The description given matches the graph of a vertical ellipse. An ellipse has a symmetric, oval shape, and the curve you describe intersects the x-axis at two points, which suggests it is centered at the origin. Additionally, since the curve reaches a maximum height at $(0, 5)$ and intersects the x-axis symmetrically, the center of this ellipse is indeed at the origin, and the major axis is vertical.

Steps to Determine the Equation

1. Identify the Center and Orientation:
   The center of the ellipse is at the origin $(0, 0)$, and it has a vertical major axis (since it peaks at $(0, 5)$).

2. Determine the Semi-Major and Semi-Minor Axes:

   • The maximum height occurs at $(0, 5)$, so the semi-major axis length is $a = 5$.
   • The ellipse crosses the x-axis at $x = -1$ and $x = 1$, giving a semi-minor axis length of $b = 1$.

3. Formulate the Equation:
   For an ellipse centered at $(0, 0)$ with a vertical major axis, the equation is: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ Substituting $a = 5$ and $b = 1$ gives: $\frac{x^2}{1^2} + \frac{y^2}{5^2} = 1$ Simplifying, we get: $x^2 + \frac{y^2}{25} = 1$

Final Equation

The equation for the ellipse is: $x^2 + \frac{y^2}{25} = 1$

Would you like a detailed breakdown or have any questions?

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1. How can you distinguish between ellipses and other conics like parabolas and hyperbolas?
2. What changes in the equation if the ellipse is centered at a different point, say $(h, k)$?
3. How do the semi-major and semi-minor axes affect the shape of an ellipse?
4. Can this equation be converted into a parametric form? How?
5. What would happen to the ellipse if the semi-minor axis were increased?

Tip: For any conic section, observing the intercepts and symmetry helps in identifying its type and orientation.