Which Is The Graph Of The Equation$\frac{(x-1)^2}{3^2} + \frac{y^2}{4^2} = 1$?A. $A$

Feb 25, 2025 by ADMIN 85 views

**Identifying the Graph of a Given Equation: A Mathematical Analysis**

Introduction

In mathematics, graphing equations is a crucial aspect of understanding various mathematical concepts, including algebra, geometry, and trigonometry. The given equation, $\frac{(x-1)^2}{3^2} + \frac{y^2}{4^2} = 1$ , represents an ellipse in the Cartesian coordinate system. In this article, we will delve into the process of identifying the graph of this equation and explore its properties.

Understanding the Equation

The given equation is in the standard form of an ellipse, which is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ represents the center of the ellipse, and $a$ and $b$ are the semi-major and semi-minor axes, respectively. By comparing the given equation with the standard form, we can identify the values of $h$ , $k$ , $a$ , and $b$ .

Identifying the Center and Axes

From the given equation, we can see that the center of the ellipse is at $(h,k) = (1,0)$ . The semi-major axis, $a$ , is equal to $4$ , and the semi-minor axis, $b$ , is equal to $3$ . This information is crucial in understanding the shape and size of the ellipse.

Graphing the Equation

To graph the equation, we can use the following steps:

Plot the Center: Start by plotting the center of the ellipse at $(1,0)$ .
Draw the Major Axis: Draw a horizontal line through the center, representing the major axis. Since the semi-major axis is $4$ , the major axis will have a length of $8$ .
Draw the Minor Axis: Draw a vertical line through the center, representing the minor axis. Since the semi-minor axis is $3$ , the minor axis will have a length of $6$ .
Plot the Endpoints: Plot the endpoints of the major and minor axes, which are $(1,4)$ and $(1,-3)$ , respectively.
Connect the Endpoints: Connect the endpoints to form the ellipse.

Properties of the Ellipse

The given equation represents an ellipse with a center at $(1,0)$ , a semi-major axis of $4$ , and a semi-minor axis of $3$ . Some of the key properties of this ellipse include:

Shape: The ellipse is a closed curve with a smooth, continuous shape.
Size: The ellipse has a major axis of length $8$ and a minor axis of length $6$ .
Orientation: The ellipse is oriented horizontally, with the major axis parallel to the x-axis.
Symmetry: The ellipse is symmetric about its center, $(1,0)$ .

Conclusion

In conclusion, the graph of the equation $\frac{(x-1)^2}{3^2} + \frac{y^2}{4^2} = 1$ is an ellipse with a center at $(1,0)$ , a semi-major axis of $4$ , and a semi-minor axis of $3$ . By understanding the properties of this ellipse, we can gain a deeper appreciation for the mathematical concepts that underlie its structure.

Final Thoughts

Graphing equations is a fundamental aspect of mathematics, and understanding the properties of various shapes, including ellipses, is crucial for success in mathematics and related fields. By following the steps outlined in this article, you can graph the equation $\frac{(x-1)^2}{3^2} + \frac{y^2}{4^2} = 1$ and gain a deeper understanding of the mathematical concepts that underlie its structure.

Introduction

In our previous article, we explored the graph of the equation $\frac{(x-1)^2}{3^2} + \frac{y^2}{4^2} = 1$ and its properties. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the center of the ellipse?

A: The center of the ellipse is at the point $(1,0)$ .

Q: What are the lengths of the major and minor axes?

A: The major axis has a length of $8$ and the minor axis has a length of $6$ .

Q: Is the ellipse oriented horizontally or vertically?

A: The ellipse is oriented horizontally, with the major axis parallel to the x-axis.

Q: Is the ellipse symmetric about its center?

A: Yes, the ellipse is symmetric about its center, $(1,0)$ .

Q: How do I graph the equation?

A: To graph the equation, follow these steps:

Plot the Center: Start by plotting the center of the ellipse at $(1,0)$ .
Draw the Major Axis: Draw a horizontal line through the center, representing the major axis. Since the semi-major axis is $4$ , the major axis will have a length of $8$ .
Draw the Minor Axis: Draw a vertical line through the center, representing the minor axis. Since the semi-minor axis is $3$ , the minor axis will have a length of $6$ .
Plot the Endpoints: Plot the endpoints of the major and minor axes, which are $(1,4)$ and $(1,-3)$ , respectively.
Connect the Endpoints: Connect the endpoints to form the ellipse.

Q: What are some of the key properties of the ellipse?

A: Some of the key properties of the ellipse include:

Shape: The ellipse is a closed curve with a smooth, continuous shape.
Size: The ellipse has a major axis of length $8$ and a minor axis of length $6$ .
Orientation: The ellipse is oriented horizontally, with the major axis parallel to the x-axis.
Symmetry: The ellipse is symmetric about its center, $(1,0)$ .

Q: How can I use this knowledge in real-world applications?

A: Understanding the properties of ellipses can be applied in various real-world scenarios, such as:

Designing shapes: Ellipses can be used to design shapes in architecture, engineering, and art.
Modeling real-world objects: Ellipses can be used to model real-world objects, such as orbits of planets, shapes of buildings, and more.
Analyzing data: Ellipses can be used to analyze data in statistics and data science.