What geometric shape is described as a two-dimensional symmetrical curve that follows a u-shape?

The geometric shape that is described as a two-dimensional symmetrical curve with a u-shape is a parabola. This is characterized by its specific mathematical definition and properties. A parabola can be represented by a quadratic equation in the form of y = ax² + bx + c, where the graph opens either upwards or downwards depending on the sign of the coefficient 'a.'

The key feature of a parabola is its symmetry about its axis, which is a vertical line that goes through its vertex (the highest or lowest point of the curve) and divides the parabola into two mirrored halves. This distinct u-shape is what differentiates a parabola from other shapes.

In contrast, other options represent different geometric figures: an ellipse is a closed curve that can look circular but has varying widths; a circle is a special case of an ellipse where the distance from the center to any point on the boundary is constant; and a hyperbola consists of two separate curves that open away from each other and do not have the smooth, continuous curve shape of a parabola. Therefore, the unique u-shaped characteristic firmly identifies the parabola as the correct answer.