Asymptotes Of Y=(2x^2+2x+3)/(4x^2-4x)

Dec 11, 2025 by Admin 38 views

Hey math whizzes! Today, we're diving deep into the fascinating world of rational functions and how to find their asymptotes. Specifically, we're going to tackle the function $y= rac{2 x^2+2 x+3}{4 x^2-4 x}$ . Understanding asymptotes is super crucial because they tell us a lot about the behavior of a graph as it heads towards infinity or zero. Think of them as invisible lines that the graph gets closer and closer to but never actually touches. We'll break down how to find both horizontal asymptotes and vertical asymptotes for this function, leaving no stone unturned. So grab your calculators, maybe a snack, and let's get this mathematical party started!

Understanding Horizontal Asymptotes

Alright guys, let's kick things off by figuring out the horizontal asymptote for our function $y= rac{2 x^2+2 x+3}{4 x^2-4 x}$ . The rule of thumb here is to look at the degrees of the polynomial in the numerator and the denominator. In our case, the degree of the numerator (the highest power of $x$ ) is 2, and the degree of the denominator is also 2. When the degrees are the same, like they are here, the horizontal asymptote is simply the ratio of the leading coefficients. The leading coefficient in the numerator is 2, and in the denominator, it's 4. So, the horizontal asymptote is $y = rac{2}{4}$ , which simplifies to $y = rac{1}{2}$ . This means that as $x$ gets really, really big (both positive and negative), the $y$ -values of our function will get super close to $rac{1}{2}$ . It's like the function is giving up on doing anything wild and just settling down towards this value. It's important to remember that a function can cross its horizontal asymptote, but it won't do so infinitely often; it just shows the end behavior. So, for our function, we have a confirmed horizontal asymptote at $y = rac{1}{2}$ . This is a key piece of information when sketching or analyzing the graph. We're halfway there, folks!

Uncovering Vertical Asymptotes

Now, let's shift our focus to finding the vertical asymptotes of $y= rac{2 x^2+2 x+3}{4 x^2-4 x}$ . Vertical asymptotes occur where the denominator of a rational function equals zero, provided that the numerator does not also equal zero at that same $x$ -value. If both are zero, you might have a hole in the graph instead of an asymptote. So, the first step is to set the denominator equal to zero: $4 x^2 - 4 x = 0$ . We can factor out a $4x$ from this expression: $4x(x - 1) = 0$ . This equation gives us two possible values for $x$ : $4x = 0 ightarrow x = 0$ and $x - 1 = 0 ightarrow x = 1$ . Now, we need to check if the numerator, $2 x^2 + 2 x + 3$ , is zero at these $x$ -values. Let's test $x=0$ : $2(0)^2 + 2(0) + 3 = 3$ . Since the numerator is not zero at $x=0$ , we have a vertical asymptote at $x=0$ . Phew! Next, let's test $x=1$ : $2(1)^2 + 2(1) + 3 = 2 + 2 + 3 = 7$ . Again, the numerator is not zero at $x=1$ . This means we also have a vertical asymptote at $x=1$ . So, our function has vertical asymptotes at both $x = 0$ and $x = 1$ . These lines represent $x$ -values where the function's output shoots off towards positive or negative infinity, indicating a discontinuity. It's like the graph is being split into different sections by these lines. Knowing these helps us understand where the function is undefined and behaves erratically. We've nailed down all the asymptotes for this function, guys!

Putting It All Together: Asymptote Summary

So, after all that mathematical detective work, let's recap what we've found for the function $y= rac{2 x^2+2 x+3}{4 x^2-4 x}$ . We discovered that there is a horizontal asymptote at $y = rac{1}{2}$ . Remember, this tells us about the graph's behavior as $x$ approaches positive or negative infinity. The graph will get closer and closer to the line $y = rac{1}{2}$ but likely won't touch it in the long run. We also identified two vertical asymptotes: one at $x = 0$ and another at $x = 1$ . These are the $x$ -values where the function's value becomes infinitely large (positive or negative), essentially breaking the graph into distinct pieces. This means the function is undefined at these points. When you're sketching this graph, these asymptotes act as guides, showing you the boundaries and the overall shape the function will take. The presence of these asymptotes helps us understand the function's domain and range limitations as well. The domain excludes $x=0$ and $x=1$ , and the graph's behavior around these lines is crucial. The horizontal asymptote indicates that the range will not include values far from $rac{1}{2}$ as $x$ becomes large. So, to summarize, we have one horizontal asymptote at $y = rac{1}{2}$ and vertical asymptotes at $x = 0$ and $x = 1$ . This gives us a really clear picture of the function's graphical structure. It's like having a map for the function's journey across the coordinate plane!

Analyzing the Options Provided

Now, let's look back at the multiple-choice options you guys were presented with and see which one perfectly matches our findings for $y= rac{2 x^2+2 x+3}{4 x^2-4 x}$ .

(A) a horizontal asymptote at $y= rac{1}{2}$ , but no vertical asymptote. This option is partially correct because it identifies the correct horizontal asymptote. However, we found two vertical asymptotes, so this option is incorrect.

(B) no horizontal asymptotes, but vertical asymptotes at $x=0$ and $x=1$ . This option correctly identifies the vertical asymptotes, which is awesome! But, we definitely found a horizontal asymptote, so this option is also incorrect.

(C) a horizontal asymptote at $y= rac{1}{2}$ , and vertical asymptotes at $x=0$ and $x=1$ . Bingo! This option perfectly aligns with our calculations. We found a horizontal asymptote at $y= rac{1}{2}$ , and we also found vertical asymptotes at $x=0$ and $x=1$ . This is the correct description of the asymptotes for the given function. It's always satisfying when the math checks out, right?

Why Asymptotes Matter in Mathematics

So, why do we even bother with asymptotes, you might ask? Well, guys, they're not just some abstract concept to confuse students; they're fundamental to understanding the behavior of functions, especially rational functions. Think about it: they show us where a function is heading in the long run (horizontal asymptotes) and where it might