Identify: a.) the x and y-intercepts; b.) the asymptotes; c.) the intervals where the function is increasing or decreasing; d.) all points of extrema; e.) the intervals of concavity; f.) inflection points, and g.) sketch the graph. Show all work in the space provided. Label each part a through g and box in your finals answers. -8x y = x² + 4 a.) g.)
Identify: a.) the x and y-intercepts; b.) the asymptotes; c.) the intervals where the function is increasing or decreasing; d.) all points of extrema; e.) the intervals of concavity; f.) inflection points, and g.) sketch the graph. Show all work in the space provided. Label each part a through g and box in your finals answers. -8x y = x² + 4 a.) g.)
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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![**Calculus Problem: Analyzing a Rational Function**
**Objective:**
Identify the following characteristics of the given rational function:
\[ y = \frac{-8x}{x^2 + 4} \]
**Tasks:**
a.) The x and y-intercepts
b.) The asymptotes
c.) The intervals where the function is increasing or decreasing
d.) All points of extrema
e.) The intervals of concavity
f.) Inflection points
g.) Sketch the graph
**Instructions:**
Show all work in the space provided. Label each part (a) through (g) and box in your final answers.
**Provided:**
**Equation:**
\[ y = \frac{-8x}{x^2 + 4} \]
**Tasks to Complete:**
a.) **Intercepts:**
- **x-intercept:**
Set \( y = 0 \) and solve for \( x \).
\[ \frac{-8x}{x^2 + 4} = 0 \]
- **y-intercept:**
Set \( x = 0 \) and solve for \( y \).
b.) **Asymptotes:**
- **Vertical asymptotes:**
Determine where the function is undefined.
- **Horizontal asymptote:**
Determine the behavior of the function as \( x \rightarrow \pm \infty \).
c.) **Intervals of Increasing/Decreasing:**
- Find the first derivative \( y' \).
- Determine the critical points by setting \( y' = 0 \).
- Use the first derivative test to determine the intervals.
d.) **Extrema:**
- Use the results from part (c) to identify the local maxima and minima.
e.) **Intervals of Concavity:**
- Find the second derivative \( y'' \).
- Determine the intervals of concavity.
f.) **Inflection Points:**
- Use the results from part (e) to identify any inflection points.
g.) **Graph:**
- Sketch the graph based on the above analysis.
**Graphical Representation:**
- A blank coordinate plane is provided for sketch purposes.
- Label the x-axis and y-axis.
- Ensure intercepts, asymptotes, and key points are marked.
**Graph:**
A blank coordinate plane is provided with labeled axes. Use this space to plot the graph based on your analysis from parts (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F966b9752-0e6e-47d7-af8f-ce4bbf704a72%2F9d87b47a-28a4-4a5a-a39d-8952a4b64634%2Fep1h569_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculus Problem: Analyzing a Rational Function**
**Objective:**
Identify the following characteristics of the given rational function:
\[ y = \frac{-8x}{x^2 + 4} \]
**Tasks:**
a.) The x and y-intercepts
b.) The asymptotes
c.) The intervals where the function is increasing or decreasing
d.) All points of extrema
e.) The intervals of concavity
f.) Inflection points
g.) Sketch the graph
**Instructions:**
Show all work in the space provided. Label each part (a) through (g) and box in your final answers.
**Provided:**
**Equation:**
\[ y = \frac{-8x}{x^2 + 4} \]
**Tasks to Complete:**
a.) **Intercepts:**
- **x-intercept:**
Set \( y = 0 \) and solve for \( x \).
\[ \frac{-8x}{x^2 + 4} = 0 \]
- **y-intercept:**
Set \( x = 0 \) and solve for \( y \).
b.) **Asymptotes:**
- **Vertical asymptotes:**
Determine where the function is undefined.
- **Horizontal asymptote:**
Determine the behavior of the function as \( x \rightarrow \pm \infty \).
c.) **Intervals of Increasing/Decreasing:**
- Find the first derivative \( y' \).
- Determine the critical points by setting \( y' = 0 \).
- Use the first derivative test to determine the intervals.
d.) **Extrema:**
- Use the results from part (c) to identify the local maxima and minima.
e.) **Intervals of Concavity:**
- Find the second derivative \( y'' \).
- Determine the intervals of concavity.
f.) **Inflection Points:**
- Use the results from part (e) to identify any inflection points.
g.) **Graph:**
- Sketch the graph based on the above analysis.
**Graphical Representation:**
- A blank coordinate plane is provided for sketch purposes.
- Label the x-axis and y-axis.
- Ensure intercepts, asymptotes, and key points are marked.
**Graph:**
A blank coordinate plane is provided with labeled axes. Use this space to plot the graph based on your analysis from parts (
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