A parabola is one of the most important curves in mathematics. It is uniquely defined by two key elements: a focus point and a directrix line. The remarkable property of a parabola is that every point on the curve maintains equal distances to both the focus and the directrix. This geometric definition gives the parabola its distinctive U-shaped form.
The mathematical definition of a parabola's focus and directrix follows specific formulas. For a parabola in the form y equals ax squared plus bx plus c, the vertex is located at h comma k, where h equals negative b over 2a. The focus is positioned at h comma k plus 1 over 4a, while the directrix is the horizontal line y equals k minus 1 over 4a. For our example parabola y equals x squared plus 1, we have a equals 1, so the focus is at 0 comma 1.25 and the directrix is y equals 0.75.
Now let's see the defining property of a parabola in action. As we move a point along the parabola, we can observe that the distance from this point to the focus is always exactly equal to the distance from the point to the directrix. This equal distance property is what creates the parabola's unique curved shape. Watch carefully as the point moves - the two orange lines representing these distances always remain equal in length.
Parabolas can open in different directions depending on their mathematical form. The most common is the upward-opening parabola with equation y equals ax squared where a is positive. When a is negative, the parabola opens downward. We can also have parabolas that open horizontally. A rightward-opening parabola has the form x equals ay squared, while a leftward-opening parabola uses x equals negative ay squared. In each case, the focus and directrix maintain their defining relationship with every point on the curve.