Graphing Rational Functions: A Step-by-Step Guide

Hey guys! Let's dive into the world of graphing rational functions. Specifically, we're gonna tackle the function $H(x)=\frac{x^2-64}{x^4-81}$. Graphing rational functions can seem a bit intimidating at first, but trust me, breaking it down step-by-step makes it totally manageable. We'll explore finding asymptotes, intercepts, and the overall behavior of the graph. Think of it like a treasure hunt, where we're looking for clues to understand the function's hidden shape. This guide will walk you through each stage, making sure you grasp the key concepts.

Understanding the Basics: What is a Rational Function?

So, what exactly is a rational function? Well, it's simply a function that can be written as the ratio of two polynomials. In other words, it's a fraction where both the numerator and the denominator are polynomials. Our example, $H(x)=\frac{x^2-64}{x^4-81}$, fits this description perfectly. The numerator, $x^2 - 64$, is a polynomial, and the denominator, $x^4 - 81$, is also a polynomial. This means $H(x)$ is a rational function. The coolest thing is that rational functions can have some really interesting behaviors, like asymptotes, which are lines that the graph gets closer and closer to but never actually touches. This is what makes them super fun to analyze. Before jumping into our specific example, let's just quickly refresh our memory on some key concepts. We’ll focus on the core ideas you’ll need to understand rational functions. We'll talk about the two main parts of rational functions which are the numerator and the denominator. The numerator tells us about the zeros of the function (where it crosses the x-axis), and the denominator tells us about the vertical asymptotes (where the function becomes undefined). We’ll also touch on horizontal asymptotes, which describe what happens to the function as x goes to positive or negative infinity. Understanding these elements helps us piece together the full picture of a rational function's graph.

Step 1: Finding the Domain and Simplifying the Function

Alright, first things first! Before we start graphing, we need to figure out the domain of our function. The domain is simply the set of all possible x-values that we can plug into the function without causing any mathematical disasters, like division by zero. For rational functions, the domain is all real numbers except for the values that make the denominator equal to zero. So, to find these values, we need to solve the equation $x^4 - 81 = 0$. Let's do it step by step. We can factor $x^4 - 81$ as a difference of squares: $(x^2 - 9)(x^2 + 9) = 0$. Then we can factor $(x^2 - 9)$ as $(x - 3)(x + 3)$. So we have $(x - 3)(x + 3)(x^2 + 9) = 0$. Setting each factor equal to zero gives us $x = 3$, $x = -3$, and $x^2 = -9$. The solutions of this are $x=3$, $x=-3$. The $x^2=-9$ part doesn't give us any real solutions. So, the domain of $H(x)$ is all real numbers except $x = 3$ and $x = -3$. These are the points where our function will be undefined, and where we'll find those all-important vertical asymptotes. Now, let's simplify our function by factoring both the numerator and the denominator. The numerator, $x^2 - 64$, factors into $(x - 8)(x + 8)$. The denominator, as we saw before, factors into $(x - 3)(x + 3)(x^2 + 9)$. This gives us $H(x) = \frac{(x - 8)(x + 8)}{(x - 3)(x + 3)(x^2 + 9)}$. This simplified form helps us identify any holes in the graph (where the function is undefined but can be “canceled out”) and makes it easier to find intercepts and asymptotes.

Step 2: Identifying Vertical Asymptotes

Vertical asymptotes are vertical lines that the graph of the function approaches but never actually touches. They occur at the x-values where the denominator of the function is equal to zero, but the numerator is not. After our simplification, we know that our function is $H(x) = \frac{(x - 8)(x + 8)}{(x - 3)(x + 3)(x^2 + 9)}$. Looking at the denominator, $(x - 3)(x + 3)(x^2 + 9)$, we can see that it equals zero when $x = 3$ and $x = -3$. Since neither of these values make the numerator zero, we have vertical asymptotes at $x = 3$ and $x = -3$. To really get a feel for what's happening near these asymptotes, you can test values close to 3 and -3. For example, if you plug in values like 2.9, 2.99, and 3.1, 3.01 into the function, you'll see that the function's values either shoot up towards positive infinity or plummet towards negative infinity as x approaches 3 and -3. So, that's it! The vertical asymptotes are at $x=3$ and $x=-3$. Remember, vertical asymptotes are like invisible barriers that the graph tries to get as close as possible to without crossing them. This information helps us understand the behavior of the function near these specific x-values.

Step 3: Finding Horizontal Asymptotes

Next up, let's find the horizontal asymptotes. Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find them, we need to compare the degrees of the numerator and the denominator. The degree of a polynomial is the highest power of the variable. In our function $H(x) = \frac{x^2-64}{x^4-81}$, the degree of the numerator is 2 (because of the $x^2$) and the degree of the denominator is 4 (because of the $x^4$). When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is always at $y = 0$. So, in our case, the horizontal asymptote is at $y = 0$. This means that as x gets very large (positive or negative), the graph of the function gets closer and closer to the x-axis. It's super important to remember that a graph can cross a horizontal asymptote, but it cannot cross a vertical asymptote. Horizontal asymptotes are like the long-term trends of the function. Now, you know the horizontal asymptote, and now you are ready to move on!

Step 4: Determining the x- and y-intercepts

Now let's find the x- and y-intercepts of our function. These are the points where the graph crosses the x-axis (x-intercepts) and the y-axis (y-intercept). To find the x-intercepts, we set $H(x) = 0$ and solve for x. So, we have $\frac{x^2-64}{x^4-81} = 0$. This means that $x^2 - 64 = 0$, which gives us $x = 8$ and $x = -8$. Therefore, the x-intercepts are at the points (8, 0) and (-8, 0). That’s where the graph crosses the x-axis. To find the y-intercept, we set $x = 0$ and solve for $H(x)$. So we have $H(0) = \frac{0^2 - 64}{0^4 - 81} = \frac{-64}{-81} = \frac{64}{81}$. Therefore, the y-intercept is at the point (0, 64/81). This tells us where the graph crosses the y-axis. Intercepts are crucial because they give us specific points on the graph, helping us get a more detailed picture of the function's shape and location in the coordinate plane.

Step 5: Sketching the Graph

Alright, time to put it all together and sketch the graph! We've already done the heavy lifting, so this part is easier than it looks. We know the following:

• Vertical Asymptotes: $x = 3$ and $x = -3$
• Horizontal Asymptote: $y = 0$
• x-intercepts: (8, 0) and (-8, 0)
• y-intercept: (0, 64/81)

1. Draw the Asymptotes: Start by drawing the vertical asymptotes as dashed lines at $x = 3$ and $x = -3$. Then, draw the horizontal asymptote as a dashed line at $y = 0$ (the x-axis). These lines will act as guides for our graph. Remember the graph should never touch the vertical asymptotes. The horizontal asymptote describes the end behavior. Meaning what is the graph doing as x approaches positive and negative infinity.

2. Plot the Intercepts: Mark the points (8, 0), (-8, 0), and (0, 64/81) on your graph. These are the points where the graph will cross the x-axis and y-axis, respectively.

3. Analyze the Behavior: Now, think about how the graph behaves in different regions. Between the vertical asymptotes and to the left and right of them. In each section, the graph will either approach positive or negative infinity or approach the horizontal asymptote. Since you know the x intercepts, you know where it crosses the x axis, and the y intercepts, you know where it crosses the y-axis. Because the vertical asymptote is at x=3 and x=-3, the function is undefined at these values. If you use a graphing calculator or software, you'll be able to confirm this. You can also test some values by picking a number between your vertical asymptote to see if the graph is above or below the x axis. If you pick a value such as x=0, you already know the y-intercept (0, 64/81). Then you can sketch the graph. It’s important to show the behavior of the graph near the asymptotes. If you are sketching by hand, be sure to label the asymptotes and the intercepts. This helps to make a really easy graph. You can also use graphing software such as Desmos, or a graphing calculator such as a TI-84 to confirm your work!

And there you have it! A complete guide to graphing rational functions. This process can be applied to other functions. Remember, the key is to break down the function into smaller, manageable steps. You've got this!