Problem 75 - College Algebra

Question

What is a parabola? Describe its shape.

Step-by-Step Solution

Answer

A parabola is a curve that is mirror-symmetrical and is approximately U-shaped. It is the graph of a quadratic function. The curve has an axis of symmetry, a vertex, and can open upwards, downwards, right or left. A distinct characteristic of a parabola is its focus, a point closer to the curve than any other point of a straight-line known as the 'directrix'.

Step 1: Understanding a Parabola

A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram, but can be in any orientation in its plane. It is often described as the graph of a quadratic function.

Step 2: Describing the Shape of a Parabola

The parabola has a line of symmetry called the axis of symmetry that splits the curve into two perfect halves. The point where the parabola makes its sharpest turn is called the vertex. If you look from the vertex, the shape of the parabola appears like an U or an inverted U, depending on its position. The parabola can either open upwards, downwards, right or left. If the parabola opens upwards or downwards for a given x, there's exactly one y. But if the parabola opens rightward or leftward for a given y, there's exactly one x.

Step 3: Distinguishing Characteristics of a Parabola

A parabola has several distinct characteristics. The ‘focus’ is a point which exists such that any point on the parabola is closer to the focus than to any straight line known as the 'directrix'. The line perpendicular to the directrix and passing through the focus is called the 'axis of symmetry', which as the name suggests, is the line about which the parabola is symmetric.