For q (x) = -5x+5x-4 x² +2 (a) Identify the horizontal asymptotes (if any). (b) If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s). Separate multiple equations of asymptotes with commas as necessary. Select "None" if applicable.
For q (x) = -5x+5x-4 x² +2 (a) Identify the horizontal asymptotes (if any). (b) If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s). Separate multiple equations of asymptotes with commas as necessary. Select "None" if applicable.
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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![### Analysis of Rational Functions
**Given the function:**
\[ q(x) = \frac{-5x^2 + 5x - 4}{x^2 + 2} \]
(a) **Identify the horizontal asymptotes (if any).**
(b) **If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s).**
**Instructions:**
Separate multiple equations of asymptotes with commas as necessary. Select "None" if applicable.
---
**Options for Horizontal Asymptotes:**
- The graph has no horizontal asymptotes.
- The graph has at least one horizontal asymptote.
**Response:**
☑️ The graph has at least one horizontal asymptote.
**Equation(s) of the horizontal asymptote(s):**
\[ y = -5 \]
Crossover point(s): [ ]
**Explanation:**
A horizontal asymptote for a rational function \( q(x) = \frac{p(x)}{d(x)}\) can be found by comparing the degrees of the polynomial in the numerator \( p(x) \) and the polynomial in the denominator \( d(x) \).
In this specific problem:
- The degree of \( p(x) \) (numerator: \( -5x^2 + 5x - 4 \)) is 2.
- The degree of \( d(x) \) (denominator: \( x^2 + 2 \)) is also 2.
When the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients of these polynomials.
\[ \text{Horizontal Asymptote: } y = \frac{\text{Leading Coefficient of Numerator}}{\text{Leading Coefficient of Denominator}} \]
Thus, for our given function:
\[ y = \frac{-5}{1} = -5 \]
Therefore, the horizontal asymptote is \( y = -5 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4830af55-84e9-44a3-a1a9-c9fad267ce9b%2F730e6c5e-12c7-4cb0-9df1-f2ae64f34108%2F0qozbo2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analysis of Rational Functions
**Given the function:**
\[ q(x) = \frac{-5x^2 + 5x - 4}{x^2 + 2} \]
(a) **Identify the horizontal asymptotes (if any).**
(b) **If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s).**
**Instructions:**
Separate multiple equations of asymptotes with commas as necessary. Select "None" if applicable.
---
**Options for Horizontal Asymptotes:**
- The graph has no horizontal asymptotes.
- The graph has at least one horizontal asymptote.
**Response:**
☑️ The graph has at least one horizontal asymptote.
**Equation(s) of the horizontal asymptote(s):**
\[ y = -5 \]
Crossover point(s): [ ]
**Explanation:**
A horizontal asymptote for a rational function \( q(x) = \frac{p(x)}{d(x)}\) can be found by comparing the degrees of the polynomial in the numerator \( p(x) \) and the polynomial in the denominator \( d(x) \).
In this specific problem:
- The degree of \( p(x) \) (numerator: \( -5x^2 + 5x - 4 \)) is 2.
- The degree of \( d(x) \) (denominator: \( x^2 + 2 \)) is also 2.
When the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients of these polynomials.
\[ \text{Horizontal Asymptote: } y = \frac{\text{Leading Coefficient of Numerator}}{\text{Leading Coefficient of Denominator}} \]
Thus, for our given function:
\[ y = \frac{-5}{1} = -5 \]
Therefore, the horizontal asymptote is \( y = -5 \).
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