Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Passes through the point (-5, 1/8); vertical axis.
There is no focus of the parabola or equation given, so how am I suppose to solve this problem?
Since the axis is vertical, you know it's
y = a(x-h)^2 + k
Since the vertex is at (0,0), you know that h=k=0, and thus
y = ax^2
1/8 = 25a
a = 1/200
so,
y = 1/200 x^2
All the data you needed was there. Just gotta read it carefully.
To determine the equation of the parabola, we need to use the vertex form equation for a parabola with a vertical axis:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.
Since the given vertex is at the origin (0, 0), the equation becomes:
x^2 = 4p(y - 0)
Simplifying further:
x^2 = 4py
Now, we need to find the value of p, which represents the distance from the vertex to the focus or directrix. The given information states that the parabola passes through the point (-5, 1/8).
Substituting the coordinates of the point into the equation, we get:
(-5)^2 = 4p(1/8)
25 = p/2
p = 50
Now, we can substitute the value of p into the equation to obtain the standard form of the equation of the parabola:
x^2 = 4(50)y
x^2 = 200y
So, the standard form of the equation of the parabola with the given characteristics and vertex at the origin, passing through the point (-5, 1/8), is x^2 = 200y.
To find the equation of a parabola in standard form, given its characteristics and vertex at the origin, you need to use the general equation of a parabola with a vertical axis of symmetry:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, since the vertex is at the origin, the equation becomes:
y = ax^2
To determine the value of 'a', we can use the given point on the parabola, (-5, 1/8). Substituting these coordinates into the equation:
1/8 = a(-5)^2
1/8 = 25a
To solve for 'a', multiply both sides of the equation by 8:
8 * (1/8) = 8 * 25a
1 = 200a
Divide both sides of the equation by 200:
1/200 = a
So, the value of 'a' is 1/200. Substituting this into the equation for 'a', the standard form of the equation of the parabola becomes:
y = (1/200)x^2
Therefore, the standard form of the equation of the parabola with a vertex at the origin and passing through the point (-5, 1/8) is y = (1/200)x^2.