The equation of a parabola is 24y=(x-2)^2-48. what is the vertex, focus and directrix of the parabola
AAAaannndd the bot gets it wrong yet again!
24y=(x-2)^2-48
(x-2)^2 = 24(y+2)
So we have a vertical axis and p=6, giving us
vertex = (2,-2)
focus = (2,-2+6) = (2,4)
directrix: y = -2-6 = -8
I'm sorry for the mistake in my previous response and thank you for providing the correct solution. Yes, you are correct that the vertex is (2,-2), the focus is (2,4), and the directrix is y = -8.
To find the vertex, focus, and directrix of the parabola, we can rewrite the given equation in the standard form of a parabola, which is:
y = (1/4a)(x-h)^2 + k
where (h, k) represents the vertex, "a" is the distance between the vertex and the focus and directrix, and the focus is given by (h, k + a), while the directrix is the line y = k - a.
In the given equation, 24y = (x-2)^2 - 48, we can start by dividing both sides by 24 to simplify:
y = (1/24)(x-2)^2 - 2
Now we can compare this equation to the standard form equation to determine the values of (h, k, a).
Comparing the equations, we can see that h = 2 and k = -2.
Next, let's find the value of "a":
1/4a = 1/24
To simplify this equation, we can cross-multiply:
1 * 24 = 4a
24 = 4a
a = 24/4
a = 6
Therefore, the values of h, k, and a are h = 2, k = -2, and a = 6.
Now we can determine the vertex, focus, and directrix:
Vertex: The vertex of the parabola is given by (h, k), which is (2, -2).
Focus: The focus of the parabola is located at (h, k + a), which is (2, -2 + 6). So, the focus is at (2, 4).
Directrix: The directrix of the parabola is the vertical line y = k - a, which is y = -2 - 6. Therefore, the directrix is y = -8.
To summarize:
- Vertex = (2, -2)
- Focus = (2, 4)
- Directrix: y = -8
To find the vertex of the parabola, we can rewrite the equation in the standard form:
y = (1/24)(x-2)^2 - 2
The vertex of the parabola is then (2, -2).
To find the focus, we need to first determine whether the parabola opens up or down. Since the coefficient of x^2 is positive, the parabola opens upwards.
The formula for finding the focus of a parabola in standard form is:
F = (h, k + 1/4a)
Where (h, k) is the vertex and a is the coefficient of x^2.
Plugging in the values for our parabola, we get:
F = (2, -2 + 1/(4*1/24))
Simplifying, we get:
F = (2, -1)
So the focus of the parabola is located at (2, -1).
To find the directrix, we need to use the formula:
y = k - 1/4a
Plugging in the values for our parabola, we get:
y = -2 - 1/(4*1/24)
Simplifying, we get:
y = -3
So the directrix of the parabola is the horizontal line y = -3.