Problem 52 - College Algebra

Question

What is a parabola? Describe its shape.

Step-by-Step Solution

Answer

A parabola is a kind of curve in geometry. It is as far from a certain point (the focus) as from a certain line (the directrix). It has a U or inverted U shape when graphed. Its main point, called the vertex, is where it turns. Parabolas also have an axis of symmetry.

Step 1: Define a Parabola

A parabola is a curve which, for any point, is as far from a certain point (the focus) as from a certain line (the directrix). This is a fundamental concept in geometry and algebra.

Step 2: Describe the Shape of a Parabola

When graphed, a parabola often takes the shape of a U or an inverted U. Its extreme point, where it turns, is called the vertex. If the parabola opens upwards or downwards, it's in the form \( y = ax^2 + bx + c \). The shape of the parabola tells us whether the leading coefficient 'a' is positive or negative. If the parabola opens to the right or to the left, it's in the form \( x = ay^2 + by + c \).

Step 3: Parabolic Features

Parabolas have an axis of symmetry. For a parabola in the form \( y = ax^2 + bx + c \), the axis of symmetry is the line \( x = -b/2a \). All points on the parabola are equidistant from the focus and the directrix. The point where the parabola crosses the axis of symmetry is called the vertex. The point on the axis of symmetry inside the parabola is called the focus. The line perpendicular to the axis of symmetry outside the parabola is called the directrix.