The equation of a parabola is 12y=(x-1)^2-48. Identify the vertex, focus, and directrix of the parabola.
the parabola x^2 = 4py has
vertex at (0,0)
focus at (0,p)
directrix at y = -p
So. Let's rearrange things a bit.
12y=(x-1)^2-48
(x-1)^2 = 12(y+4)
So, we can see that we now have
vertex at (1,-4)
focus at y=(-4+3) = (1,-1)
directrix at y=(-4-3) or y = -7
to confirm our work, see
https://www.wolframalpha.com/input/?i=parabola+12y%3D(x-1)%5E2-48
To identify the vertex, focus, and directrix of the parabola with the equation 12y=(x-1)^2-48, we need to rewrite it in the standard form for a parabola.
Step 1: Expand the equation.
12y = (x-1)^2 - 48
12y = x^2 - 2x + 1 - 48
12y = x^2 - 2x -47
Step 2: Divide both sides by 12.
y = (1/12)x^2 - (1/6)x - (47/12)
Now we have the equation in the form y = ax^2 + bx + c, which is the standard form for a parabola.
Comparing the equation with y = ax^2 + bx + c, we can find the values of a, b, and c.
a = 1/12
b = -1/6
c = -47/12
The general form of a parabola is given as (x - h)^2 = 4a(y - k), where (h, k) is the vertex.
Step 3: Find the vertex.
To find the vertex, we need to complete the square. Considering only the x terms,
x^2 - 2x = (x - 1)^2 - 1.
Now we can rewrite the equation as:
y = (1/12)(x^2 - 2x) - (47/12)
= (1/12)((x - 1)^2 - 1) - (47/12)
= (1/12)(x - 1)^2 - 1/12 - (47/12)
= (1/12)(x - 1)^2 - 48/12 - 47/12
= (1/12)(x - 1)^2 - 95/12
The vertex of the parabola is (h, k) = (1, -95/12).
Step 4: Find the focus.
The focus of the parabola is given by the equation (h, k + 1/4a), where a = 1/12.
So, the focus is located at (1, -95/12 + 1/(4*(1/12)) = (1, -95/12 + 12)
= (1, -95/12 + 144/12)
= (1, 49/12).
Therefore, the focus of the parabola is at (1, 49/12).
Step 5: Find the directrix.
The directrix of the parabola can be found using the equation y = k - 1/4a.
For a = 1/12,
the directrix is given by y = -95/12 - 1/(4*(1/12))
= -95/12 - 12/4
= -95/12 - 36/12
= -131/12.
So, the equation of the directrix is y = -131/12.
To summarize:
- The vertex of the parabola is (1, -95/12).
- The focus of the parabola is (1, 49/12).
- The directrix of the parabola is y = -131/12.
Oh, the joys of parabolas! Now let's unravel this equation excitement together.
The equation of your parabola is given by 12y = (x - 1)^2 - 48. To identify the vertex, focus, and directrix, we need to put the equation into the standard form: y = a(x - h)^2 + k, where (h, k) represents the vertex.
So, let's rewrite the equation: 12y = (x - 1)^2 - 48
Dividing both sides by 12, we have y = (1/12)(x - 1)^2 - 4.
Now we can see that the vertex (h, k) is given by (1, -4).
To find the focus and directrix, we'll use the formula:
For a parabola of the form y = a(x - h)^2 + k,
the focus is at (h, k + 1/(4a)) and the directrix is y = k - 1/(4a).
In our equation, a = 1/12, h = 1, and k = -4.
So, the focus is located at (1, -4 + 1/(4*(1/12))) = (1, -4 + 3) = (1, -1).
And the directrix is given by y = -4 - 1/(4*(1/12)) = -4 - 3 = -7.
To sum it up, the vertex of your parabola is (1, -4), the focus is (1, -1), and the directrix is y = -7. Voilà!
To identify the vertex, focus, and directrix of the given parabola equation, let's first rewrite it in standard form. The standard form of a parabola equation is y^2 = 4px.
Given equation: 12y = (x - 1)^2 - 48
Dividing both sides of the equation by 12, we get:
y = (1/12)(x - 1)^2 - 4
Now, comparing this equation with y^2 = 4px, we can determine the values of p (which is the distance from the vertex to the focus) and the coordinates of the vertex.
Comparing the given equation with y^2 = 4px, we can deduce:
4p = 1/12
p = 1/48
Since p is positive, this means the parabola opens upward. Now, let's find the vertex.
The coordinates of the vertex (h, k) are given by: (h, k) = (h, -p)
Using the equation above, we can determine the x-coordinate of the vertex (h):
h - 1 = 0
h = 1
Substituting the x-coordinate (h) into the equation, we can find the y-coordinate (k) of the vertex:
k = -p
k = -1/48
Therefore, the vertex is (1, -1/48).
To find the focus, we use the formula (h, k + p):
The x-coordinate of the focus will be the same as the x-coordinate of the vertex since p is positive.
The focus is located at (1, -1/48 + 1/48).
This simplifies to the focus being at (1, 0).
To find the directrix, we use the formula y = k - p:
The directrix is given by the equation y = -1/48 - 1/48.
This simplifies to y = -1/24.
Therefore, the vertex is (1, -1/48), the focus is (1, 0), and the directrix is the line y = -1/24.
12y = (x-1)^2 - 48.
Vertex form: Y = a(x-h)^2 + k.
Y = (1/12)(x-1)^2 - 4.
V(h, k) = V(1, -4).
F(h, k+1/(4a)) = F(1, -1).
D(h, k-1/(4a)) = D(1, -7).