Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Use the Distance Formula to find the coordinate of If necessary, round your answer to decimal places.
Is it possible to find a point directly above so that the distance from to and the distance from to are the same?
Consider the previous parabola, this time drawn on a coordinate plane. The focus of the parabola is and its directrix is the line with the equation Consider also a point with an coordinate of lying on the parabola.
Use the Distance Formula to express the distance from the focus to the point
Note that the directrix of the parabola is a horizontal line. Therefore, the distance between and the directrix is given by the length of the vertical segment that connects and the line. This length is the difference between the coordinate of and which is the coordinate of every point on the line. Recall that a parabola is the set of all the points in a plane that are equidistant from a point called focus and a line called directrix. Since point is on the parabola, it is equidistant from the directrix and the focus.
By definition, any point on the parabola must be equidistant from the focus and the directrix. This means that and are congruent segments. Therefore, they have the same length. The Distance Formula can be used to write an expression for each length.
In the graph, a parabola with the focus at and directrix is shown.
Since is equidistant from and the line it is known that and are equal. Therefore, is also By substituting this information together with the points and into the Distance Formula, the equation of the parabola can be obtained.
Therefore, the equation of the parabola with focus at and directrix is
Recall the general form for translations of functions.
Izabella is making an original video game character. She wants a force field in the shape of a parabola. When the character is at and the opposing team's army is on vertical line the force field will appear as shown in the graph.
What is the purpose of designing a satellite dish in the form of a paraboloid, or a three-dimensional parabola?The shape of a parabola brings along an important reflective property. This property is used to collect or project light, sound, or radio waves. For this reason, satellite dishes are designed in the form of paraboloids — surfaces generated by the rotation of a parabola around its axis of symmetry.