ind the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the ot exist, enter DNE.) 2x²+3
ind the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the ot exist, enter DNE.) 2x²+3
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Finding Horizontal and Vertical Asymptotes**
To determine the horizontal and vertical asymptotes of the given function, follow these steps.
**Function:**
\[ y = \frac{2x^2 + 3}{7x^2 + 48x - 7} \]
**Steps to Find Vertical Asymptotes:**
Vertical asymptotes occur where the denominator is equal to zero (and the numerator is not zero at the same point). Set the denominator equal to zero and solve for \( x \):
\[ 7x^2 + 48x - 7 = 0 \]
**Answer:**
\[ x = \_\_\_ \]
**Steps to Find Horizontal Asymptotes:**
Horizontal asymptotes depend on the degrees of the polynomial in the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Since both the numerator and the denominator have degree 2:
\[ \text{Horizontal asymptote: } y = \frac{2}{7} \]
**Answer:**
\[ y = \frac{2}{7} \]
Enter this data in the provided fields on the educational website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e135d5c-f867-4afd-9f0d-b505f4f19664%2F6c617b39-61d8-4388-b770-037f704d8850%2F5ohmsfh_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Horizontal and Vertical Asymptotes**
To determine the horizontal and vertical asymptotes of the given function, follow these steps.
**Function:**
\[ y = \frac{2x^2 + 3}{7x^2 + 48x - 7} \]
**Steps to Find Vertical Asymptotes:**
Vertical asymptotes occur where the denominator is equal to zero (and the numerator is not zero at the same point). Set the denominator equal to zero and solve for \( x \):
\[ 7x^2 + 48x - 7 = 0 \]
**Answer:**
\[ x = \_\_\_ \]
**Steps to Find Horizontal Asymptotes:**
Horizontal asymptotes depend on the degrees of the polynomial in the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Since both the numerator and the denominator have degree 2:
\[ \text{Horizontal asymptote: } y = \frac{2}{7} \]
**Answer:**
\[ y = \frac{2}{7} \]
Enter this data in the provided fields on the educational website.
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