a) What is the y-intercept of r (x) ? b) What is/are the x-intercepts of r(x)? c) Find the vertical asymptote/s. d) What is the horizontal asymptote, any? if e) What is the domain of r(x)? f) Sketch a rough graph of r(x). Don't forget to clearly label the intercepts, holes if any, and asymptotes.
a) What is the y-intercept of r (x) ? b) What is/are the x-intercepts of r(x)? c) Find the vertical asymptote/s. d) What is the horizontal asymptote, any? if e) What is the domain of r(x)? f) Sketch a rough graph of r(x). Don't forget to clearly label the intercepts, holes if any, and asymptotes.
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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Question
![**Given the polynomial**
\[ r(x) = \frac{x^2 - 25}{x^3 - 6x^2 + 5x} \]
This expression is a rational function where the numerator is \(x^2 - 25\), and the denominator is \(x^3 - 6x^2 + 5x\).
- **Numerator:** \(x^2 - 25\) is a difference of squares, which can be factored as \((x-5)(x+5)\).
- **Denominator:** \(x^3 - 6x^2 + 5x\) can be factored by taking the common factor \(x\), giving \(x(x^2 - 6x + 5)\). The quadratic can be further factored to \((x-1)(x-5)\).
**Key Features:**
- **Zeros of the numerator:** At \(x = 5\) and \(x = -5\), provided these are not also zeros of the denominator.
- **Vertical asymptotes:** At values where the denominator equals zero, which are \(x = 0\), \(x = 1\), and \(x = 5\) unless they are also solutions of the numerator.
- **Hole:** Located at \(x = 5\) because it is a zero of both the numerator and the denominator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4adc11d5-605b-4e54-a139-b421352267a6%2F763c38c2-718c-4e68-b6c5-95e5df53c5da%2F8trra3g_processed.png&w=3840&q=75)
Transcribed Image Text:**Given the polynomial**
\[ r(x) = \frac{x^2 - 25}{x^3 - 6x^2 + 5x} \]
This expression is a rational function where the numerator is \(x^2 - 25\), and the denominator is \(x^3 - 6x^2 + 5x\).
- **Numerator:** \(x^2 - 25\) is a difference of squares, which can be factored as \((x-5)(x+5)\).
- **Denominator:** \(x^3 - 6x^2 + 5x\) can be factored by taking the common factor \(x\), giving \(x(x^2 - 6x + 5)\). The quadratic can be further factored to \((x-1)(x-5)\).
**Key Features:**
- **Zeros of the numerator:** At \(x = 5\) and \(x = -5\), provided these are not also zeros of the denominator.
- **Vertical asymptotes:** At values where the denominator equals zero, which are \(x = 0\), \(x = 1\), and \(x = 5\) unless they are also solutions of the numerator.
- **Hole:** Located at \(x = 5\) because it is a zero of both the numerator and the denominator.
Transcribed Image Text:Certainly! Here's the transcription formatted as it might appear on an educational website:
---
### Exercise on Rational Functions
Consider the function \( r(x) \). Answer the following questions based on its graph and analytical properties.
a) **What is the \( y \)-intercept of \( r(x) \)?**
___________________________________________________________________________
b) **What is/are the \( x \)-intercepts of \( r(x) \)?**
___________________________________________________________________________
c) **Find the vertical asymptote(s).**
d) **What is the horizontal asymptote, if any?**
e) **What is the domain of \( r(x) \)?**
___________________________________________________________________________
f) **Sketch a rough graph of \( r(x) \). Don’t forget to clearly label the intercepts, holes if any, and asymptotes.**
---
### Instructions
- Analyze the function or given data to find intercepts and asymptotes.
- Use the information to construct a graph that accurately represents \( r(x) \).
- Ensure all key features such as intercepts, asymptotes, and any discontinuities or holes are clearly marked.
This exercise aids in understanding the graphing and properties of rational functions, which are crucial for comprehending their behavior and applications.
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