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2 days ago · A parabola is a U-shaped curve in which all points are equidistant from a fixed point and a fixed straight line. The point is the focus of the parabola, and the line is the directrix. The focus lies on the axis of symmetry, and the directrix is parallel to either the x-axis or the y-axis. However, the focus never lies on the directrix. Formulas
The red point in the pictures below is the focus of the parabola and the red line is the directrix. As you can see 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 interactive ...
Feb 13, 2022 · A parabola consists of three parts: Vertex, Focus, and Directrix. The vertex of a parabola is the maximum or minimum of the parabola and the focus of a parabola is a fixed point that lies inside the parabola. The directrix is outside of the parabola and parallel to the axis of the parabola. Related Topic. How to Write the Equation of Parabola
The directrix of a parabola can be found, by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the parabola in standard form y 2 = 4ax, with focus at (a, 0), axis as the x-axis, the equation of the directrix of this parabola is x + a = 0 .
- Overview
- What terms (like “parabola”) should I know?
- Which equations/formulas do I use?
- Example Walk-Through: y=x^2
- More Examples
1 What terms (like “parabola”) should I know?
2 Which equations/formulas do I use?
If you’ve ever cooked food with a parabolic oven in science class or seen the Death Star’s laser in
, you have an idea of what the focal point (or focus) of a parabola is. But how do you calculate the focus mathematically? We’ve provided the formulas and equations you need to find the focus of any parabola, and added several helpful sample problems that you might see on your next algebra exam!
Depending on the type of parabola, use the equation
to determine the parabola’s vertex coordinates.
defined as the graph of a quadratic equation
—that is, the curved line you’ll get if you plot the equation on graph paper. Or, if you want to be more technical, it’s a curved line in which all coordinate points along the line are equidistant from a specific focal point and a specific line called a directrix.
In practical terms, it’s often easier to recognize parabolas in three dimensions. For example, think of large parabolic satellite dishes, or the clear plastic parabolic microphones you see on the sidelines of football games. Both of these direct waves (radio, sound, etc.) toward a single point—the focal point (or focus).
The vertex is the “turning point” of a parabola—the point along the curve at which it changes direction. So, in a classic “U”-shaped parabola, the vertex is at the very bottom of the “U” shape. You need to know the coordinates of the vertex in order to find the coordinates of the focus.
use one of these “vertex form of a parabola” equations
based on the type of parabola you’re dealing with. A “regular” parabola that opens upward or downward (like a right-side up or upside-down “U”) needs to be converted into the form of the first equation, while a “sideways” parabola that opens to the side (like a forwards or backwards “C”) must be converted to the second.
If you have a graph of the parabola, it’s easy to tell which equation to use. But what if you’re only given the parabola in equation form? Here’s the trick to use:
component is squared in the parabola’s equation—for example
—convert it into the form
component is squared (like in
Put the equation into the vertex form of a parabola.
Because the portion of the equation is squared, the correct vertex form is , meaning this is a “regular” parabola (it opens either up or down).
is positive, the parabola opens upward.
Identify the vertex and the focus equation.
In , both and equal zero .
That means the focus equation is
Find the focus of the parabola .
This is a “sideways” parabola because the component is squared, so use the vertex form and the focus equation .
Answer: The focus is located at
Find the focus of the parabola .
Here’s another “sideways” parabola because the y component is squared, so use the vertex form and the focus equation .
Answer: The focus is located at
Steps to Find Vertex Focus and Directrix Of The Parabola. Step 1. Determine the horizontal or vertical axis of symmetry. Step 2. Write the standard equation. Step 3. Compare the given equation with the standard equation and find the value of a. Step 4.
People also ask
What is the vertex focus and directrix of a parabola?
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What is the focus of a parabola centered at 0?
How do you find the vertex and focus of a parabola?
What are the 3 parts of a parabola?
Why is a parabola called a focus?
The simplest equation for a parabola is y = x2. Turned on its side it becomes y2 = x. (or y = √x for just the top half) A little more generally: y 2 = 4ax. where a is the distance from the origin to the focus (and also from the origin to directrix) Example: Find the focus for the equation y 2 =5x.
