Problem 57 - College Algebra

Question

What is a parabola?

Step-by-Step Solution

Answer

A parabola is a U-shaped symmetrical curve, also known as a conic section, formed by the intersection of a cone and a plane. It's represented by the equation \( y = ax^2 + bx + c \) in Cartesian coordinates. A distinguishing property of a parabola is that any point on the parabola is equidistant from the focus and the directrix.

Step 1: Definition

A parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It is a specific type of curve called a conic section, formed by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

Step 2: Mathematical Description

In the Cartesian coordinate system, the graph of a quadratic equation \( y = ax^2 + bx + c \) is a parabola. The property that distinguishes a parabola is that, for any point on the parabola, the distance from that point to the focus is equal to the distance from that point to the directrix.

Step 3: Applications and Properties

Parabolas can be seen in many real-world applications such as the design of car headlights, satellite dishes, and suspension bridge cables. A parabola has the property that, if it is made of a reflective material, all rays of light that enter it parallel to its axis of symmetry are reflected to its focus, regardless of where on the parabola they hit.