A parabola is a locus of points equidistant from both 1) a azygos point, called the focus on of the parabola, and 2) a line, called the directrix of the parabola.
What is the Focus and Directrix?
The red point in the pictures below is the focus of the parabola and the scarlet pipeline is the directrix. Equally you can find out from the diagrams, when the focus is above the directrix Example 1, the parabola opens upwards. In the next section, we will explain how the focus and directrix relate to the actual parabola. Explore this more with our interactional app below.
When the focus is below, the directrix , then the parabola opens down.
How do Focus/Directrix relate to the Parabola?
The majestic lines in the picture below represent the distance between the focus and different points along the directrix . Every repoint on the parabola is sporting as far away (equidistant) from the directrix and the concentre.
In other words, line $$ l_1 $$ from the directrix to the parabola is the same length atomic number 3 $$ l_1 $$ from the parabola back to the focus . The same goes for all of the otherwise distances from a point on the parabola to the focus and directrix ( $$ l_2, l_3 \text{ etc.. } $$). See animation below
Explore this more with our interactive app below.
Parabola Locale Animation
Exploring Focus/Directrix relation to Graph
You probably know that the smaller |a| in the standard form equation of a parabola, the wider the parabola. In unusual words y = .1x² is a wider parabola than y = .2x² and y = -.1x² is a wider parabola than y = .-2x². You hind end understand this 'widening' effect in damage of the focus and directrix. Equally the distance between the focus and directrix increases, |a| decreases which means the parabola widens. See the pictures below to infer.
|a| = 1
|a| = .6
|a| =.3
|a| = .2
Centre and Directrix Applet
Search how the rive and directrix relate to the graph of a parabola with the interactive computer program below.
oy2%3Asvi1y2%3Artzy2%3Afdi1y2%3Alci1y2%3Aaxi1y2%3Ayizy2%3Asgzy2%3Agazy3%3Apoai1y3%3Apobi2y3%3Apoci-3y3%3Ashxi51y3%3Ashyi-176y1%3Azi2g
Try this interactive parabola applet on its own page.
- Related Links:
- Parabola home
- Axis of Symmetry
- Parabola Grapher
- Center and Directrix
- Standard and Apex Form
- Acme
- Very World Applications
find standard equation of parabola given focus and directrix
Source: https://www.mathwarehouse.com/quadratic/parabola/focus-and-directrix-of-parabola.php
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