Write the standard equation for the parabola with the given characteristics.
vertex:(0,0)
directrix:y= -1.
The equation x^2 = 4 py is a parabola with vertex at (0,0) and a directrix below it at y = -p. You want p to be 1.
Therefore 4p = 1 and p = 1/4
y = x^2 /4 is the equation
To write the standard equation for a parabola, we need to determine its focus and its axis of symmetry.
The vertex of the parabola is given as (0,0), which means the axis of symmetry is the y-axis (vertical line passing through the vertex).
The directrix is given as y = -1, which means the focus of the parabola is at a distance of 1 unit above the directrix (since the parabola opens upwards).
The focus of the parabola is at (0, 1).
Now we can proceed with writing the standard equation for the parabola.
For a parabola with a vertical axis of symmetry, the standard equation can be written as:
(y - k) = 4p(x - h)^2
Where (h, k) represents the vertex of the parabola and p represents the distance between the vertex and the focus (also the distance between the vertex and the directrix).
Substituting the known values into the equation, we get:
(y - 0) = 4(1)(x - 0)^2
Simplifying:
y = 4x^2
Therefore, the standard equation for the parabola with the given characteristics is y = 4x^2.