The method to find the focus depends on the form of the parabola equation you are given. Here's how to find the focus, focusing primarily on parabolas that open upwards or downwards:
Understanding the Basics
A parabola is defined as the set of all points equidistant to a point (the focus) and a line (the directrix). The vertex is the turning point of the parabola, located exactly halfway between the focus and the directrix.
Finding the Focus from Vertex Form
The vertex form of a parabola equation (with a vertical axis of symmetry) is:
y = a(x - h)² + k
where:
- (h, k) is the vertex of the parabola
- a determines the direction and "width" of the parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
Steps to Find the Focus:
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Identify the vertex (h, k) from the equation.
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Determine the value of 'a' from the equation.
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Calculate the distance 'p' between the vertex and the focus using the formula: p = 1 / (4a).
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Find the coordinates of the focus:
Since the parabola opens upwards or downwards, the x-coordinate of the focus is the same as the x-coordinate of the vertex (h). The y-coordinate of the focus is found by adding 'p' to the y-coordinate of the vertex (k). Therefore, the focus is located at (h, k + p) or (h, k + 1/(4a)).
Example:
Consider the equation: y = 2(x - 1)² + 3
- Vertex: (h, k) = (1, 3)
- a = 2
- p = 1 / (4 * 2) = 1/8
- Focus: (1, 3 + 1/8) = (1, 25/8)
Finding the Focus from Standard Form
The standard form of a parabola equation (with a vertical axis of symmetry) is:
y = ax² + bx + c
Steps to Find the Focus:
- Convert to Vertex Form: Complete the square to rewrite the equation in vertex form: y = a(x - h)² + k. You can find the vertex (h, k) using the formulas: h = -b / (2a) and k = f(h) where f(x) = ax² + bx + c.
- Follow the steps outlined above for Vertex Form to determine the location of the focus (h, k + 1/(4a)).
Parabolas with a Horizontal Axis of Symmetry
If the parabola opens to the left or right, the equation will have the form:
x = a(y - k)² + h
In this case:
- The vertex is still (h, k).
- The focus is located at (h + 1/(4a), k). The x-coordinate changes and the y-coordinate remains the same.
Example:
x = (1/4)(y-2)^2 + 1
- Vertex: (1,2)
- a = 1/4
- 1/(4a) = 1/(4*(1/4)) = 1
- Focus: (1+1, 2) = (2,2)
Summary
Finding the focus of a parabola involves identifying the vertex and using the 'a' value in the appropriate formula, depending on whether the parabola has a vertical or horizontal axis of symmetry. Always remember to express the parabola in either vertex or standard form to easily derive the vertex and the 'a' value.
