LESSON 13.1
Skills Practice
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The Coordinate Plane
Circles and Polygons on the Coordinate Plane
Problem Set
Use the given information to show that each statement is true. Justify your answers by using theorems
and by using algebra.
1. The center of circle O is at the origin. The
coordinates of the given points are A(⫺4, 0),
B(4, 0), and C(0, 4). Show that n ABC is a right triangle.
y
8
6
n ABC is an inscribed triangle in circle O with the
hypotenuse as the diameter of the circle, therefore the
triangle is a right triangle by the Right Triangle Diameter
Theorem.
___________________
AB 5 √ (24 2 4)2 1 (0 2 0)2
_____
___
5 √ (28)2 5 √ 64 5 8
4 C
2
A
–8
–6
–4
B
O
–2
0
2
4
x
6
8
–2
–4
D
–6
___________________
AC 5 √(24 2 0)2 1 (0 2 4)2
–8
_____________
5 √ (24)2 1 (24)2
___
__
5 √ 32 5 4√2
_________________
BC 5 √ (4 2 0)2 1 (0 2 4)2
__________
5 √ 42 1 (24)2
___
__
5 √ 32 5 4√2
__
__
© Carnegie Learning
(4√ 2 )2 1 (4√ 2 )2 0 82
32 1 32 5 64
Therefore, by the Converse of the Pythagorean Theorem, nABC is a right triangle.
Chapter 13 Skills Practice
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LESSON 13.1
Skills Practice
page 2
‹___›
2. The
center of circle O is at the origin. AZ and
‹___›
AT are tangent to circle O. The coordinates of the given
points are ___
A(⫺10, 30),
T(8, 6), and Z(⫺10, 0). Show that the
___
lengths of AT and AZ are equal.
y
A 30
20
10
O
Z
–30
–20
T
0
–10
x
10
20
30
–10
–20
–30
‹___›
3. The center of circle C is at the origin. AB is
tangent to circle C
at (3, 4). Show that the tangent line is
___
perpendicular to CA .
y
8
B (–1, 7)
6
4
A (3, 4)
2
–10 –8
–6
–4
–2
2
4
0
–2 C (0, 0)
6
8
x
10
–4
–8
13
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Chapter 13 Skills Practice
© Carnegie Learning
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LESSON 13.1
Skills Practice
page 3
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4. The center of circle O is at the origin. The coordinates
of the given points are A(3, 4), B(⫺4, ⫺3), D(0, ⫺5), E(⫺3, 4),
and F(⫺1.5, ⫺0.5). Show that EF ⭈ FD ⫽ AF ⭈ FB.
10
y
8
6
E
A
4
2
O
–10 –8
–6
–4
B
–2 F0
–2
2
4
6
8
x
10
–4
–6 D
–8
© Carnegie Learning
–10
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LESSON 13.1
Skills Practice
page 4
5. The center of circle O is at the origin. The coordinates
of the given points are A(⫺5, 5), B(⫺4, 3), C(0, ⫺5), and
D(0, 5). Show that AD2 ⫽ AB ⭈ AC.
10
y
8
6
A
D
4
B
2
O
–10 –8
–6
–4
–2
0
2
4
6
8
x
10
–2
–4
C
–6
–8
© Carnegie Learning
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Chapter 13 Skills Practice
LESSON 13.1
Skills Practice
page 5
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___
6. The
center of circle O is at the origin. BD is perpendicular
___
to OA at point C. The coordinates of the given points are
A(⫺5,
0), B(⫺4,
3), C(⫺4, 0), and D(⫺4, ⫺3). Show that
___
___
OA bisects BD .
10
y
8
6
4
B
A
–10 –8
–6
2
O
C
–4
–2
0
2
4
6
8
x
10
–2
D
–4
–6
–8
© Carnegie Learning
–10
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Chapter 13 Skills Practice
879
LESSON 13.1
Skills Practice
page 6
Classify the polygon formed by connecting the midpoints of the sides of each quadrilateral. Show all
your work.
7. The rectangle shown has vertices A(0, 0), B(x, 0),
C(x, y), and D(0, y).
___
x, 0
Midpoint AB : U __
2
(
y
)
___
y
Midpoint BC : S x, __
2
( )
___
x, y
Midpoint CD : T __
2
___
y
Midpoint DA : R 0, __
2
(
T
D (0, y)
)
R
( )
A (0, 0)
y
y
y 2 __ __ y
2 5 __
2 5 __
Slope RT 5 ______
x
_x_ 2 0
_x_
2
2
y
y
0 2 __ 2 __ y
___ ______
2 __
Slope US 5 x 2 5 ____
x 5x
__ 2 x
2__
2
2
___
C (x, y)
S
x
U
B (x, 0)
y
y
__
2 y 2__
y
2
2 5 2__
5 ____
Slope TS 5 ______
x
x
x
__
_
_
x2
___
2
2
y
y
__
__
20
___ ______
y
2 5 2__
Slope RU 5 2 x 5 ____
x
x
__
_
_
2
02
2
2
___
___
y
RT and US are parallel since they have the same slope of __
x.
___
___
y
TS and RU are parallel since they have the same slope of 2__
x.
There are no perpendicular sides. The slopes are not opposite reciprocals.
__________________
y
1 __ 2 y
2
√( 2 ) ( 2 )
y
x
5 √ ( 2__ ) 1 ( 2__ )
2
2
RT 5
2
_____________
2
__2
4
__________________
x 2 _x_
13
y
1 __ 2 0
2
√( 2 ) ( 2 )
y
5 √ ( _x_ ) 1 ( __ )
2
2
US 5
2
__________
2
2
__
2
2
__
2
_______
y
x2 1 __
__
√4
__
__
_______
5
√( 2x 2 x ) 1 ( y 2 2y )
y
x
5 √( 2 ) 1 ( )
2
2
y
5 √x 1
4
4
TS 5
____________
2
2
__________________
2
_______
y2
x2 1 __
5 __
4
4
√
2
__
__________________
√( 0 2 _2x_ ) 1 ( __2y 2 0 )
y
x
5 √ ( 2__ ) 1 ( __ )
2
2
2
RU 5
2
____________
2
2
_______
y2
x2 1 __
5 __
4
4
√
All four sides of RTSU are congruent.
Opposite sides are parallel and all sides are congruent, so the quadrilateral formed by connecting
the midpoints of the rectangle is a rhombus.
880
Chapter 13 Skills Practice
© Carnegie Learning
x
0 2 __
LESSON 13.1
Skills Practice
page 7
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8. The isosceles trapezoid shown has vertices
A(0, 0), B(6, 0), C(4, 3), and D(2, 3).
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y
D (2, 3)
C (4, 3)
x
B (6, 0)
© Carnegie Learning
A (0, 0)
13
Chapter 13 Skills Practice
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LESSON 13.1
Skills Practice
page 8
9. The parallelogram shown has vertices A(4, 5),
B(7, 5), C(3, 0), and D(0, 0).
10
y
8
6
A
B
4
2
–10 –8
–6
–4
–2
D
0
2 C4
6
8
x
10
–2
–4
–6
–8
© Carnegie Learning
–10
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Chapter 13 Skills Practice
LESSON 13.1
Skills Practice
page 9
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10. The rhombus shown has vertices A(4, 10), B(8, 5),
C(4, 0), and D(0, 5).
10
y
A
8
6
D
B
4
2
C
–10 –8
–6
–4
–2 0
–2
2
4
6
8
x
10
–4
–6
–8
© Carnegie Learning
–10
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Chapter 13 Skills Practice
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LESSON 13.1
Skills Practice
page 10
11. The square shown has vertices A(0, 0), B(4, 0),
C(4, 4), and D(0, 4).
10
y
8
6
D
C
4
2
B
–10 –8
–6
–4
–2
0A
2
4
6
8
x
10
–2
–4
–6
–8
© Carnegie Learning
–10
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Chapter 13 Skills Practice
LESSON 13.1
Skills Practice
page 11
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12. The kite shown has vertices A(0, 2), B(2, 0),
C(0, ⫺ 3), and D(⫺2, 0).
10
y
8
6
4
2 A
D
–10 –8
–6
–4
B
–2
0
2
4
6
8
x
10
–2
–4
C
–6
–8
© Carnegie Learning
–10
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Chapter 13 Skills Practice
885
© Carnegie Learning
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886
Chapter 13 Skills Practice
LESSON 13.2
Skills Practice
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Bring On the Algebra
Deriving the Equation for a Circle
Problem Set
Write an equation in standard form of each circle.
1. a circle with center point at the origin when r 5 4
x2 1 y2 5 r2
x2 1 y2 5 42
x2 1 y2 5 16
2
2. a circle with center point at the origin when r 5 __
3
© Carnegie Learning
3. a circle with center point (6, 5) when r 5 1
4. a circle with center point (28, 212) when r 5 7
13
Chapter 13 Skills Practice
887
LESSON 13.2
Skills Practice
5.
10
page 2
y
8
6
4
2
(4, 0)
0
210 28 26 24 22
2
4
6
8
x
10
6
8
x
10
22
24
(4, 24)
26
28
210
6.
10
(23, 7)
y
8
6
4
(0, 3)
2
210 28 26 24 22 0
22
2
4
24
26
28
13
Write the equation of each circle in standard form. Then identify the center point and radius of the circle.
7. x2 1 y2 1 6x 2 2y 1 1 5 0
x2 1 y2 1 6x 2 2y 1 1 5 0
x2 1 6x 1 y2 2 2y 5 21
(x2 1 6x 1 9) 1 (y2 2 2y 1 1) 5 21 1 9 1 1
(x 1 3)2 1 (y 2 1)2 5 9
center: (23, 1), radius: 3
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Chapter 13 Skills Practice
© Carnegie Learning
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LESSON 13.2
Skills Practice
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8. x2 1 y2 2 14x 1 4y 1 49 5 0
9. x2 1 y2 2 2x 2 2y 1 1 5 0
© Carnegie Learning
10. 81x2 1 81y2 1 36x 2 324y 1 327 5 0
11. 9x2 1 9y2 1 72x 2 12y 1 147 5 0
13
Chapter 13 Skills Practice
889
LESSON 13.2
Skills Practice
page 4
12. 36x2 1 36y2 2 36x 1 72y 1 29 5 0
Determine if each equation represents a circle. If so, describe the location of the center and radius.
13. x2 1 y2 1 4x 1 4y 2 17 5 0
x2 1 y2 1 4x 1 4y 2 17 5 0
x2 1 4x 1 y2 1 4y 5 17
(x2 1 4x 1 4) 1 (y2 1 4y 1 4) 5 17 1 4 1 4
(x 1 2)2 1 (y 1 2)2 5 25
center: (22, 22), radius: 5
14. 2x2 1 y2 1 2x 2 6y 1 6 5 0
13
16. x2 1 y2 2 8x 2 10y 1 5 5 0
890
Chapter 13 Skills Practice
© Carnegie Learning
15. x2 1 4y2 2 3x 1 3y 2 9 5 0
LESSON 13.2
Skills Practice
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Determine an equation of the circle that meets the given conditions.
17. Same center as circle A, (x 1 3)2 1 (y 1 5)2 5 9, but with a circumference that is twice that of circle A
__
The radius of circle A is √ 9 , or 3. To calculate the circumference of A, substitute 3 for r in the
formula for the circumference of a circle.
C 5 2␲r
5 2␲(3)
C 5 6␲
A circle with twice the circumference of circle A has circumference 2(6␲), or 12␲ units. To
calculate its radius, substitute 12␲ for C in the formula for the circumference of a circle, and then
solve for r.
C 5 2␲r
12␲ 5 2␲r
65r
The radius of the circle is 6. So an equation of the circle with the same center as circle A but with
a circumference that is twice that of circle A is
(x 1 3)2 1 (y 1 5)2 5 62, or (x 1 3)2 1 (y 1 5)2 5 36.
© Carnegie Learning
18. Same center as circle B, (x 2 6)2 1 (y 1 4)2 5 49, but with a circumference that is three times that of
circle B
13
Chapter 13 Skills Practice
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© Carnegie Learning
13
892
Chapter 13 Skills Practice
LESSON 13.3
Skills Practice
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Is That Point on the Circle?
Determining Points on a Circle
Problem Set
For each circle A, determine whether the given point P lies on the circle. Explain your reasoning.
1.
10
a2 1 b2 5 c2
y
8
52 1 32 5 r2
6
25 1 9 5 r2
4
34 5 r2
P (5, 3)
2
A (0, 0)
210 28 26 24 22 0
22
r
25
___
r 5 √ 34 < 5.8
3
4
6
8
x
10
24
26
Because the length of line segment AP is
approximately 5.8 units instead of 6 units, line
segment AP is not a radius of circle A; therefore,
point P does not lie on circle A.
28
210
2.
10
y
P (6, 8)
8
6
© Carnegie Learning
4
2
A (0, 0)
210 28 26 24 22
0
22
2
4
6
8
x
10
13
24
26
28
210
Chapter 13 Skills Practice
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LESSON 13.3
Skills Practice
3.
10
page 2
y
8
P (5, Œ39)
6
4
2
A (0, 0)
210 28 26 24 22
0
22
2
4
6
8
x
10
8
x
10
24
26
28
210
10
4.
y
8
P (4, 7)
6
A (0, 4)
4
210 28 26 24 22
0
22
2
24
26
13
894
28
210
Chapter 13 Skills Practice
4
6
© Carnegie Learning
2
LESSON 13.3
Skills Practice
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5.
y
5
P (3, 112Œ3 )
4
3
2
1
A (1, 1)
25 24 23 22 21
0
21
1
2
3
4
5
x
22
23
24
25
6.
10
y
P (7, 8)
8
6
4
2 A (2, 3)
© Carnegie Learning
210 28 26 24 22
0
2
4
6
8
x
10
22
24
26
13
28
210
Chapter 13 Skills Practice
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LESSON 13.3
Skills Practice
page 4
Use symmetry to determine the coordinates of each labeled point on the circle. Give exact values,
not approximations.
7.
10
y
8.
20
16
8
12
6
4
P (0, 4)
4
B (0, 0)
210 28 26 24 22
0
2
(4, 0)
4
6
8
B (6, 0)
x
10
28
216
210
220
y
10.
10
27
16
x
20
y
(313Œ2, 413Œ2 )
8
21
6
(3, 15)
C (3, 4)
4
9
2
3
227 221 215 29 23 0
12
8
28
(0, 24)
212
15
4
24
26
9.
C(26, 3)
0
220 216 212 28 24
22
24
P (17, 5Œ3 )
8
2
(24, 0)
y
3
15
9
21
27
x
210 28 26 24 22 0
22
29
2
6
4
8
x
10
24
215
26
221
28
227
210
13
20
y
12.
16
16
12
12
8
8
4
(0, 0)
210 26 22 0
24
2
(14, 0)
6
10
14
28
216
18
22
26
x
30
212
(5, –12)
220
Chapter 13 Skills Practice
y
4
(0, 0)
28 24 0
24
28
212
896
20
216
220
(4, 7.5)
(14, 0)
4
8
12
16
20
24
x
© Carnegie Learning
11.
LESSON 13.4
Skills Practice
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The Parabola
Equation of a Parabola
Vocabulary
a. Ax2 1 Dy 5 0 or By2 1 Cx 5 0
2. parabola
b. x2 5 4py or y2 5 4px
3. focus of a parabola
c. describes the orientation of the curvature of the
parabola
4. directrix of a parabola
d. a set of points in a plane that are equidistant from
a fixed point and a fixed line
5. general form of a parabola
e. the maximum or minimum point of a parabola
6. standard form of a parabola
f. a set of points that share a property
7. axis of symmetry
g. a line that passes through the parabola and divides
the parabola into two symmetrical parts that are
mirror images of each other
8. vertex of a parabola
h. the fixed point from which all points of a parabola
are equidistant
9. concavity
i. the fixed line from which all points of a parabola
are equidistant
© Carnegie Learning
1. locus of points
13
Chapter 13 Skills Practice
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LESSON 13.4
Skills Practice
page 2
Problem Set
Determine the equation of the parabola.
1.
5
_________________
__________________
2
1 (1 2 y)2 5 √(x 2 x)2 1 (21 2 y)2
√(0 2 x)____________
_________
√x2 1 (1 2 y)2 5 √(21 2 y)2
y
4
x2 1 (1 2 y)2 5 (21 2 y)2
3
(x, y)
2
x2 1 1 2 2y 1 y2 5 1 1 2y 1 y2
x2 5 4y
1
(0, 1)
25 24 23 22 21 0
21
1
2
3
4
5
x
(x, –1)
22
23
24
25
2.
5
y
4
(x, y)
(3, y)
3
2
1
25 24 23 22 21 0
21
1
22
23
13
24
25
898
Chapter 13 Skills Practice
2
3
4
5
x
© Carnegie Learning
(23, 0)
LESSON 13.4
Skills Practice
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page 3
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3.
5
y
4
(x, y)
3
(22, y)
2
1
(2, 0)
25 24 23 22 21
0
2
3
4
5
1
2
(0, 20.5)
3
4
5
1
x
21
22
23
24
25
4.
5
y
4
3
2
(x, 0.5)
© Carnegie Learning
25 24 23 22 21
1
0
21
x
22
(x, y) 23
13
24
25
Chapter 13 Skills Practice
899
LESSON 13.4
Skills Practice
page 4
Identify the vertex, axis of symmetry, value of p, focus and directrix for each parabola.
5.
y
5
25 24 23 22 21
Vertex: (0, 0)
4
Axis of symmetry: x 5 0
3
Value of p: 20.75
2
Focus: (0, 20.75)
1
Directrix: y 5 0.75
0
1
21
2
3
x
4
5
8
x
10
(0, 27.5)
22
23
24
25
6.
10
y
8
6
4
2
(6, 0)
0
210 28 26 24 22
22
2
4
6
24
28
210
13
900
Chapter 13 Skills Practice
© Carnegie Learning
26
LESSON 13.4
Skills Practice
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page 5
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7.
y
10
8
6
4
Parabola c
2
0
210 28 26 24 22
2
22
6
4
y 5 21
8
x
10
24
26
28
210
8.
10
y
8
6
4
x52
2
© Carnegie Learning
210 28 26 24 22
0
2
4
6
8
10
22
24
26
13
28
210
Chapter 13 Skills Practice
901
LESSON 13.4
Skills Practice
page 6
Sketch each parabola.
9. x2 5 24y
y
5
4
3
2
1
0
25 24 23 22 21
1
2
3
4
5
x
21
22
23
24
25
10. y2 5 16x
10
y
8
6
4
2
210 28 26 24 22
0
2
4
6
8
x
10
22
26
13
902
28
210
Chapter 13 Skills Practice
© Carnegie Learning
24
LESSON 13.4
Skills Practice
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page 7
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11. y2 5 220x
10
y
8
6
4
2
210 28 26 24 22
0
2
4
6
8
x
10
2
4
6
8
x
10
22
24
26
28
210
12. x2 5 6y
10
y
8
6
4
© Carnegie Learning
2
210 28 26 24 22 0
22
24
13
26
28
210
Chapter 13 Skills Practice
903
© Carnegie Learning
13
904
Chapter 13 Skills Practice
LESSON 13.5
Skills Practice
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Simply Parabolic
More With Parabolas
Problem Set
Determine the equation of the parabola.
1.
__________________
2
x2 1 (22 2 y)2 5 (x 2 2)2
1
(x, y)
0
25 24 23 22 21
_________________
2
1 (22 2 y)2 5 √ (x 2 2)2 1 (y 2 y)2
√(0 2 x)_____________
_______
√x2 1 (22 2 y)2 5 √(x 2 2)2
y
3
3
(2, y)
4
5
x
x2 1 4 1 4y 1 y2 5 x2 2 4x 1 4
y2 2 4x 1 4y 5 0
(0, 22)
22
23
24
25
26
27
2.
10
y
8
6
(3, 5)
© Carnegie Learning
4
(x, y)
2
0
210 28 26 24 22
(x, 23)
2
4
6
8
x
10
13
22
24
26
28
210
Chapter 13 Skills Practice
905
LESSON 13.5
3.
10
Skills Practice
page 2
y
8
6
4
2
26 24 22
(5, 1)
0
2
4
6
8 10
(x, y)
22
12
x
14
4
6
24
(x, –5)
26
28
210
4.
10
y
8
(x, 6)
6
4
2
(x, y)
214 212 210 28 26 24 22 0
22
(23, 24)
2
x
24
28
210
13
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Chapter 13 Skills Practice
© Carnegie Learning
26
LESSON 13.5
Skills Practice
page 3
Name
me
e
Date
ate
te
Identify the vertex, axis of symmetry, value of p, focus and directrix for each parabola.
5.
14
y
Vertex: (3, 4)
12
Axis of symmetry: x 5 3
10
Value of p: 2
Focus: (3, 6)
8
Directrix: y 5 2
(3, 6)
6
4
2
0
26 24 22
2
4
6
8
10
12
6
8
x
14
22
24
26
6.
10
y
8
6
4
2
210 28 26 24 22
0
2
4
x
10
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(0, 22)
24
26
13
28
210
Chapter 13 Skills Practice
907
LESSON 13.5
Skills Practice
7.
12
page 4
y
10
8
6
4
2
212 210 28 26 24 22
y 5 22
0
2
4
6
8
x
22
24
26
28
8.
12
y
10
8
x55
6
4
2
212 210 28 26 24 22 0
22
2
4
6
8
x
26
28
13
908
Chapter 13 Skills Practice
© Carnegie Learning
24
LESSON 13.5
Skills Practice
page 5
Name
me
e
Date
ate
te
Complete the table for each equation. Then, plot the points and graph the curve on the coordinate plane.
9. x2 2 6x 2 y 1 11 5 0
y
x
y
10
1
6
8
3
2
5
6
x
y
12
6
4
2
28 26 24 22 0
22
2
4
6
8
10
x
12
24
26
28
10. y2 2 x 1 6y 1 8 5 0
10
y
8
21
6
3
4
3
2
© Carnegie Learning
210 28 26 24 22 0
22
2
4
6
8
x
10
24
26
13
28
210
Chapter 13 Skills Practice
909
LESSON 13.5
Skills Practice
page 6
11. x2 1 4x 1 y 1 5 5 0
y
10
x
8
24
6
22
4
y
0
2
210 28 26 24 22 0
22
2
4
6
8
x
10
24
26
28
210
12. y2 2 6x 2 y 1 11 5 0
10
y
x
8
21
6
1
4
y
1
2
0
2
26
910
6
8
x
10
22
24
13
4
28
210
Chapter 13 Skills Practice
© Carnegie Learning
210 28 26 24 22
LESSON 13.5
Skills Practice
page 7
Name
me
e
Date
ate
te
Rewrite the equations in standard form.
13. x2 2 4x 1 3y 1 10 5 0
14. y2 2 6x 2 6y 1 15 5 0
x2 2 4x 1 3y 1 10 5 0
x2 2 4x 5 23y 2 10
x2 2 4x 1 4 5 23y 2 10 1 4
(x 2 2)2 5 23(y 1 2)
16. y2 2 2x 1 6y 1 3 5 0
© Carnegie Learning
15. x2 1 2x 1 2y 1 1 5 0
13
Chapter 13 Skills Practice
911
© Carnegie Learning
13
912
Chapter 13 Skills Practice