With the given information below, write the standard form equation for the parabola.
1.) Vertex: (-1,2)
Focus(-1,0)
2.) Vertex: (-2,1)
Directrix x=1
the parabola is vertical, and opens downwards
standard form x^2 = 4py
but vertex is (-1,2)
(x+1)^2 = 4p(y-2)
but p = -2
(x+1)^2 = -8(y-2)
(x+1)^2 = -8y + 16
8y = -(x+1)^2 + 16
y = (-1/8)(x+1)^2 - 2
for the 2nd , use y^2 = 4px and follow the above steps
To find the standard form equation of a parabola, we need to use the coordinates of the vertex and either the focus or the directrix. Let's take a look at both cases:
1) Vertex: (-1,2) Focus: (-1,0)
To determine whether the parabola opens upwards or downwards, we can compare the y-coordinate of the vertex with the y-coordinate of the focus. Since the focus is below the vertex, the parabola opens downwards.
Step 1: Determine the equation for a downward-opening parabola:
(x - h)^2 = -4p(y - k)
Step 2: Substitute the vertex coordinates into the equation:
(x + 1)^2 = -4p(y - 2)
Step 3: Substitute the coordinates of the focus to find p:
(-1 + 1)^2 = -4p(0 - 2)
0 = -8p
Since p = 0, the equation simplifies further:
Step 4: Simplify the equation to standard form:
(x + 1)^2 = 0
Therefore, the standard form equation for the parabola is x^2 + 2x + 1 = 0.
2) Vertex: (-2,1) Directrix: x = 1
Similar to the previous case, we need to determine whether the parabola opens upwards or downwards. In this case, the directrix is a vertical line, so the parabola opens horizontally.
Step 1: Determine the equation for a horizontally-opening parabola:
(y - k)^2 = 4p(x - h)
Step 2: Substitute the vertex coordinates into the equation:
(y - 1)^2 = 4p(x + 2)
Step 3: Substitute the equation of the directrix to find p:
(y - 1)^2 = 4p(x + 2) = 4p(1 - h)
(y - 1)^2 = 4p(-1)
Since p = -1/4, the equation simplifies further:
Step 4: Simplify the equation to standard form:
4(y - 1)^2 = -(x + 2)
Therefore, the standard form equation for the parabola is 4y^2 - 8y + 4 = -(x + 2).