A parabola can be drawn given a focus of (7, -7) and a directrix of x=11. What can be said about the parabola? The Parabola has a vertex a ( ? , ? ), has a p-value of ? and it opens (Up, down, left, right)
y^2 = 4px has
vertex at (0,0)
focus at (p,0)
directrix at x = -p
Now for yours, the directrix is 4 units to the right of the focus, so p = -2
The vertex will be 2 units to the right, at (9,-7)
That makes the equation
(y+7)^2 = -8(x-9)
To determine the vertex, p-value, and direction of opening of the given parabola, we need to use the focus-directrix definition of a parabola. In this case, the given focus is (7, -7) and the directrix is x=11.
1. Focus-Directrix Definition: A parabola is the set of all points that are equidistant to the focus and the directrix.
2. For a parabola with a vertical axis, the vertex is the midpoint between the focus and the directrix. In this case, the directrix is a vertical line x=11, so the x-coordinate of the vertex will be 11.
3. To find the y-coordinate of the vertex, we subtract the same value from the y-coordinate of the focus and the directrix. The y-coordinate of the focus is -7, and the y-coordinate of the directrix remains the same. So the y-coordinate of the vertex is (-7-7)/2 = -14/2 = -7.
Therefore, the vertex of the parabola is (11, -7).
4. The p-value represents the distance between the focus and the vertex (or directrix and the vertex). From the given information, we can calculate the p-value by finding the distance between the focus and the vertex:
p-value = distance between (7, -7) and (11, -7) = 11 - 7 = 4.
Therefore, the p-value of the parabola is 4.
5. Finally, to determine the direction of opening, we look at the position of the focus with respect to the vertex. Since the given focus is at (7, -7) and the vertex is at (11, -7), the focus is to the left of the vertex. This means the parabola opens to the left.
In summary:
- The vertex of the parabola is (11, -7).
- The p-value of the parabola is 4.
- The parabola opens to the left.
To determine what can be said about the parabola given the focus (7, -7) and the directrix x=11, we can use the definition of a parabola.
The vertex of a parabola is equidistant from the focus and the directrix. Since the directrix is a vertical line x=11, the vertex will have the same x-coordinate and the y-coordinate of the focus. Therefore, the vertex is (11, -7).
The distance between the vertex and the focus is known as the p-value. The p-value represents the distance from the vertex to either the focus or the directrix. In this case, the distance from the vertex (11, -7) to the focus (7, -7) is 4. Therefore, the p-value is 4.
Based on the p-value, we can determine if the parabola opens up, down, left or right. Since the directrix x=11 is vertical, the parabola opens either upward or downward. In this case, the parabola will open downward since the focus is below the vertex. So, the parabola opens down.
To summarize:
- The vertex of the parabola is (11, -7).
- The p-value is 4.
- The parabola opens downward.