find and sketch important points ( i.e, x, y intercept, vertex, etc.)3x^2 - 2y^2-6x-12y-27=0

To sketch  3x^2 -2y^2-6x-12y-27=
0.

We bring this to the standard form (x-h)a^2+(y-k)^2/b^2 =
1.

3x^2-6x  - (2y^2 +12y) =
27.

3(x^2-2y )  -  2(y^2 +6y) =
27.

3(x^2-2y+1)-3 - 2(y^2+6y+3^2) + 18 =
27

3(x-1)^2 -2(y+1)^2 = 27 -18+3 =
12.

3(x-1)^2/(12) - 2(y+1)/12 =
12.

(x-1)/(12/3) - 2(y+1)/(12/2) =
1.

(x-1)^2/2^2 - (y+1)^2 / 6 = 1.

This
is a hyperbola of the form (x-h)^2/a^2 -(y-k)^2/k^2 = 1.

Its centre
is (h,k) = (1, -1).

The length of axes are 2a and 2b  and in our
case 2*2 = 4 and 2sqrt6.

The vertices are at (1+4 ,-1 ) and (1-4 ,
-1) or at (5,-1) and (-3 , -1).

Accentricity is e given by e^2 =
1+b^2/a^2 = 1+6/4 = 5/2. So e = sqrt(10/4) = +or- sqrt(5/2)

Focii S
ans S' are  are at  (ae+h , k) = (4sqrt(5/2) + 1 , -1) = (10 , -1)  and S' = (-ae+1 , -1) = (-
4sqrt(5/2) , -1).

The equations of the directrices : x = a/e Or x=
4/sqrt(5/2) . And x = -a/e . Or x = -4/sqrt(5/2).

The x intercepts
are given by putting y = 0 and solving for x in the given equation of the
hyperbola:

3x^2-6x-27 = 0.

x^2 -2x-9 =
0

x1 = {2+sqrt(2^2+36}/2 = 1+2sqrt10
and

x2 = 1- 2sqrt10.

y intercepts are
got by putting y = 0 in the given parabola and solving for
y:

-2y^2-12y-27 = 0.

2y^2+12y+27 =
0.

 y1 = {-12 +sqrt(12^2-4*27)}/2*2 = - 3 + 3sqrt
3.

 y2 = -(3+3sqrt3).

How did the Constitution itself reflect issues concerning distribution of power?

The Constitution tried to distribute power out as much as
possible -- to give power to as many different groups or people as possible.  This is
one of the fundamental ideas of American Government.

For
example, the Constitution gave some amount of legislative power to each branch of
government.  The Congress gets to pass the laws, but the President may veto them if he
(or someday she) sees fit to do so.  The judicial branch can even get in on the act by
saying that a law passed by Congress and signed by the President is unconstitutional and
therefore invalid.

The Constitution also gave power to the
states as well as to the federal government.  This, too, was meant to make sure that
there would be various centers of power so that no one part of the government got too
much power.

I need to find the recurrence for integral from 0 to pi/2 sin^n(x) * cos(x) dx.I made int 0-pi/2 sin^n(x) *cosx dx=int sin^n (x) *...

We write the integral as:

Int
(sin x)^n*cos xdx

We notice that the derivative of sin x is
cos x. We'll substitute:

sin x =
t

We'll differentiate both
sides:

cos xdx = dt

We'll
re-write the integral in t:

Int t^n*dt = t^(n+1)/(n+1) +
C

But t = sin x

Int (sin
x)^n*cos xdx = (sin x)^(n+1)/(n+1) + C

We'll determine F(b)
for b = pi/2:

F(pi/2) = (sin
pi/2)^(n+1)/(n+1)

F(pi/2) =
1^(n+1)/(n+1)

F(pi/2) =
1/(n+1)

F(0) = 0, for sin 0 =
0

Int (sin x)^n*cos xdx = F(pi/2) -
F(0)

Int (sin x)^n*cos xdx =
1/(n+1)

In Oedipus Rex, select an object and explain how that object could be used to symbolize one of the following characters: Oedipus, Creon, Iocaste,...

I love this question! It really helps you to think
creatively about the various characters in the play and their function. I would pick
Teiresias and the object I would pick would be a mirror. Let me explain: in the play, as
a prophet, he is called upon to explain to Oedipus the reason for the plague that is
causing so much trouble in Thebes and to point a finger at who is responsible. In
response to his reluctance to reveal his information, Oedipus goads him, insults him and
accuses him of betraying him. Yet, in a highly significant dialogue, Teiresias again and
again confronts Oedipus with the truth in Scene 1, and yet Oedipus is blind to it or
does not want to accept the truth of Teiresias's words. Consider the following words
from Teiresias:

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I say you live in hideous shame with
those

Most dear to you. You cannot see the
evil.

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You are the madman. There is no one
here

Who will not curse you soon, as you curse
me.

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But I say that you, with both your eyes, are
blind:

You cannot see the wretchedness of your
life,

Nor in whose house you live, no, nor with
whom.

Note how in each of
these exchanges Teiresias takes what is thrown at him by a desperate Oedipus and turns
it back on to him, confronting him with the unyielding truth just as a mirror confronts
us with our own reflection which cannot be ignored or denied. The tragedy of this play
lies in the fact that Oedipus is only able to accept the reality of these "reflections"
at the end.

What do the illustrations of slavery and freedom suggest about slavery, abolition and freedom?

Photographs or any visual representation of slavery can have
incredible effects on the viewer that other representations, particularly text, cannot have. One
might describe in gruesome detail the horrible practices of abuse and torture that existed on
plantations in the United States or in Jamaica or in South America, anywhere that slavery
existed, and the horror is palpable and disturbing. But they lack the power and focus of visual
images for a variety of reasons.

One, and it was mentioned in the
previous answer, is the way that a visual representation allows the viewer to identify as another
human being with the slave or slaves in the picture. Sometimes this can inspire more empathy than
any description or account, no matter how well written. As visual animals, humans tend to have
more of a visceral reaction to visuals than to other forms of
presentation.

The second important thing that pictures can
contribute are the incredible contrasts between the manner and posture of people enslaved and
free people. The juxtaposition of free man and slave that often occurs in visual representations
of slavery is often an extremely powerful one because the differences are often understated but
also often quite powerful.

Why might a business invest in another company's stock?

A major reason for one firm to buy the stock of another
firm is as a means to move towards merging with or acquiring that
company.

It is not unusual for two firms to merge or for
one to acquire the other.  Firm A, for example, might believe that it can make more
money if it acquires Firm B.  It might believe that it can acquire Firm B, fire many of
its workers (by getting rid of people who are duplicated in Firm A's work force), and
still produce as much revenue as A and B together used to
produce.

As a step towards acquiring Firm B, Firm A would
typically try to buy up as many of Firm B's shares as it could.

What is Thomas Randolph's "A Song" about?

Thomas Randolph's "A Song," is a poem about
nature.

The speaker calls on Music—personifying it, as if
it could create music on an instrument)—and asks that it ("the queen of souls") play a
song on its "powerful lute" (a musical instrument), but asks for a sad "requiem" (which
is a Mass said in the hope that a soul that has passed on will rest
peacefully).

However, the direction and action of the poem
changes when the speaker then asks Music to 'run gentle, nimble hands' over the strings
to change the tempo and mood of the sounds it creates so that even the forest will come
alive: the trees of various kinds (cedar, elm, palm, myrtle, and oak) will dance in a
"jolly train" or line.

Until, and here the tempo and mood
of the poem changes a second time, a sad note is played and the trees become "fixed" and
unmoving once again.

By the way, the poem is written in
rhyming couplets: pairs of lines are joined that have the same rhyme at the end of their
lines ("string" and "sing").