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Precalculus
Convert equations of conic sections from general to standard form
In the right triangle shown, the length of
A
C
‾
=
5
\overline{AC} =5
A
C
=
5
and the length of
B
C
‾
=
7
\overline{BC} =7
BC
=
7
. What is the length of
A
B
‾
\overline{AB}
A
B
?
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Write the equation in standard form for the ellipse
x
2
+
7
y
2
=
21
x^2 + 7y^2 = 21
x
2
+
7
y
2
=
21
.
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Write the equation in standard form for the ellipse
x
2
+
8
y
2
=
16
x^2 + 8y^2 = 16
x
2
+
8
y
2
=
16
.
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Express the following in vertex form
\newline
f
(
x
)
=
x
2
+
6
x
−
2
f(x)=x^{2}+6 x-2
f
(
x
)
=
x
2
+
6
x
−
2
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A triangle has side lengths of
(
5.1
g
−
6.9
h
)
(5.1 g-6.9 h)
(
5.1
g
−
6.9
h
)
centimeters,
(
6.4
g
−
8.8
k
)
(6.4 g-8.8 k)
(
6.4
g
−
8.8
k
)
centimeters, and
(
9.3
k
+
3.4
h
)
(9.3 k+3.4 h)
(
9.3
k
+
3.4
h
)
centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
\newline
−
3.5
h
+
0.5
k
+
11.5
g
-3.5 h+0.5 k+11.5 g
−
3.5
h
+
0.5
k
+
11.5
g
\newline
−
1.8
g
h
−
2.4
g
k
+
12.7
h
k
-1.8 g h-2.4 g k+12.7 h k
−
1.8
g
h
−
2.4
g
k
+
12.7
hk
\newline
−
12.3
h
k
+
20.8
g
k
-12.3 h k+20.8 g k
−
12.3
hk
+
20.8
g
k
\newline
−
5.4
k
+
11.5
g
+
2.4
h
-5.4 k+11.5 g+2.4 h
−
5.4
k
+
11.5
g
+
2.4
h
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The graph shows the parabola
y
=
−
(
x
+
4
)
2
+
5
y=-(x+4)^2+5
y
=
−
(
x
+
4
)
2
+
5
. Consider the following linear equation:
y
=
−
3
x
+
b
y=-3x+b
y
=
−
3
x
+
b
for some constant
b
b
b
. If one of the solutions to the system of equations formed by the parabola and the linear equation is
(
−
4
,
5
)
(-4,5)
(
−
4
,
5
)
, which of the following is the other solution?
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Complete the square to re-write the quadratic function in vertex form:
\newline
y
=
x
2
+
9
x
+
1
y=x^{2}+9 x+1
y
=
x
2
+
9
x
+
1
\newline
Answer:
y
=
y=
y
=
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Complete the square to re-write the quadratic function in vertex form:
\newline
y
=
x
2
+
3
x
+
8
y=x^{2}+3 x+8
y
=
x
2
+
3
x
+
8
\newline
Answer:
y
=
y=
y
=
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Complete the square to re-write the quadratic function in vertex form:
\newline
y
=
x
2
+
9
x
+
10
y=x^{2}+9 x+10
y
=
x
2
+
9
x
+
10
\newline
Answer:
y
=
y=
y
=
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Write the equation in standard form for the hyperbola
−
x
2
+
4
y
2
−
100
=
0
-x^{2}+4y^{2}-100=0
−
x
2
+
4
y
2
−
100
=
0
.
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A hyperbola centered at the origin has vertices at
(
±
7
,
0
)
( \pm \sqrt{7}, 0)
(
±
7
,
0
)
and foci at
(
±
27
,
0
)
( \pm \sqrt{27}, 0)
(
±
27
,
0
)
.
\newline
Write the equation of this hyperbola.
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A hyperbola centered at the origin has vertices at
(
±
33
,
0
)
( \pm \sqrt{33}, 0)
(
±
33
,
0
)
and foci at
(
±
59
,
0
)
( \pm \sqrt{59}, 0)
(
±
59
,
0
)
.
\newline
Write the equation of this hyperbola.
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(
x
−
7
)
2
+
(
y
+
5
)
2
=
16
(x-7)^{2}+(y+5)^{2}=16
(
x
−
7
)
2
+
(
y
+
5
)
2
=
16
\newline
The given equation represents a circle in the
x
y
x y
x
y
-plane. If
(
x
,
−
5
)
(x,-5)
(
x
,
−
5
)
is a point on the circle, what is a possible value of
x
x
x
?
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(
x
+
4
)
2
+
(
y
+
2
)
2
=
25
(x+4)^{2}+(y+2)^{2}=25
(
x
+
4
)
2
+
(
y
+
2
)
2
=
25
\newline
The given equation represents a circle in the
x
y
x y
x
y
-plane. If
(
−
4
,
y
)
(-4, y)
(
−
4
,
y
)
is a point on the circle and
y
>
0
y>0
y
>
0
, what is the value of
y
y
y
?
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