Find the equation for the parabola that has its vertex at the origin and has directrix at .
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To find the equation of the parabola with a vertex at the origin (0,0) and a directrix at , we note that this is a vertically oriented parabola that opens upwards. The distance from the vertex to the directrix is . The standard form for such a parabola is given by . Substituting into the equation gives us:

So the equation of the parabola is:

Now you’ve got a snazzy parabola that bounces towards the sky!

Parabolas have fascinating properties, such as reflective symmetry. Any light that enters a parabola parallel to its axis of symmetry will reflect off and pass through the focus! In our case, the focus is at , just above the vertex, making it ideal for applications like satellite dishes or car headlights, where you want to control light paths effectively.

Fun fact: The concept of parabolas dates back to ancient Greece! The mathematician Apollonius of Perga was one of the first to study these beautiful curves systematically. His work laid down the foundational principles that mathematicians use today, making parabolas truly a timeless geometry treasure!