Rational Function Analysis: Finding Asymptotes

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Rational Function Analysis: Finding Asymptotes

Hey guys! Let's dive into the fascinating world of rational functions. Today, we're going to break down the rational function r(x) = (2x² - 3x + 7) / (x - 4). Our mission? To figure out which statement about this function is correct. We'll be exploring asymptotes – those invisible lines that the graph of a function gets closer and closer to, but never quite touches. This is like the North Star for our function, guiding its behavior as x goes to positive or negative infinity, or as x approaches a specific value where the function is undefined. Knowing how to identify and classify these asymptotes is a key skill in understanding the overall behavior of the function. We will focus on two specific types: horizontal and slant asymptotes. Horizontal asymptotes tell us what y-value the function approaches as x gets incredibly large (positive or negative). Slant asymptotes, also known as oblique asymptotes, appear when the degree of the numerator is exactly one greater than the degree of the denominator. They are linear functions that the graph tends towards as x approaches infinity. Ready to explore? Let's get started, and I'll walk you through each step to determine the correct statement about our function r.

Decoding the Statements and Our Strategy

Alright, let's take a look at the statements we need to evaluate. We're given three options, each making a claim about the asymptotes of r(x):

A. The graph of r has a horizontal asymptote of y = 2. B. The graph of r has a slant asymptote of y = 2x - 3. C. The graph

To figure out the truth, we need a solid strategy, right? Here's how we'll do it. First, we'll quickly dismiss option A by understanding the rules governing horizontal asymptotes in rational functions. Then, because the degree of the numerator (2) is exactly one more than the degree of the denominator (1), we will use polynomial long division to find the slant asymptote. This method is like a mathematical magic trick that reveals the hidden linear function that our original rational function is trying to get close to. Remember, a slant asymptote only exists if the degree of the numerator exceeds the degree of the denominator by 1. Once we have the result of the division, we can compare it to the statements and reveal our answer! Ready to get started?

Analyzing Horizontal Asymptotes

Let's tackle the first statement, which suggests a horizontal asymptote at y = 2. Horizontal asymptotes for rational functions depend on the degrees of the numerator and denominator. When the degree of the numerator is greater than the degree of the denominator, there's no horizontal asymptote. Instead, as x approaches positive or negative infinity, the function either goes to infinity or negative infinity, or there's a slant asymptote. If the degrees are equal, the horizontal asymptote is at the ratio of the leading coefficients. Now, in our function, r(x) = (2x² - 3x + 7) / (x - 4), the degree of the numerator (2) is greater than the degree of the denominator (1). Because of this, we know there is no horizontal asymptote, which means option A is incorrect. The function's behavior at the extremes of x is not governed by a constant y-value. It's time to shift our focus to the possibility of a slant asymptote and delve deeper into the function's behavior. We know that as x grows, the function doesn't settle at y = 2, so let's move on!

Calculating the Slant Asymptote via Polynomial Long Division

Now, for the main event! Because our function has a numerator of degree 2 and a denominator of degree 1, we can have a slant asymptote. To find this slant asymptote, we will employ polynomial long division. This method is very similar to the long division of numbers, but we work with polynomials. Let's get into the step-by-step process of polynomial long division.

  1. Set up the division: Write the numerator (2x² - 3x + 7) inside the division symbol and the denominator (x - 4) outside.
  2. Divide the leading terms: Divide the leading term of the numerator (2x²) by the leading term of the denominator (x). This gives us 2x. Write this above the division symbol.
  3. Multiply: Multiply the quotient (2x) by the entire denominator (x - 4). This results in 2x² - 8x. Write this below the numerator.
  4. Subtract: Subtract the result from the numerator: (2x² - 3x) - (2x² - 8x) = 5x. Bring down the +7 from the original numerator to get 5x + 7.
  5. Repeat: Now, divide the new leading term (5x) by the leading term of the denominator (x). This gives us 5. Write this above the division symbol, next to the 2x.
  6. Multiply again: Multiply the new quotient (5) by the entire denominator (x - 4). This results in 5x - 20. Write this below 5x + 7.
  7. Subtract a second time: Subtract the result from the previous line: (5x + 7) - (5x - 20) = 27.

After we perform the polynomial long division, we get a quotient of 2x + 5 and a remainder of 27. The slant asymptote is given by the quotient, which means the slant asymptote for r(x) is y = 2x + 5. The remainder (27) doesn't affect the slant asymptote's equation.

Choosing the correct statement and why

So, after a good amount of work and analysis, we know our slant asymptote is y = 2x + 5. Looking back at our options:

A. The graph of r has a horizontal asymptote of y = 2. (Incorrect) B. The graph of r has a slant asymptote of y = 2x - 3. (Incorrect) C. The graph

Since we determined a slant asymptote of y = 2x + 5 and we know that Option A is not correct because a horizontal asymptote does not exist, we are left to select the correct answer. The correct statement is not mentioned. However, since we've done our math, we can now conclude the correct statement. The closest statement to the correct answer is option B. However, based on our calculations, the correct slant asymptote is y = 2x + 5, and not y = 2x - 3.

Therefore, considering our analysis, the most accurate conclusion is that the graph of r has a slant asymptote of y = 2x + 5. The original options given didn't have the correct answer, but we've successfully derived the correct asymptote for the given function. Excellent work, everyone! We've successfully navigated the world of rational functions, horizontal asymptotes, and slant asymptotes using polynomial long division. Knowing how to analyze these concepts really helps us understand the behavior of functions. Keep practicing, and you'll become a pro at these problems!