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What is an equation of a parabola with the given vertex and focus? Show all the steps that you use to solve this problem.

vertex: (-2,5); focus: (-2,6)

the axis is vertical, since the vertex and the focus are on x = -2

The vertex is 1 unit away from the focus, so
(x+2)^2 = 4(y - 5)

Did you know?

Did you know that the equation of a parabola with a given vertex and focus can be found by analyzing the horizontal or vertical shifts and stretching/compression of the standard parabola equation? To find the equation with the given vertex (-2,5) and focus (-2,6), we know that the vertex and focus share the same x-coordinate, which in this case is -2. This indicates that the parabola is symmetric along the y-axis. To determine the horizontal or vertical orientation of the parabola, we can compare the y-coordinate of the vertex with the y-coordinate of the focus. In this case, the y-coordinate of the vertex is 5, and the y-coordinate of the focus is 6. Since the y-coordinate of the focus is greater, the parabola opens upwards. Now, we can use the standard form of the parabola equation, which is y = a(x - h)^2 + k, where (h,k) represents the vertex. Substituting the vertex coordinates into the equation, we have y = a(x - (-2))^2 + 5, which simplifies to y = a(x + 2)^2 + 5. Since the parabola opens upwards, the coefficient "a" must be positive. To find the value of "a," we use the distance formula, which states that the distance between the vertex and focus is equal to 1/(4a). In this case, the distance between (-2,5) and (-2,6) is 1. Therefore, 1 = 1/(4a), implying that 4a = 1. Solving for "a," we find a = 1/4. Finally, substituting the value of "a" in the equation, we obtain the equation of the parabola: y = (1/4)(x + 2)^2 + 5. By following these steps, we can determine the equation of a parabola with a given vertex and focus.