Final answer:
The equation of the parabola with the given vertex and focus is y = (1 / 8)(x - 4)^2 + 5.
Step-by-step explanation:
The equation of a parabola with a given vertex and focus can be found using the formula:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the vertex is (4,5) which means h = 4 and k = 5.
The focus of the parabola is (6,5) which means the distance from the vertex to the focus is 2 units. From this information, we can determine the value of a in the equation.
The equation becomes:
y = a(x - 4)^2 + 5
To find the value of a, we use the equation:
a = 1 / (4p)
where p is the distance from the vertex to the focus. In this case, p = 2, so a = 1 / (4 * 2) = 1 / 8.
Substituting the value of a in the equation, we get the final form of the equation:
y = (1 / 8)(x - 4)^2 + 5