What are the verteix, focus, and directrix of the parabola with equation
y = x^2 - 6x + 15?
To find the vertex, focus, and directrix of a parabola with equation y = ax^2 + bx + c, we can use the following formulas:
Vertex: To find the x-coordinate of the vertex, we use the formula x = -b / (2a), and to find the y-coordinate of the vertex, we substitute the x-coordinate into the equation.
Focus: The focus of the parabola is given by the point (h, k + (1 / (4a))), where (h, k) is the vertex.
Directrix: The directrix of the parabola is a horizontal line y = k - (1 / (4a)), where (h, k) is the vertex.
For the given equation y = x^2 - 6x + 15:
a = 1, b = -6, c = 15
First, let's find the x-coordinate of the vertex:
x = -(-6) / (2 * 1) = 6 / 2 = 3
Next, let's find the y-coordinate of the vertex by substituting x = 3 into the equation:
y = (3)^2 - 6(3) + 15 = 9 - 18 + 15 = 6
Therefore, the vertex is (3, 6).
Next, let's find the focus:
The x-coordinate of the focus is h = 3, and the y-coordinate of the focus is k + (1 / (4a)).
k = 6 and a = 1, so the y-coordinate of the focus is 6 + (1 / (4 * 1)) = 6 + 1/4 = 6.25
Therefore, the focus is (3, 6.25).
Finally, let's find the directrix:
The directrix is a horizontal line y = k - (1 / (4a)).
k = 6 and a = 1, so the directrix is y = 6 - (1 / (4 * 1)) = 6 - 1/4 = 5.75
Therefore, the directrix is y = 5.75.
To summarize:
The vertex of the parabola is (3, 6), the focus is (3, 6.25), and the directrix is y = 5.75.