Finding The Focus and Directrix of a Parabola - Conic Sections
The Organic Chemistry Tutor・2 minutes read
The equation for a parabola with the vertex at the origin is y squared = 4px or x squared = 4py, with the focus and directrix located accordingly. The orientation and parameters determine whether the parabola opens up, down, left, or right, with the focus, vertex, and directrix being key components in graphing the curve.
Insights
- The equation for a parabola with the vertex at the origin differs based on its orientation: y squared = 4px for left-opening and x squared = 4py for right-opening.
- The position of the focus determines the direction in which the parabola opens, with a positive p leading to an upward opening parabola and a negative p resulting in a downward opening one.
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Recent questions
How do you determine the focus of a parabola?
By calculating the distance p from the vertex.
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Summary
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Parabola Equations, Focus, and Directrix Explained
- The equation for a parabola with the vertex at the origin on the left is y squared = 4px, while for the one on the right with the vertex at the origin, it is x squared = 4py.
- The focus of the parabola is to the right where it opens towards, with the distance p between the vertex and the focus.
- Moving p units in the opposite direction leads to the directrix, x = -p for the left-opening parabola.
- The lattice rectum, a segment passing through the focal point with a length of 4p, connects the focus and the curve.
- A positive p leads to an upward opening parabola, while a negative p results in a downward opening one.
- To graph an equation like x squared = 4py with the vertex at the origin, calculate p (in this case, 2) to find the focus and directrix.
- The focus is at (0, 2), and moving 2p units to the right and left gives additional points for graphing.
- The directrix is at y = -2, and the equation of the parabola is y squared = 4px with p = 2.
- For a parabola with a focus at (-3, 0) and a directrix at x = 3, the equation becomes y squared = -12x.
- Identifying the vertex at (-2, 3), the focus at (-1, 3), and the directrix at x = -3, the length of the lattice rectum is 4 units, and the domain is from -2 to infinity while the range is all real numbers.
20:05
Parabola Properties and Vertex Coordinates
- A horizontal parabola that opens to the left or right will have a range of real numbers.
- For a vertical parabola that opens up or down, the domain will be all real numbers from negative infinity to positive infinity.
- The standard form of the parabola is x minus h squared equals 4p times y minus k, with the vertex at h, k.
- The coordinates of the vertex for the given parabola are (3, 2), with p calculated as 2.
- The focus of the parabola is at (3, 0), determined by h, k, and p.
- The length of the lattice rectum for this parabola is 8 units.
- The domain for a vertical parabola is all real numbers, while the range is from negative infinity to the y-coordinate of the vertex, which is 2.
