6. Find the equation of each parabola described below.

a) parabola with vertex (0,0) and the focus (0,7)

b) parabola with focus (-3,0) and directrix x=3

c) parabola with vertex (3,3) and directrix x=-1

d) parabola with focus (-2,-1) and directrix y=5

e) ysquared=-6x translated according to ((x,y) -->(x-2,y+4))

could someone help me with these??

Note that the vertex is halfway between the focus and the directrix by the way.

You need to know the following:

vertex at (h,k)
if it opens up or down:
4 a (y-k) = (x-h)^2
a negative a opens down
if it opens right or left:
4 a (x-h) = (y-k)^2
a negative a opens left
distance vertex to focus = a
distance vertex to directrix = a
ALWAYS SKETCH THEM
================================
for example a is vertex (0,0) so h = k = 0 and focus (0,7)
the sketch shows it with axis along the y axis opening up and distance a = 7
so
4 * 7 * y = x^2
or y = x^2/28

Sure, I can help you with those. Let's go through each one step by step:

a) For a parabola with vertex (0,0) and focus (0,7), we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus.
Plugging in the given values, we have:
(x - 0)^2 = 4(7)(y - 0)
x^2 = 28y

So, the equation of the parabola is x^2 = 28y.

b) For a parabola with focus (-3,0) and directrix x = 3, we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus (or directrix).
Plugging in the given values, we have:
(x + 3)^2 = 4p(y - 0)

To find the value of p, we can use the distance formula between the focus and directrix:
p = (3 - (-3))/2 = 6/2 = 3
Substituting the value of p, we have:
(x + 3)^2 = 4(3)(y - 0)
(x + 3)^2 = 12y

So, the equation of the parabola is (x + 3)^2 = 12y.

c) For a parabola with vertex (3,3) and directrix x = -1, we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus (or directrix).
Plugging in the given values, we have:
(x - 3)^2 = 4p(y - 3)

To find the value of p, we can use the distance formula between the vertex and directrix:
p = (3 - (-1))/2 = 4/2 = 2
Substituting the value of p, we have:
(x - 3)^2 = 4(2)(y - 3)
(x - 3)^2 = 8(y - 3)

So, the equation of the parabola is (x - 3)^2 = 8(y - 3).

d) For a parabola with focus (-2,-1) and directrix y = 5, we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus (or directrix).
Plugging in the given values, we have:
(x + 2)^2 = 4p(y - (-1))

To find the value of p, we can use the distance formula between the focus and directrix:
p = (5 - (-1))/2 = 6/2 = 3
Substituting the value of p, we have:
(x + 2)^2 = 4(3)(y + 1)
(x + 2)^2 = 12(y + 1)

So, the equation of the parabola is (x + 2)^2 = 12(y + 1).

e) To translate the equation y^2 = -6x according to ((x,y) -->(x-2,y+4)), we can simply substitute (x-2) for x and (y+4) for y in the original equation:
(y+4)^2 = -6(x-2)

Expanding and simplifying, we have:
y^2 + 8y + 16 = -6x + 12

Rearranging terms, we get:
6x + y^2 + 8y + 4 = 0

So, the equation of the translated parabola is 6x + y^2 + 8y + 4 = 0.

I hope this helps! Let me know if you have any further questions.

Sure! I'd be happy to help you with these problems. Let's go through each question one by one and find the equation of each parabola.

a) For a parabola with vertex (0,0) and the focus (0,7), we can use the formula for the equation of a parabola with a vertical axis of symmetry:
(y - k)^2 = 4a(x - h)
where (h, k) is the vertex and a is the distance between the focus and the vertex. In this case, (h, k) = (0,0) and a = 7.

Substituting these values into the formula, we get:
(y - 0)^2 = 4(7)(x - 0)
y^2 = 28x

So, the equation of the parabola is y^2 = 28x.

b) For a parabola with focus (-3,0) and directrix x=3, we can use the formula for the equation of a parabola with a horizontal axis of symmetry:
(x - h)^2 = 4a(y - k)
where (h, k) is the vertex and a is the distance between the focus and the vertex. In this case, (h, k) = (-3,0) and a = 3.

Substituting these values into the formula, we get:
(x + 3)^2 = 4(3)(y - 0)
(x + 3)^2 = 12y

So, the equation of the parabola is (x + 3)^2 = 12y.

c) For a parabola with vertex (3,3) and directrix x=-1, we can use the formula for the equation of a parabola with a vertical axis of symmetry:
(y - k)^2 = 4a(x - h)
where (h, k) is the vertex and a is the distance between the focus and the vertex. In this case, (h, k) = (3,3) and a = 2.

Substituting these values into the formula, we get:
(y - 3)^2 = 4(2)(x - 3)
(y - 3)^2 = 8(x - 3)

So, the equation of the parabola is (y - 3)^2 = 8(x - 3).

d) For a parabola with focus (-2,-1) and directrix y=5, we can use the formula for the equation of a parabola with a horizontal axis of symmetry:
(x - h)^2 = 4a(y - k)
where (h, k) is the vertex and a is the distance between the focus and the vertex. In this case, (h, k) = (-2,-1) and a = 6.

Substituting these values into the formula, we get:
(x + 2)^2 = 4(6)(y - (-1))
(x + 2)^2 = 24(y + 1)

So, the equation of the parabola is (x + 2)^2 = 24(y + 1).

e) For the equation y^2 = -6x translated according to ((x,y) -->(x-2,y+4)), we can plug in the shifted values into the equation. The translation (x-2, y+4) moves the graph 2 units to the right and 4 units up.

Substituting these shifted values into the equation y^2 = -6x, we get:
(y+4)^2 = -6(x-2)
(y+4)^2 = -6x + 12

So, the equation of the translated parabola is (y+4)^2 = -6x + 12.

I hope this explanation helps! Let me know if you have any further questions.