9.2 Graph and Write Equations of Parabolas
Goal  Graph and write equations of parabolas that open left or right.
VOCABULARY
Focus
A fixed point that lies on the axis of symmetry of a parabola
Directrix
A line that is perpendicular to the axis of symmetry of a parabola
STANDARD EQUATION OF A PARABOLA WITH VERTEX AT THE ORIGIN
The standard form of the equation of a parabola with vertex at (0, 0) is as follows:
Equation
x2 = 4py
Focus
(0, p)
Directrix
y=-p
Axis of Symmetry
Vertical ( x = 0 )
y2 = 4px
(p, 0)
x=-p
Horizontal ( y = 0 )
Example 1
Graph an equation of a parabola
Graph x =
1 2
y . Identify the focus, directrix, and axis of symmetry.
2
Checkpoint Complete the following exercise.
1. Graph y = 
1 2
x .Identify the focus, directrix, and axis of symmetry.
4
Example 2
Write an equation of a parabola
Write an equation of the parabola shown.
Checkpoint Complete the following exercise.
2.
Write the standard form of the equation of the parabola with vertex at (0, 0) and the directrix
3
x 
4.
9.6 Translate and Classify Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
In the following equations, the point (h, k) is the vertex of the parabola and the center of the other conics
Parabola (y  k)2 = 4p(x  h)
(x  h)2 = 4p(y  k)
Horizontal axis
Vertical axis
Example 1
Graph the equation. Identify the vertex, focus, directrix of the parabola.
 x  3
2
 4  y  4
Example 2
Write an equation of a translated parabola
Write an equation of the parabola whose vertex is at (2,1) and whose focus is at (5,1).
Checkpoint
1. Graph the equation. Identify the vertex,
focus, directrix of the parabola.
2
 y  2  8  x  1
2. Write an equation of the parabola whose vertex is at (3, 1) and whose focus is at (3, 1).
Example 3
Write the equation in standard
form and graph the equation.
y 2  4 y  4 x  28  0
Checkpoint 3
Write the equation in standard
form and graph the equation.
x2  2 x  8 y  31  0
9.3 Graph and Write Equations of Circles
Goal  Graph and write equations of circles.
VOCABULARY
Circle
The set of all points (x, y) that are equidistant from a fixed point
Center
The fixed point that is equidistant from all the points on a circle
Radius
The distance r between the center and any point (x, y) on a circle
STANDARD EQUATION OF A CIRCLE WITH CENTER AT THE ORIGIN
The standard form of the equation of a circle with center at (0, 0) and radius r is as follows:
x2 + y2 = _r2_
Example 1
Graph an equation of a circle
Graph y2 = x2 + 16. Identify the radius of the circle.
Solution
Checkpoint Graph the equation. Identify the radius.
1. x2 = 4  y2
Example 2
Write an equation of a circle
The point (3, 4) lies on a circle whose center is the origin. Write the standard form of the equation of the
circle.
Checkpoint Complete the following exercises.
2. Write the standard form of the equation of the circle with center at the origin that passes through the point
(6, 3).
9.6 Translate and Classify Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
In the following equations, the point (h, k) is the center of the circle.
Circle
(x  h)2 + (y  k)2 = r2
Example 1
Graph the equation of a translated circle
Graph (x + 3)2 + (y  2)2 = 4.
Example2
Write an equation of a circle
with a center at (2,-6) and a radius of 4.
Checkpoint Complete the following exercises.
1. Graph (x  2)2 + (y + 3)2 = 9.
Identify the center and radius.
2. Write an equation of a circle
with a center ar (-3,0) and a radius of 5.
Example 3
Write the equation of the circle in standard form. Graph the equation.
x2  y 2  12 x  2 y  15  0
9.4 Graph and Write Equations of Ellipses
Goal  Graph and write equations of ellipses.
VOCABULARY
Ellipse
The set of all points P such that the sum of the distances between P and two fixed points, called the foci, is a
constant
Foci
Two fixed points in an ellipse
Vertices
The points at which the line through the foci intersect the ellipse
Major axis
The line segment that joins the vertices
Center
The midpoint of the major axis
Co-vertices
The points of intersection of an ellipse and the line perpendicular to the major axis at the center
Minor axis
The line segment that joins the co-vertices
STANDARD EQUATION OF AN ELLIPSE WITH CENTER AT THE ORIGIN
Equation
2
2
x +y =
1
a 2 b2
x2 y2

1
b2 a 2
Major Axis
Horizontal
Vertices
( _a_, 0)
Co-Vertices
(0,  _b_)
Vertical
(0,  _a_)
( _b_, 0)
The major and minor axes are of lengths 2a and 2b, respectively, where a > b >
0. The foci of the ellipse lie on the major axis at a distance of c units from the
center, where c2 = _a2  b2_
Example 1
Graph an equation of an ellipse
Graph the equation 9x2 + 36y2 = 324. Identify the vertices, co-vertices, and foci of the ellipse.
Example 2
Write an equation given a vertex and a co-vertex
Write an equation of the ellipse that has a vertex at (0, 7), a co-vertex at (4, 0), and center at (0, 0).
Example 3
Write an equation given a vertex and a focus
Write an equation of the ellipse that has a vertex at (6, 0) and a focus at (5, 0).
Checkpoint Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.
1. x2 +
y2
=1
25
Checkpoint Write an equation of the ellipse with the given characteristics and center at (0, 0).
2. Vertex: (9, 0)
Co-vertex: (0, 4)
3. Vertex: (0, 7)
Focus: (0, 3)
9.6 Translate and Classify Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
In the following equations, the point (h, k) is the center of the ellipse
( x  h) ( y  k )

1
2
2
a
b
2
Ellipse
2
( x  h) ( y  k )

1
2
2
b
a
2
2
Horizontal axis
Vertical axis
Example 1
Graph the ellipse. Identify the center, vertices, co-vertices, and foci
 x  4
16
2
 y  2

4
2
1
Example 2
Write an equation of a translated ellipse
Write an equation of the ellipse with foci at (2, 3) and (4, 3) and co-vertices at (1, 4) and (1, 2).
Checkpoint Complete the following exercise.
1. Graph the ellipse. Identify the center, vertices, co-vertices, and foci
 x  2
9
2

y2
1
1
2. Write an equation of the ellipse with foci at (3, 0) and (3, 6) and
co-vertices at (1, 3) and (5, 3)
Example 3
Write the equation in standard form and graph.
9 x2  36 y 2  54 x 144 y  99  0
9.5 Graph and Write Equations of Hyperbolas
Goal  Graph and write equations of hyperbolas.
VOCABULARY
Hyperbola
The set of all points P such that the difference of the distances between P and two fixed points, called the foci,
is a constant
Foci
Two fixed points in a hyperbola
Vertices
The points of intersection of a hyperbola and the line through the foci
Transverse Axis
The line segment that connects the vertices of a hyperbola
Center
The midpoint of the transverse axis
STANDARD EQUATION OF A HYPERBOLA WITH CENTER AT THE ORIGIN
Equation
Transverse Axis
Asymptotes
Vertices
x2 y2

1
a2 b2
Horizontal
y
b
x
a
( _a_, 0)
y2 x 2

1
a2 b2
Vertical
y
a
x
b
(0, _a_)
The foci lie on the transverse axis, c units from the center, where c2  _a2 + b2.
Example 1
Graph an equation of a hyperbola
Graph 36y2  9x2  324. Identify the vertices, foci, and asymptotes of the hyperbola.
Checkpoint Graph the equation. Identify the vertices, foci, and asymptotes of the.hyperbola.
x2
1.
49

y2
9
1
Example 2
Write an equation of a hyperbola
Write an equation of the hyperbola with foci at (5, 0) and (5, 0) and vertices at (4, 0) and (4, 0).
Checkpoint Write an equation of the hyperbola with the given foci and vertices.
2.
Foci: (0, 8), (0, 8)
Vertices: (0, 5), (0, 5)
Example 3
Solve a multi-step problem
Lamp The diagram shows the hyperbolic cross section of a lamp. Write an equation for the cross section
of the lamp. The lamp is 10 inches high. How wide is the base?
Solution
9.6 Translate and Classify Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
In the following equations, the point (h, k) is the center of the hyperbola.
2
2
( x  h) ( y  k )

1
Hyperbola
2
2
a
b
Horizontal axis
( y  k ) ( x  h)

1
2
2
a
b
2
2
Vertical axis
Example 1
Graph the equation of a translated hyperbola. List the vertices, foci and center.
Graph ( y  2)  ( x  1)  1
16
4
2
2
Checkpoint Complete the following exercises. List the vertices, foci and center.
2
Graph
2
( x  3) ( y  1)

1
9
25
.
Example 2
Write an equation of the hyperbola with vertices (2,4) and (8,4) and foci at (-2,4) and (12,4)
Checkpoint
Write an equation of the hyperbola with vertices (2,5) and (2,-1) and foci at (2,7) and (2,-3)
Example 3:
Write the equation in standard form and graph.
y 2  9 y  54 y  90  0