Certainly! Below is the transcription and detailed explanation of the images intended for an educational website:---### Understanding the Translation of Circles on a Cartesian Plane#### Diagram 1: Circle Centered at (2, 1)The first diagram depicts a circle centered at the coordinates (2, 1) on a Cartesian coordinate plane. The key elements include:- **Center Point**: The circle's center is marked with a dot at (2, 1).- **Radius**: The circle surrounds the center point with equal radius on all sides, indicating it is equidistant from the center.- **Axes**: The x-axis and y-axis are clearly labeled, intersecting at the origin (0, 0).The grid makes it easy to identify the precise location and spacing along the axes.#### Diagram 2: Circle Centered at (-2, -1)The second diagram showcases a circle centered at the coordinates (-2, -1) on a Cartesian coordinate plane. Key aspects of the diagram include:- **Center Point**: The circle's center is noted by a dot at the location (-2, -1).- **Radius**: The circle extends equally from the center point to form a perfect circle.- **Axes**: The x-axis and y-axis are clearly defined, meeting at the origin (0, 0).The grid format aids in the accurate visualization of the circle's placement relative to the coordinate system.### Analysis and ComparisonThese diagrams help illustrate the concept of translating geometric shapes, such as circles, across the Cartesian plane. By observing how the center shifts from one location to another, students can gain a better understanding of how to manipulate and transform circles in a coordinate system.### Practical ApplicationUnderstanding the translation of circles is essential in various fields of mathematics and physics, particularly in coordinate geometry. It also has practical applications in computer graphics, robotics, and engineering, where precise movements and positioning are crucial.---This comprehensive explanation is designed to help students and educators alike understand the fundamental principles of translating circles on a Cartesian plane. **Title: Understanding the Graph of a Circle from its Equation****Content:**In this section, we will explore how to determine the correct graph of a circle given its equation. We will review two graphs and identify which one accurately represents the equation \( (x - 2)^2 + (y - 1)^2 = 12.25 \).### Identifying the Circle's EquationThe general form of a circle's equation is:\[ (x - h)^2 + (y - k)^2 = r^2 \]where:- \((h, k)\) is the center of the circle.- \(r\) is the radius of the circle.Given:\[ (x - 2)^2 + (y - 1)^2 = 12.25 \]We can identify:- The center of the circle \((h, k) = (2, 1)\).- The radius \(r = \sqrt{12.25} = 3.5\).### Analyzing the Graphs#### Graph 1:- **Appearance:** The first graph shows a circle centered at \((2, 1)\). The radius appears to be approximately 3.5 units.- **Origin Check:** Plotting the center of the circle reveals it is placed accurately at \( (2, 1) \).#### Graph 2:- **Appearance:** The second graph shows a circle centered at a completely different point, not \((2, 1)\). It does not correspond to the given circle's equation.- **Origin Check:** The center of this circle is not at \( (2, 1) \), making it not the correct graph for the given equation.### Conclusion:The correct graph representing the equation \( (x - 2)^2 + (y - 1)^2 = 12.257 \) is the first graph. This graph correctly places the circle's center at \((2, 1)\) with a radius of 3.5 units.This tutorial demonstrates how to verify the graph of a given equation by identifying the center and radius accurately.

Answered: Which of the following is the graph of…

Math

Calculus

Which of the following is the graph of (x-2)(y-1)² = 12.257 17 19 & + 17 24 47 7 M -10 IN 2 D

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

Certainly! Below is the transcription and detailed explanation of the images intended for an educational website:

---

### Understanding the Translation of Circles on a Cartesian Plane

#### Diagram 1: Circle Centered at (2, 1)
The first diagram depicts a circle centered at the coordinates (2, 1) on a Cartesian coordinate plane. The key elements include:
- **Center Point**: The circle's center is marked with a dot at (2, 1).
- **Radius**: The circle surrounds the center point with equal radius on all sides, indicating it is equidistant from the center.
- **Axes**: The x-axis and y-axis are clearly labeled, intersecting at the origin (0, 0).
The grid makes it easy to identify the precise location and spacing along the axes.

#### Diagram 2: Circle Centered at (-2, -1)
The second diagram showcases a circle centered at the coordinates (-2, -1) on a Cartesian coordinate plane. Key aspects of the diagram include:
- **Center Point**: The circle's center is noted by a dot at the location (-2, -1).
- **Radius**: The circle extends equally from the center point to form a perfect circle.
- **Axes**: The x-axis and y-axis are clearly defined, meeting at the origin (0, 0).
The grid format aids in the accurate visualization of the circle's placement relative to the coordinate system.

### Analysis and Comparison
These diagrams help illustrate the concept of translating geometric shapes, such as circles, across the Cartesian plane. By observing how the center shifts from one location to another, students can gain a better understanding of how to manipulate and transform circles in a coordinate system.

### Practical Application
Understanding the translation of circles is essential in various fields of mathematics and physics, particularly in coordinate geometry. It also has practical applications in computer graphics, robotics, and engineering, where precise movements and positioning are crucial.

---

This comprehensive explanation is designed to help students and educators alike understand the fundamental principles of translating circles on a Cartesian plane.

**Title: Understanding the Graph of a Circle from its Equation**

**Content:**

In this section, we will explore how to determine the correct graph of a circle given its equation. We will review two graphs and identify which one accurately represents the equation \( (x - 2)^2 + (y - 1)^2 = 12.25 \).

### Identifying the Circle's Equation

The general form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the circle.

Given:
\[ (x - 2)^2 + (y - 1)^2 = 12.25 \]

We can identify:
- The center of the circle \((h, k) = (2, 1)\).
- The radius \(r = \sqrt{12.25} = 3.5\).

### Analyzing the Graphs

#### Graph 1:
- **Appearance:** The first graph shows a circle centered at \((2, 1)\). The radius appears to be approximately 3.5 units.
- **Origin Check:** Plotting the center of the circle reveals it is placed accurately at \( (2, 1) \).

#### Graph 2:
- **Appearance:** The second graph shows a circle centered at a completely different point, not \((2, 1)\). It does not correspond to the given circle's equation.
- **Origin Check:** The center of this circle is not at \( (2, 1) \), making it not the correct graph for the given equation.

### Conclusion:
The correct graph representing the equation \( (x - 2)^2 + (y - 1)^2 = 12.257 \) is the first graph. This graph correctly places the circle's center at \((2, 1)\) with a radius of 3.5 units.

This tutorial demonstrates how to verify the graph of a given equation by identifying the center and radius accurately.

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