Graph Of 1 X 2 Y 2

Understanding the Graph of 1x² + 2y²

The equation 1x² + 2y² represents an ellipse, a specific type of conic section. This article will explore its characteristics, how to graph it, and its key features. Understanding this equation is crucial for anyone studying conic sections, analytic geometry, or related mathematical fields.

Key Features and Characteristics of the Ellipse

The general equation for an ellipse centered at the origin is:

(x²/a²) + (y²/b²) = 1

Where 'a' and 'b' represent the lengths of the semi-major and semi-minor axes, respectively. Comparing this to our equation, 1x² + 2y² = 1, we can rewrite it in the standard form:

x²/1 + y²/(1/2) = 1

From this, we can identify:

• a² = 1, therefore a = 1 (length of the semi-major axis along the x-axis)
• b² = 1/2, therefore b = 1/√2 ≈ 0.707 (length of the semi-minor axis along the y-axis)

This means the ellipse is elongated along the x-axis. The major axis has a length of 2a = 2, and the minor axis has a length of 2b = √2. The foci are located along the major axis, inside the ellipse.

Calculating the Foci

The distance from the center to each focus (c) can be calculated using the relationship:

c² = a² - b² = 1 - (1/2) = 1/2

Therefore, c = 1/√2 ≈ 0.707. The foci are located at approximately (±0.707, 0).

Graphing the Ellipse

To graph the ellipse, follow these steps:

1. Plot the center: The center of the ellipse is at the origin (0, 0).
2. Plot the vertices: The vertices are located at (±a, 0), which are (±1, 0) in this case.
3. Plot the co-vertices: The co-vertices are located at (0, ±b), which are approximately (0, ±0.707).
4. Sketch the ellipse: Using the vertices and co-vertices as guides, sketch a smooth, oval-shaped curve. Remember the ellipse is symmetrical about both the x and y axes.

Understanding the Equation's Implications

The coefficients of x² and y² in the equation dictate the shape and orientation of the ellipse. A larger coefficient indicates a shorter axis. In this case, the larger coefficient for x² indicates that the ellipse is wider along the x-axis.

Further Exploration

This analysis provides a foundational understanding of the graph of 1x² + 2y² = 1. Further exploration could involve:

• Examining transformations of the equation (e.g., translations, rotations).
• Exploring the relationship between the ellipse's equation and its eccentricity.
• Applying this knowledge to solve real-world problems involving elliptical shapes.

By understanding the standard form of the ellipse equation and applying the steps outlined above, you can accurately graph and analyze this specific conic section. This knowledge is essential for various mathematical applications.