DATE
NAME
7-1
Student Edition
Pages 408–414
Study Guide
Integration: Geometry
The Distance and Midpoint Formulas
For number lines, you can use absolute value and averages to
find distances and locate midpoints. You can do the same in the
coordinate plane, though to find distances you need to use the
Pythagorean theorem.
Number Line
A: coordinate a
B: coordinate b
Coordinate Plane
P: coordinates (x1, y1)
Q: coordinates (x2, y2)
2
2
distance: AB 5 ) a 2 b ) or ) b 2 a ) distance: PQ 5 Ï(x2 2 x1) 1 (y2 2 y1)
midpoint:
a1b
(average)
2
midpoint:
( x 12 x , y 12 y )
1
2
1
2
Example: Find the distance from P to Q and the midpoint of PQ if
P has coordinates (22, 7) and Q has coordinates (9, 3).
You can choose either point for (x1, y1).
Use the other point for (x2, y2 ). Let (x1, y1 ) be (22, 7).
Then (x2, y2 ) is (9, 3).
(x 12 x , y 12 y )
(22) 1 9 7 1 3
5( 2 , 2 )
7
5 (2, 5)
PQ 5 Ï(x2 2 x1)2 1 (y2 2 y1)2
midpoint of PQ 5
5 Ï(9 2 (22)) 2 1 (3 2 7) 2
5 Ï121 1 16
1
2
1
2
5 Ï137
Find the distance between each pair of points with the given
coordinates.
1. (23, 4), (6,211)
3Ï34
2. (13, 9), (11, 15)
3. (215, 27), (2, 12)
2Ï10
(1 ) (
1
)
4. 2, 2 , 22, 1
5Ï26
(1 1) (1 1)
5. 4, 2 , 2, 4
6. (1.0, 20.31), (20.2, 0.19)
1.3
1
Ï2
4
Ï2
Find the midpoint of each line segment if the coordinates of
the endpoints are given.
7. (3, 5), (26, 11)
(
3
2 ,
2
)
8
10. (27,26), (21, 24)
(24, 9)
© Glencoe/McGraw-Hill
8. (8, 215), (27, 13)
(
1
,
2
)
21
11. (3, 210), (30, 220)
33
,
2
215)
T49
9. (2.5, 26.1), (7.9, 13.7)
(5.2, 3.8)
12. (29, 1.7), (211, 1.3)
(210, 1.5)
Algebra 2
DATE
NAME
7-2
Student Edition
Pages 415–422
Study Guide
Parabolas
A parabola is a curve consisting of all points in the coordinate
plane that are the same distance from a given point (the focus)
and a given line (the directrix). The chart summarizes
important information about parabolas.
Information about Parabolas
Form of equation
y 5 a (x 2 h) 2 1 k
x 5 a( y 2 k ) 2 1 h
Axis of symmetry
x5h
y5k
Vertex
(h, k )
(h, k )
Focus
(h, k 1 4a1 )
(h 1 4a1 , k)
Directrix
y5k2
Direction of opening
up (a . 0); down (a , 0)
right (a . 0); left (a , 0)
Length of latus rectum
u uunits
u 1a uunits
1
4a
x5h2
1
a
1
4a
1
Example: Graph y 5 4 (x 2 2)2 2 3.
vertex: (2, 23)
axis of symmetry: x 5 2
focus: (2, 23 1 1) or (2, 22)
directrix: y 5 23 2 1 or y 5 24
direction of opening: upward, since a . 0
1
length of latus rectum: or 4 units
y
x
O
uu
1
4
Name the coordinates of the vertex and focus, the equations of the axis
of symmetry and directrix, and the direction of opening of the parabola
with the given equation. Then find the length of the latus rectum.
1. x2 5 2y
( );
(0, 0); x 5 0; 0,
1
2
1
2
y 5 2 ; upward; 2
2. x2 5 y 1 2
(
(0, 22); x 5 0; 0,
1
4
3
21
4
y 5 22 ; upward; 1
The coordinates of the focus and the equation of the
directrix of a parabola are given. Write an equation for
each parabola. Then draw the graph.
4. (3, 5), y 5 1
5. (4, 24), y 5 26
(22, 21); x 5 22;
(22, 2 ); y 5 21 ;
3
4
upward; 1
y
x
O
x
O
O
1
4
6. (5, 21), x 5 3
y
y
);
3. y 5 x2 1 4x 1 3
x
1
8
y 5 (x 2 4)2 2 5
© Glencoe/McGraw-Hill
T50
y 5 (x 2 3)2 1 3
1
4
1
4
x 5 ( y 1 1)2 1 4
Algebra 2
DATE
NAME
7-3
Student Edition
Pages 423–429
Study Guide
Circles
A circle is the set of all points in a plane that are equidistant
from a given point, called the center. The distance from the
center to any point on the circle is called the radius.
Equation of Circle with Center at (h, k), radius r
(x 2 h)2 1 ( y 2 k )2 5 r 2
Example: Find the center and radius of the circle whose equation
is x2 1 2x 1 y2 1 4y 5 11. Then graph the circle.
Complete the square
for each variable.
Write the equation in
the form
(x 2 h)2 1 (y 2 k)2 5 r 2
y
x2 1 2x 1 y2 1 4y 5 11
x2 1 2x 1 j 1 y2 1 4y 1 h 5 11 1 j 1 h
x2 1 2x 1 1 1 y2 1 4y 1 4 5 11 1 1 1 4
x
O
(x 1 1)2 1 (y 1 2)2 5 16
The circle has its center at (21, 22) and a radius of 4.
Find the coordinates of the center and the radius of each
circle whose equation is given. Then draw the graph.
1. (x 2 3) 2 1 y2 5 9
2. x2 1 ( y 1 5) 2 5 4
y
y
O
y
3. (x 2 1) 2 1 (y 1 3) 2 5 9
x
O
x
x
O
(3, 0), r 5 3
(0, 25), r 5 2
4. (x 2 2)2 1 (y 1 4)2 5 16
y
O
(1, 23), r 5 3
5. x2 1 14x 1 y 2 1 2y 5 240
y
6. x2 1 y2 2 10x 1 8y 1 16 5 0
y
x
O x
Write an equation for each circle if the coordinates of the
center and length of the radius are given.
(5, 24), r 5 5
8. (24, 26), 5 units
7. (23, 5), 7 units
(x 1 3) 1 ( y 2 5) 5 49
© Glencoe/McGraw-Hill
x
(27, 21), r 5 Ï10
(2, 24), r 5 4
2
O
(x 1 4)2 1 (y 1 6)2 5 25
2
T51
Algebra 2
DATE
NAME
7-4
Student Edition
Pages 431–439
Study Guide
Ellipses
An ellipse is the set of all points in a plane such that the sum
of the distances from two given points in the plane, called the
foci, is constant. An ellipse has two axes of symmetry. The
intersection of the two axes is the center of the ellipse. The
ellipse intersects the axes to define two segments whose
endpoints lie on the ellipse. The longer segment is called the
major axis, and the shorter segment is called the minor axis.
Standard Equations for Ellipses with Center at (h, k)
Horizontal Major Axis:
Vertical Major Axis:
Example:
y
(–5, 0)
(5, 0)
(–3, 0)
(3, 0)
O
x
(x 2 h) 2
a2
(x 2 h) 2
b2
1
1
(y 2 k) 2
b2
(y 2 k) 2
a2
51
(a 2 . b 2)
51
(a 2 . b 2)
Write the equation of the ellipse.
First find the length of the major axis. The
distance between (25, 0) and (5, 0) is 10 units.
2a 5 10
a 5 5 so a2 5 25
Since the foci are at (23, 0) and (3, 0), c 5 3.
b 2 5 a2 2 c2
b2 5 52 2 32 so b2 5 16
x2
y2
The equation is 25 1 16 5 1.
Write an equation for each ellipse.
1.
y
8
y
2.
(–3, 0)
–4 O
4
(4, 0)
(–4, 0)
(3, 0)
x
y
3.
x
O
O
x
–8
x2
9
1
y2
64
x2
16
51
1
y2
4
(x 1 3)2
4
51
1
( y 1 1)2
9
11
Find the coordinates of the center and foci, and the lengths of
the major axis and minor axis for each ellipse whose equation
is given. Then draw the graph.
x2
y2
4. 4 1 25 5 1
5. 9x2 1 16y2 5 144
y
y
O
x
(0, 0); (0, 6Ï21 ); 10, 4
© Glencoe/McGraw-Hill
6. x2 1 4y2 1 24y 5 232
y
O
x
(0, 0); (6Ï7, 0); 8, 6
T52
O
x
(0, 23); (6Ï3, 0); 4, 2
Algebra 2
DATE
NAME
7-5
Student Edition
Pages 440–447
Study Guide
Hyperbolas
Equation of the Hyperbola
Slopes of the Asymptotes
(x 2 h) 2
a2
b
6
a
2
(y 2 k) 2
b2
51
( y 2 k) 2
a2
a
6
b
2
(x 2 h) 2
b2
y
te
pto
m
sy
a
conjugate axis
A hyperbola is the set of all points in a plane such
as
ym
that the absolute value of the difference of the distances
pt
ot
e
from any point on the hyperbola to two given points in
the plane, called the foci, is constant. Key features
transverse axis
of a hyperbola are the foci, vertex, asymptotes,
F1
transverse axis, and conjugate axis, shown in the
figure. The center of a hyperbola is the midpoint of
the segment connecting the foci. The lengths a, b, and
c are related by the formula c2 5 a2 1 b2.
b
c
a
x
F2
51
Transverse Axis
Horizontal
Vertical
Foci
(h 2 c, k ), (h 1 c, k )
(h, k 2 c), (h, k 1 c)
Vertices
(h 2 a, k ), (h 1 a, k )
(h, k 2 b), (h, k 1 b)
Find the coordinates of the vertices and foci and the slopes of
the asymptotes for each hyperbola whose equation is given.
Then draw the graph.
1.
x2
4
2
y2
16
51
2.
(y 2 3) 2
1
(x 1 2) 2
9
2
51
3. 36x2 2 25y2 5 900
y
y
y
4
O
x
(2, 0), (22, 0);
(2Ï5, 0), (22, Ï5); 62
4.
y2
16
2
x2
9
51
O
(22, 4), (22, 2);
(22, 3 1 Ï10 ),
1
(22, 3 2 Ï10 ); 6
(5, 0), (25, 0);
(Ï61 , 0), (2Ï61 , 0);
6
6
3
5
5. 6(x 2 3) 2 4(y 1 1) 5 96
2
y
x
–4 O
x
2
6. y 2 2x2 1 6y 1 4x 5 9
2
y
y
4
O
x
(0, 4), (0, 24);
4
(0, 5), (0, 25), 6
3
© Glencoe/McGraw-Hill
O
O
x
–4
(7, 21), (21, 21);
(3 2 2Ï10 , 21),
Ï6
(3 1 2Ï10 , 21); 6
T53
2
x
(1, 1), (1, 27);
(1, 23 1 2Ï6),
(1, 23 2 2Ï6); 6Ï2
Algebra 2
DATE
NAME
7-6
Student Edition
Pages 450–455
Study Guide
Conic Sections
Parabolas, circles, ellipses, and hyperbolas
are known as conic sections. Any conic
section in the coordinate plane can be
described by an equation of the form
Ax2 1 Bxy 1 Cy2 1 Dx 1 Ey 1 F 5 0, where
A, B, and C are not all zero. When B 5 0, the
coefficients of x2 and y2 tell you what kind of
conic section the equation will have for its
graph.
A5C
circle
A Þ C, but have same sign
ellipse
A Þ C, but have opposite signs
hyperbola
A 5 0 or C 5 0, but not both
parabola
Example: Write x2 5 4y2 1 16 in the form Ax2 1 Bxy 1 Cy2 1 Dx 1 Ey 1 F 5 0.
Tell what kind of conic section the graph will be. Then change the equation
to the standard form for that conic section and graph the equation.
x2 5 4y2 1 16
x 2 4y2 5 16
y
2
Since A and C have opposite signs, the graph
will be a hyperbola.
x
O
Next change x2 2 4y2 5 16 to the standard
form for a hyperbola. Divide each side by 16.
x2
y2
2
51
16
4
Write each equation in standard form. State whether the
graph of the equation is a parabola, a circle, an ellipse, or
a hyperbola. Then graph the equation.
1. x2 2 2x 1 y2 1 8y 5 8
2. y 5 x2 2 2x 2 8
y
y
2
2
x
O 2
(x 2 1)2 1 (y 1 4)2 5 25; circle
4. x2 5 2x 1 y2 2 4y 1 7
y
y
(x 1 4) 2
4
1
O
x
( y 2 1) 2
9
© Glencoe/McGraw-Hill
x
y 5 (x 2 1)2 2 9; parabola
3. 9(x 1 4)2 1 4(y 2 1)2 5 36
O
O
–2
(x 2 1) 2
4
5 1; ellipse
T54
2
x
( y 2 2) 2
4
5 1; hyperbola
Algebra 2
DATE
NAME
7-7
Student Edition
Pages 460–467
Study Guide
Solving Quadratic Systems
You can use algebra to find exact solutions for systems of
quadratic equations. For systems of inequalities, it is usually
best to show the solution set with a graph.
x2 1 y 2 5 25
Example: Use algebra to find the solutions of the system y 2 x 5 1.
{
Solve y 2 x 5 1 to get y 5 x 1 1.
x2 1 (x 1 1)2 5 25
Substitute x 1 1 for y.
2
2x 1 2x 2 24 5 0
Simplify. Add 225 to both sides.
2(x 1 4)(x 2 3) 5 0
Factor.
x 1 4 5 0 or x 2 3 5 0
Zero Product Property
x 5 24
x53
Solve for x.
y 5 23
y54
Substitute for x in y 5 1 1 x.
The solutions are (24, 23) and (3, 4).
Example: Solve the system
x2 1 y2 # 25
{(
x2
5
2
2
) 1y $
2
25
4
y
by graphing.
The graph of x 2 1 y 2 # 25 consists of all points on or inside the
5 2
circle with center (0, 0) and radius 5. The graph of x 2
1
2
25
y2 $
consists of all points on or outside the circle with center
(
5
,
2
( )
4
)
x
O
5
0 and radius . The solution of the system is the set of
2
points in both regions.
Solve each system of equations, algebraically. Check your
solutions with a graphing calculator.
1. x2 1 y2 5 9
x2 1 y 5 3
2. x2 1 (y 2 5)2 5 25
y 5 2x2
(0, 3), (Ï5, 22),
(2Ï5, 22)
3. y 5 x2 2 1
y5x23
(2, 21), (21, 24)
(0, 0)
Solve each system of inequalities by graphing.
x2
4. x2 1 y2 # 169
x2 1 9y2 $ 225
y2
5. 16 1 4 # 1
1
y . 2x 2 2
y
y
(–12.7, 2.6)
3
O 3
(–12.7, –2.6)
© Glencoe/McGraw-Hill
(12.7, 2.6)
x
O
(12.7, –2.6)
T55
x
Algebra 2