determine vertex, focus and directrix of parabola. then graph the parabola.
second equation:
y^2 -4y -12x= 8
Please help me this is do 2day bc i forget to turn it in to my teacher
12 x + 8 = y^2 - 4 y
add 4 to both sides
12 x + 12 = y^2 - 4 y + 4
so
(y-2)^2 = 12 (x+1) = 4*3 (x+1)
opens right
vertex at (-1,2)
focus to vertex = 3 so focus at (2,2)
directrix to vertex = 3 so directrix at x = -4
To determine the vertex, focus, and directrix of a parabola, we can start by rearranging the equation into the standard form of a parabola, which is given by:
(y - k)^2 = 4a(x - h)
Here, (h, k) represents the vertex of the parabola. Once we determine the values for a, the focus can be found at the point (h + a, k), and the directrix will be a horizontal line with the equation y = k - a.
Now let's work through the given equation step by step:
1. Start with the given equation: y^2 - 4y - 12x = 8.
2. We will complete the square for the y terms by adding 4 to both sides to balance the equation: y^2 - 4y + 4 = 12x + 8 + 4.
3. Simplify the right side: y^2 - 4y + 4 = 12x + 12.
4. Rewrite the left side as a perfect square: (y - 2)^2 = 12x + 12.
Now the equation is in standard form. By comparing this equation to (y - k)^2 = 4a(x - h), we can see that the vertex is (h, k) = (0, 2).
To determine the focus, we can use the formula (h + a, k), where a is the coefficient of x in the standard form equation. In our case, a = 12, so the focus is at (0 + 12, 2) = (12, 2).
The directrix is a horizontal line with the equation y = k - a. Substituting the values, we find y = 2 - 12, which simplifies to y = -10.
To graph the parabola, we can plot the vertex, focus, and directrix on a coordinate plane, and then sketch the curve of the parabola passing through these points.
The vertex (0, 2) represents the lowest point on the parabola. The focus (12, 2) will be located to the right of the vertex, and the directrix y = -10 will be a horizontal line below the vertex.
Here is a rough sketch of the parabola:
```
-10 -5 0 5 10 15 20
--------------------------------------------
-10|
|
-5 |
|
0 | •
|
5 |
|
10|
|
--------------------------------------------
```
Note: The actual graph may appear different based on the scaling and proportions of the axes.