Easy 7-Step Guide: Graphing $f(x)=(x+2)/(x^2+x-20)$

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Easy 7-Step Guide: Graphing $f(x)=\frac{x+2}{x^2+x-20}$\n\nHey there, math explorers! Are you ready to **tackle the awesome world of rational functions** and learn how to graph them like a pro? You've landed in the right place, because today, we're going to break down the process of *graphing rational functions* into seven super easy-to-follow steps. We'll be using a specific example, $f(x)=\frac{x+2}{x^2+x-20}$, to make sure every concept is crystal clear. Trust me, once you get the hang of these steps, graphing any rational function will feel much less intimidating. We're going to cover everything from finding asymptotes and intercepts to plotting points and sketching the final masterpiece. So, grab your pencil, some paper, and let's get ready to make some incredible graphs! This guide is designed to be friendly, engaging, and *packed with value*, ensuring you understand not just *what* to do, but *why* you're doing it. By the end of this article, you'll have a solid strategy for graphing complex functions and impressing everyone with your math skills!\n\n## Understanding Rational Functions: The Core Concepts\n\nBefore we dive into the specific steps for *graphing rational functions*, let's chat a bit about what these mathematical creatures actually are. Simply put, a **rational function** is any function that can be written as a fraction where both the numerator and the denominator are polynomials. Think of it like this: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, and here's the crucial part, $Q(x)$ cannot be zero! Why is that so important, you ask? Because dividing by zero is a big no-no in mathematics; it creates undefined points, which often show up as *vertical asymptotes* or *holes* in our graph. These aren't just abstract ideas, guys; they are key features that shape the entire visual representation of our function. The specific rational function we're focusing on today, $f(x)=\frac{x+2}{x^2+x-20}$, is a fantastic example to illustrate all these concepts. Notice how the numerator, $x+2$, is a simple linear polynomial, while the denominator, $x^2+x-20$, is a quadratic. The degree of these polynomials (the highest power of $x$) plays a massive role in determining the behavior of the graph, especially when we start looking for horizontal or slant asymptotes. Understanding these foundational elements – the definition, the role of polynomials, and the critical importance of the denominator not equaling zero – is your first step towards truly mastering rational function graphing. It's not just about memorizing steps; it's about grasping the underlying logic that makes these functions behave the way they do. With this solid base, every subsequent step will make a lot more sense, and you'll be able to graph rational functions with confidence and precision.\n\n## Step 1: Simplify Our Rational Function – $f(x)=\frac{x+2}{x^2+x-20}$\n\nThe very first thing you want to do, guys, when you're faced with *graphing rational functions*, is to **simplify the function** as much as possible. This step is absolutely crucial because it helps us identify any *holes* in the graph versus *vertical asymptotes*, which behave very differently. For our specific function, $f(x)=\frac{x+2}{x^2+x-20}$, simplification means factoring both the numerator and the denominator. The numerator, $x+2$, is already as simple as it gets – it's a linear factor. So, our main task here is to factor the quadratic denominator, $x^2+x-20$. To factor a quadratic expression like this, we're looking for two numbers that multiply to give us -20 and add up to give us 1 (the coefficient of the $x$ term). After a bit of thought, or maybe a quick mental run-through, you'll find that 5 and -4 fit the bill perfectly: $5 \times (-4) = -20$ and $5 + (-4) = 1$. Awesome! This means we can rewrite our denominator as $(x+5)(x-4)$. So, our rational function now looks like this in its factored form: $f(x)=\frac{x+2}{(x+5)(x-4)}$. Now, here's where the magic of simplification comes in: we check for *common factors* between the numerator and the denominator. If there were a common factor, say $(x+2)$ was also in the denominator, then that would indicate a hole in the graph at $x=-2$. However, in our case, $(x+2)$ is *not* a factor in the denominator. This is actually a good thing for our first graph, as it means we won't have any holes to worry about right now. This *simplification step* is fundamental for accurately identifying the true nature of the discontinuities in your graph, differentiating between a removable discontinuity (a hole) and an essential discontinuity (a vertical asymptote). Don't ever skip this first, vital step when you're learning how to graph rational functions, as it sets the stage for everything that follows!\n\n## Step 2: Discovering the Domain and Vertical Asymptotes\n\nAlright, mathletes, once our function is simplified, the next critical step in *graphing rational functions* is to **determine the domain** and identify any **vertical asymptotes**. These are essentially the