Master Graphing H(x)=(x^3-27)/(x^2-100) Step-by-Step

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Master Graphing $H(x)=\frac{x^3-27}{x^2-100}$ Step-by-Step\n\nHey there, math enthusiasts! Ever looked at a rational function like $H(x)=\frac{x^3-27}{x^2-100}$ and thought, "Whoa, where do I even begin to graph this beast?" Well, you're in the right place! Graphing rational functions might seem *intimidating* at first glance, but I promise you, with a solid, step-by-step approach, it becomes much more manageable – even *enjoyable*! Think of it like a puzzle; each piece you solve brings you closer to seeing the full, beautiful picture. This article is all about giving you the tools, tips, and tricks to confidently graph *any* rational function, specifically walking through the nuances of our example, $H(x)=\frac{x^3-27}{x^2-100}$. We're going to break down every single aspect, from finding those tricky asymptotes to nailing down the intercepts, making sure you not only *know* the steps but *understand* the *why* behind them. So, grab your pencils, some graph paper (or your favorite graphing software!), and let's dive into mastering this essential skill that’s super valuable in algebra, calculus, and beyond.\n\nUnderstanding how to graph these functions isn't just about passing a test; it's about developing a deeper intuition for how mathematical expressions translate into visual representations. Rational functions, with their unique behaviors like holes, vertical asymptotes, and interesting end behaviors (sometimes horizontal, sometimes slant!), offer a fantastic playground for exploring advanced graphing concepts. We'll make sure to use a friendly, conversational tone, just like we're chatting over coffee, to make this learning experience as *smooth* and *stress-free* as possible. By the end of this guide, you'll be able to look at a rational function and practically *see* its graph forming in your mind, which is a pretty cool superpower, if you ask me. Let's get started on this exciting graphing adventure, shall we?\n\n## Why Graphing Rational Functions Matters (And a Heads-Up About H(x))\n\nBefore we roll up our sleeves and get into the nitty-gritty of *graphing rational functions*, let's take a quick moment to appreciate *why* this skill is so crucial in mathematics and various real-world applications. These functions pop up everywhere, from modeling population growth and concentrations of medicines in the bloodstream to analyzing electrical circuits and economic trends. So, understanding their behavior isn't just an academic exercise; it's a practical tool for interpreting data and predicting outcomes. When you graph a rational function, you're essentially creating a visual story of how one quantity changes in relation to another, especially when dealing with scenarios where there are limitations or specific breaking points. For instance, think about a scenario where the cost per item decreases as you produce more, but only up to a certain point – that kind of relationship can often be described and visualized with a rational function. Knowing where the function goes towards infinity (asymptotes) or where it crosses the axes (intercepts) gives us critical insights into the system being modeled.\n\nNow, let's turn our attention to our specific challenge today: $H(x)=\frac{x^3-27}{x^2-100}$. This particular function is a fantastic example because it presents a few *really interesting challenges* that we’ll tackle head-on. It's not your basic rational function; it's got a higher degree in the numerator, which immediately signals something *different* about its end behavior – hinting at a slant asymptote rather than a horizontal one. Plus, the numerator is a difference of cubes, and the denominator is a difference of squares, meaning factoring will be key to unlocking all its secrets. We'll discover its specific vertical asymptotes and, importantly, its x and y-intercepts. By working through this example diligently, you'll not only graph $H(x)$ but also build a robust framework for approaching *any* complex rational function. So, prepare to see how factoring, finding asymptotes, and pinpointing intercepts all come together to paint a clear, accurate picture of this function's unique journey on the coordinate plane. This isn't just about memorizing steps; it's about truly *understanding* the mathematical landscape these functions create. Let's make this journey both educational and genuinely enjoyable, shall we?\n\n## Step-by-Step Guide to Graphing Rational Functions\n\nAlright, folks, it's time to get down to business! Graphing a rational function can feel like navigating a complex maze, but with a structured approach, you'll find it's totally doable. We're going to break it down into several clear, easy-to-follow steps, applying each one directly to our function, $H(x)=\frac{x^3-27}{x^2-100}$. Each step builds upon the last, so pay close attention, and don't worry, we'll explain *everything* in a friendly, no-jargon way. Let's dive in and conquer this graph!\n\n### Step 1: Factor Everything! (Numerator and Denominator)\n\nOur very first and perhaps *most crucial* step in understanding any rational function, especially one as intriguing as $H(x)=\frac{x^3-27}{x^2-100}$, is to **factor both the numerator and the denominator completely**. Why is this so important, you ask? Well, factoring is like shining a spotlight on the hidden characteristics of your function. It helps us identify potential holes, vertical asymptotes, and x-intercepts with much greater clarity. Without proper factoring, you're essentially trying to solve a puzzle with half the pieces missing! It’s the foundational step that unlocks all subsequent analysis, so let's take our time and do it right.\n\nFor our numerator, we have $x^3-27$. Does that expression ring a bell? It should! This is a classic example of a ***difference of cubes***. Remember the formula for factoring a difference of cubes, $a^3 - b^3 = (a-b)(a^2+ab+b^2)$? Here, $a=x$ and $b=3$ (since $3^3=27$). So, factoring $x^3-27$ gives us: $(x-3)(x^2+3x+9)$. Now, a quick check on the quadratic factor, $x^2+3x+9$: can it be factored further? We can use the discriminant, $b^2-4ac$. In this case, $3^2-4(1)(9) = 9-36 = -27$. Since the discriminant is negative, this quadratic factor has no real roots, meaning it cannot be factored into simpler linear terms with real coefficients. This is important because it tells us that $x-3$ is the *only* real root of the numerator, which will directly lead us to our x-intercept.\n\nNext, let’s tackle the denominator: $x^2-100$. This one is perhaps even more recognizable than the numerator's factoring pattern! It's a classic ***difference of squares***. The formula for factoring a difference of squares is $a^2 - b^2 = (a-b)(a+b)$. Here, $a=x$ and $b=10$ (since $10^2=100$). So, factoring $x^2-100$ yields: $(x-10)(x+10)$. Both of these are linear factors and cannot be simplified further. They are also crucial because they will tell us where our function is undefined, leading to vertical asymptotes.\n\nSo, after factoring both parts, our function $H(x)$ now looks much more revealing: $H(x)=\frac{(x-3)(x^2+3x+9)}{(x-10)(x+10)}$. See how much clearer that is? This factored form is our roadmap. It helps us immediately spot the terms that define domain restrictions and potential intercepts. Importantly, we also check for any common factors between the numerator and denominator. If there were any common factors, they would indicate a "hole" in the graph, where the function is undefined at a single point, rather than approaching infinity. In our case, $(x-3)$, $(x^2+3x+9)$, $(x-10)$, and $(x+10)$ are all unique. There are no common factors to cancel out, which means *no holes* in this graph! That simplifies things a bit, but it also means every factor in the denominator will lead to a vertical asymptote. This comprehensive factoring step is truly the gateway to understanding the full behavior of $H(x)$, so always make sure you nail it down completely and confidently before moving on.\n\n### Step 2: Uncover Those Domain Restrictions, Holes, and Vertical Asymptotes\n\nNow that we have our function in its beautifully factored form, $H(x)=\frac{(x-3)(x^2+3x+9)}{(x-10)(x+10)}$, it's time to dig into some of the most dramatic features of a rational function's graph: its **domain restrictions**, **holes**, and **vertical asymptotes**. These elements define where the function *cannot* exist or where it behaves in an extreme manner, often shooting off towards positive or negative infinity. Ignoring these aspects is like trying to draw a landscape without knowing where the cliffs and canyons are – you're bound to fall off! So, let's meticulously identify them for $H(x)$.\n\nFirst, let's talk about **domain restrictions**. The domain of any rational function is all real numbers *except* for the values of $x$ that make the denominator equal to zero. Why? Because division by zero is mathematically undefined. It creates a mathematical singularity, a point where the function simply doesn't exist. Looking at our factored denominator, $(x-10)(x+10)$, we can see that it will be zero if $x-10=0$ or if $x+10=0$. This means our domain restrictions are $x=10$ and $x=-10$. These are the values of $x$ where our graph will have some kind of break or discontinuity. It's *critical* to identify these right away because they dictate the boundaries of our function's existence.\n\nNext up, **holes**. A hole in the graph occurs when a factor in the numerator *exactly cancels out* a factor in the denominator. If a factor $(x-c)$ appears in both the numerator and the denominator, it means that at $x=c$, the function is undefined, but the graph doesn't shoot off to infinity; instead, there's just a "missing point" – a hole. We already performed this check in Step 1, and we found that there are no common factors between $(x-3)(x^2+3x+9)$ and $(x-10)(x+10)$. Since nothing cancels, our function $H(x)$ ***has no holes***. This simplifies our graphing process a little, as we don't have to worry about finding the exact y-coordinate of a missing point. If there *were* a hole, you'd find its y-coordinate by plugging the x-value of the hole into the *simplified* function (after cancellation).\n\nFinally, we arrive at **vertical asymptotes (VAs)**. These are vertical lines that the graph approaches but never actually touches. They occur at the x-values where the denominator is zero *after* all common factors (which would cause holes) have been cancelled. Since we determined there are no holes, every factor remaining in the denominator directly corresponds to a vertical asymptote. For $H(x)$, our denominator factors are $(x-10)$ and $(x+10)$. Setting each to zero gives us: $x-10=0 \Rightarrow x=10$ and $x+10=0 \Rightarrow x=-10$. Therefore, $H(x)$ has **two vertical asymptotes**: one at $x=10$ and another at $x=-10$. These are invisible fences that guide the extreme behavior of our graph, showing us where the function values will shoot up or down infinitely. Understanding where these asymptotes lie is paramount, as they divide our coordinate plane into distinct regions, and the graph's behavior within each region needs to be analyzed separately. Think of them as the pillars around which the graph dramatically bends and turns; knowing their exact locations ($x=10$ and $x=-10$) is a major win for sketching an accurate graph.\n\n### Step 3: Pinpoint the Intercepts (x and y-intercepts)\n\nAfter nailing down the factoring and identifying those critical asymptotes, our next mission is to find the **intercepts**. These are the points where our graph crosses or touches the x-axis and the y-axis. Think of them as the anchor points of your graph, providing concrete locations that are easy to plot and give us vital information about where the function starts and where it crosses the zero line. For our function $H(x)=\frac{x^3-27}{x^2-100}$, let’s find both types of intercepts.\n\nFirst, let's find the **y-intercept**. The y-intercept is the point where the graph crosses the y-axis. This happens when $x=0$. To find it, we simply plug $x=0$ into our original function: \n$H(0) = \frac{(0)^3-27}{(0)^2-100} = \frac{-27}{-100} = \frac{27}{100}$.\nSo, our y-intercept is at $(0, \frac{27}{100})$. This means when $x$ is zero, $y$ is a small positive value, about $0.27$. This is a specific point that we can plot on our graph, and it tells us exactly where the function intersects the vertical y-axis. It's typically one of the easiest points to find, and it provides a great starting reference for our sketch.\n\nNow, for the **x-intercepts**. These are the points where the graph crosses or touches the x-axis. This occurs when $H(x)=0$. For a rational function, $H(x)=\frac{P(x)}{Q(x)}$, the function is zero only when its numerator, $P(x)$, is zero (and the denominator, $Q(x)$, is not zero at that same point, which would create a hole). So, we set our numerator equal to zero: \n$x^3-27=0$\n$x^3=27$\nTo solve for $x$, we take the cube root of both sides:\n$x=\sqrt[3]{27}$\n$x=3$\nSo, our only real x-intercept is at $(3,0)$. This is another critical anchor point for our graph. It shows where the function value is exactly zero, which is often a point of interest in any real-world application of such functions. Remember how we factored the numerator as $(x-3)(x^2+3x+9)$? The factor $x-3=0$ directly gave us $x=3$, confirming this result. The quadratic factor $x^2+3x+9$ does not have any real roots, as its discriminant was negative, so it doesn't contribute any other real x-intercepts.\n\nLet's address the statement from the prompt directly: