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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![**Mathematical Expression and Graph Analysis**
**Expression:**
The equation given is:
\[ y = \frac{1}{\sqrt{16 - x^2}} \]
**Graph Description:**
- The graph is presented on a coordinate plane with the x-axis ranging approximately from -4 to 4 and the y-axis ranging from -1 to 3.
- The curve is symmetric about the y-axis, consistent with the symmetry of the function y = \(\frac{1}{\sqrt{16 - x^2}}\), which represents a hyperbola.
- The shape of the graph features two branches that curve upwards as they approach vertical asymptotes near x = ±4. These asymptotes are due to the denominator approaching zero, making the function undefined as x approaches ±4.
- The y-values of the graph do not include the negative region, as the function remains positive for the valid range of x-values.
- A shaded area is present between x = -2 and x = 2, marking a region of interest or possible integration bounds.
**User Interface Elements:**
- There is a text input box showing the number "8" next to it, indicating a user's input.
- A red "X" next to the input suggests an incorrect submission.
- Below the graph, there is a "Submit Answer" button for user interaction.
This interactive graph likely forms part of an educational tool designed to help users explore the properties of functions and graph behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff998db76-2065-477f-b7ac-e73b653f1166%2F87b52b8b-85ed-48a4-b526-8d0714fed801%2Feeeioj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Expression and Graph Analysis**
**Expression:**
The equation given is:
\[ y = \frac{1}{\sqrt{16 - x^2}} \]
**Graph Description:**
- The graph is presented on a coordinate plane with the x-axis ranging approximately from -4 to 4 and the y-axis ranging from -1 to 3.
- The curve is symmetric about the y-axis, consistent with the symmetry of the function y = \(\frac{1}{\sqrt{16 - x^2}}\), which represents a hyperbola.
- The shape of the graph features two branches that curve upwards as they approach vertical asymptotes near x = ±4. These asymptotes are due to the denominator approaching zero, making the function undefined as x approaches ±4.
- The y-values of the graph do not include the negative region, as the function remains positive for the valid range of x-values.
- A shaded area is present between x = -2 and x = 2, marking a region of interest or possible integration bounds.
**User Interface Elements:**
- There is a text input box showing the number "8" next to it, indicating a user's input.
- A red "X" next to the input suggests an incorrect submission.
- Below the graph, there is a "Submit Answer" button for user interaction.
This interactive graph likely forms part of an educational tool designed to help users explore the properties of functions and graph behavior.
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