To find the equation of a parabola given the vertex and the focus, we can start by identifying the key components from the given data.
In this case, the vertex is at the point (2, 5) and the focus is at (2, 6). Since the x-coordinates of both the vertex and focus are the same, we can conclude that the parabola opens either upwards or downwards.
Here’s a breakdown of the steps:
- Identify the vertex: The vertex is the point (h, k), which is (2, 5) in our case. Thus, h = 2 and k = 5.
- Determine the direction of opening: Since the focus (2, 6) is above the vertex, the parabola opens upwards.
- Calculate the distance (p): The distance from the vertex to the focus is denoted as p. Since our focus is at (2, 6) and our vertex is at (2, 5), we find p = 6 – 5 = 1.
- Formulate the equation: The standard form of the equation for an upward-opening parabola is (x – h)2 = 4p(y – k).
Substituting the values we have:
h = 2, k = 5, and p = 1.
Thus, the equation becomes:
(x – 2)2 = 4(1)(y – 5).
Simplifying this, we have:
(x – 2)2 = 4(y – 5).
Therefore, the equation of the parabola with vertex (2, 5) and focus (2, 6) is:
(x – 2)2 = 4(y – 5).