To determine the standard form of the equation of a parabola, we need additional information such as the vertex and the focus of the parabola.
Given that the parabola passes through the point (2, -4), we can infer that this point lies on the parabola. However, without any other information, we cannot determine the standard form of the equation of the parabola.
What is the correct standard form of the equation of the parabola?
Enter your answer below. Be sure to show each step of your work.
Where the parabola is (2,-4)
3 answers
the diretix is (2,-5)
To find the equation of a parabola given its vertex (h, k) and directrix, we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and (h, k + p) is the focus, and p is the distance between the vertex and the focus.
The given vertex is (2, -4), and the given directrix is the point (2, -5).
Since the directrix is a horizontal line, the parabola opens either upwards or downwards. Let's assume it opens upwards.
The vertex form of the equation becomes:
(x - 2)^2 = 4p(y + 4)
To find the value of p, we need to calculate the distance between the vertex and the directrix. Since the parabola opens upwards, p is positive and is equal to the distance between the vertex and the directrix.
Using the distance formula, we have:
p = |-5 - (-4)| = |-5 + 4| = |1| = 1
Substituting this value of p into the equation, we get:
(x - 2)^2 = 4(1)(y + 4)
(x - 2)^2 = 4y + 16
x^2 - 4x + 4 = 4y + 16
Expanding and rearranging the equation, we have:
x^2 - 4x - 4y - 12 = 0
Therefore, the standard form of the equation of the parabola with the vertex (2, -4) and directrix (2, -5) is:
x^2 - 4x - 4y - 12 = 0
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and (h, k + p) is the focus, and p is the distance between the vertex and the focus.
The given vertex is (2, -4), and the given directrix is the point (2, -5).
Since the directrix is a horizontal line, the parabola opens either upwards or downwards. Let's assume it opens upwards.
The vertex form of the equation becomes:
(x - 2)^2 = 4p(y + 4)
To find the value of p, we need to calculate the distance between the vertex and the directrix. Since the parabola opens upwards, p is positive and is equal to the distance between the vertex and the directrix.
Using the distance formula, we have:
p = |-5 - (-4)| = |-5 + 4| = |1| = 1
Substituting this value of p into the equation, we get:
(x - 2)^2 = 4(1)(y + 4)
(x - 2)^2 = 4y + 16
x^2 - 4x + 4 = 4y + 16
Expanding and rearranging the equation, we have:
x^2 - 4x - 4y - 12 = 0
Therefore, the standard form of the equation of the parabola with the vertex (2, -4) and directrix (2, -5) is:
x^2 - 4x - 4y - 12 = 0
1 answer