numerator to zero, giving us (5, 0) and (-3, 0), while the y-intercept was found by setting x = 0, resulting in (0, 15/4). Determining asymptotes was a critical step in understanding the function's behavior as x approaches infinity and at points of discontinuity. Vertical asymptotes were located at x = 4 and x = -2, where the denominator is zero. The horizontal asymptote was found to be y = 2, as the degrees of the numerator and denominator were equal, and we took the ratio of the leading coefficients. We also checked for holes by looking for common factors in the numerator and denominator, but found none in this case. Analyzing intervals and test points helped us determine the sign of the function in different regions of the x-axis. This allowed us to sketch the graph accurately, knowing whether the function was above or below the x-axis in each interval. Finally, we sketched the graph, connecting the intercepts and asymptotes while adhering to the sign analysis. The resulting graph provides a comprehensive visual representation of the function's behavior. Mastering these steps equips you with the skills to graph a wide range of rational functions. By understanding the interplay between algebraic properties and graphical representations, you can gain a deeper appreciation for the beauty and power of mathematics. Graphing rational functions is not just a mechanical process; it's an art that combines precision with insight, allowing you to visualize complex mathematical relationships.