provides a factorization of quadratics.
7. (10 pts.)
Very carefully sketch the graph of the equation
y
2
 x
2
= 1 below. [
See trt4g7.pdf.
]
y
x
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
TEST04/MAC1114
Page 4 of 4
8. (10 pts.)
Find all the complex cube roots of 3
1/2
+ i. Leave
your answer in polar form with the arguments given in degrees.
If z = 3
1/2
+ i, then we can write z in polar form as
z = 2[cos(30°) + sin(30°)i]. The three cube roots are
w
0
= 2
1/3
[cos(10°) + sin(10°)i],
w
1
= 2
1/3
[cos(130°) + sin(130°)i], and
w
2
= 2
1/3
[cos(250°) + sin(250°)i]
9. (5 pts.)
Find the vertex, focus, and directrix of the
parabola that has the equation given below.
y
2
 4y = x + 4.
By performing the usual algegraic magic you can transform
the equation above into the standard form equation
(y  2)
2
= 4(1/4)(x  (8)).
Using this, you can easily see that the vertex is (8,2), the
focus is (8 + (1/4), 2) = (31/4, 2), and the directrix is the
line defined by x = 8  (1/4) = 33/4.
10. (5 pts.)
Find the center, foci, and vertices of the ellipse
that has the equation given below.
4x
2
+ y
2
+ 4y = 0.
Again, by playing the completethesquare game carefully, you
should obtain the standard form equation
(x  0)
2
+ (1/4)(y  (2))
2
= 1.
From this you should have c = (4  1)
1/2
. Clearly the center is
(0,2), the two vertices are (0,0) and (0,4), and the two foci
are (0,2 + 3
1/2
) and (0,2  3
1/2
).
11. (5 pts.)
Find the center, foci, and vertices of the
hyperbola that has the equation given below.
y
2
 4x
2
16x  2y  19 = 0
Finally, by being very puntilious in your algebraic
prestidigitation, you can obtain the standard form equation
(1/4)(y  1)
2
 (x  (2))
2
= 1.
The center is (2,1), the two vertices are (2,3) and (2,1),
and since c = (4 + 1)
1/2
, the two foci are (2, 1 + 5
1/2
) and
(2, 1  5
1/2
).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Storfer
 Cartesian Coordinate System, 5 pts, Polar coordinate system, 15 pts, 10°, prestidigitation

Click to edit the document details