## Understanding the Equation: (x – 3)2/25 + (y + 1)2/16 = 1

This equation represents an **ellipse**, a fundamental geometric shape. Let's break down the equation and explore its key features.

### Standard Form of an Ellipse

The general standard form for an ellipse centered at (h, k) is:

**(x - h)2/a2 + (y - k)2/b2 = 1**

Where:

**(h, k)**represents the center of the ellipse.**a**represents the distance from the center to the vertices along the major axis.**b**represents the distance from the center to the vertices along the minor axis.

### Analyzing our Equation

Comparing our given equation **(x – 3)2/25 + (y + 1)2/16 = 1** to the standard form, we can identify the following:

**Center:**(h, k) = (3, -1)**a2 = 25, so a = 5**(This is the distance from the center to the vertices along the horizontal axis)**b2 = 16, so b = 4**(This is the distance from the center to the vertices along the vertical axis)

### Key Features of the Ellipse

Based on the information we have gathered:

**Center:**The ellipse is centered at the point (3, -1).**Major Axis:**The major axis is horizontal, with length 2a = 10.**Minor Axis:**The minor axis is vertical, with length 2b = 8.**Vertices:**The vertices are located 5 units to the left and right of the center: (3 - 5, -1) = (-2, -1) and (3 + 5, -1) = (8, -1).**Co-vertices:**The co-vertices are located 4 units above and below the center: (3, -1 + 4) = (3, 3) and (3, -1 - 4) = (3, -5).

### Visualizing the Ellipse

To visualize the ellipse, you can plot the center, vertices, and co-vertices on a coordinate plane. Then, sketch a smooth curve that passes through these points, forming the shape of an ellipse.

**Remember:** Ellipses are symmetrical about both their major and minor axes. This symmetry aids in creating a more accurate visual representation.

### Conclusion

By understanding the standard form and analyzing the given equation, we can easily identify the center, major and minor axes, and vertices of the ellipse represented by **(x – 3)2/25 + (y + 1)2/16 = 1**. This knowledge allows us to accurately visualize and understand the properties of this specific ellipse.