1. 9.1 Conic Sections
Conic sections – curves that result from the intersection of
a right circular cone and a plane.
Circle Ellipse Parabola Hyperbola

2. 9.2 Quadratic Function: Any function of the form
y = ax2 + bx + c
a, b, c are real numbers & a 0
A quadratic Equation: y = x2 + 4x + 3 a = _____ b = _____ c = ______ 1 4 3
x y
Formula for
Vertex:
X = -b 2a
Plug x in to
Find y
The graph is a
parabola.
Where is the vertex?
Where is the axis of symmetry?

3. 9.2 Parabola x = a ( y - k ) + h Vertex Form of a Parabola
A Parabola is the set of all points in a plane that are equidistant from a
fixed line (the directrix) and a fixed point (the focus) that is not on the line.
Directrix
Parabola
Vertex
Focus
Axis of Symmetry
Vertex Form of a Parabola
(opens up or down) x = a ( y - k ) 2 + h
(opens right or left)
Vertex = (h, k)
Focus & Directrix = +/ from vertex 4a
Plotting Points of the parabola can be done by:
finding x/y intercepts and/or
choosing values symmetric to the vertex x
Directrix
Parabola
Vertex
Focus

4. Parabola Graphing h k y a x + - = ) (
Step 1: Place equation in one of two vertex forms:
(opens up or down) h k y a x + - = 2 ) (
(opens right or left)
Step 2: Find the vertex at : (h, k) & determine which direction the parabola opens
Step 3: Use an x/y chart to plot points symmetric to the vertex & draw the parabola
Step 4: Find the focus/directrix displacement with 1/(4a)
Try the following Examples:
#1) y2 = 4x
#2) x2 = -16y
#3) (y + 4)2 = 12(x + 2)
#4) x2 – 2x – 4y + 9 = 0

5. 1.1 Circles Standard circle equation (x – h)2 + (y – k)2 = r2
center = (h, k) radius = r
Graph the circle: (x – 2)2 + (y + 3)2 = 9
Step 1: Find the center & radius
Step 2: Graph the center point
Step 3: Use the radius to find 4 points
directly north, south, east, west
of center.
h is 2.
(x – 2)2 + (y – (-3))2 = 32
k is –3.
r is 3.
Center = (2, -3)
Radius = 3

6. General Form of a Circle Equation (Changing from General to Standard Form)
Standard circle equation : (x – h)2 + (y – k)2 = r2 center = (h, k) radius = r
General Form of a circle : x2 + y2 + Dx + Ey + F = 0
Change from General Form to Standard Form
General Form: x2 + y2 –4x +6y +4 = 0
Move constant : x2 + y2 –4x +6y = -4
X’s & Y’s together : (x2 – 4x ) + (y2 + 6y ) = -4
Complete the Square: (x2 – 4x ) + (y2 + 6y +9 )=
Factor (x – 2) (x –2) + (y + 3) (y + 3) = 9
Standard Form (x – 2) (y + 3) = 9
Setup to Complete Square
½ (-4) = ½ (6) = 3
(-2)2 = (3)2 = 9

7. 9.3 Ellipse
Ellipse - the set of all points in a plane the sum of whose distances from
two fixed points, is constant. These two fixed points are called the foci.
The midpoint of the segment connecting the foci is the center of the ellipse. F1 F2 P
A circle is a special kind of ellipse. Since we know about circles, the
equation of a circle can be used to help us understand the equation
and graph of an ellipse.

8. Comparing Ellipses to Circles
circle: (x – h)2 + (y – k)2 = r2 Ellipse: (x – h)2 + (y – k)2 = 1
a b2
center = (h, k) center = (h, k)
radius = r radiushorizontal = a radiusvertical = b
(0, 5)
(0, -5)
(0, 3)
(0, -3)
(0, 0)
(4, 0)
(-4, 0)
We do not usually call ‘a’ and ‘b’ radii
since it is not the same distance
from the center all around the ellipse,
but you may think of them as
such for the purpose of locating
vertices located at : (h + a, k)
(h – a, k)
(h, k + b)
(h, k – b)
Example1: Graph the ellipse:
25x2 + 16y2 = 400
Finding Ellipse Foci:
(foci-radius)2 = (major-radius)2 – (minor-radius)2
Major and Minor axis is determined
by knowing the longest and shortest
horizontal or vertical distance
across the ellipse.
Example2: Calculate/Find the foci above.

9. Converting to Ellipse Standard Form
Example: Graph the ellipse: 9x2 + 4y2 – 18x + 16y – 11 = 0
Move constant : x2 + 4y2 –18x +16y = 11
X’s & Y’s together : (9x2 – 18x ) + (4y2 + 16y ) = 11
9(x2 – 2x ) + 4(y2 + 4y ) = 11
Complete the Square: (x2 – 2x ) + 4(y2 + 4y + 4) =
Factor (x – 1) (x –1) (y + 2) (y + 2) = 36
9 (x – 1) (x + 2)2 = 36
Standard Form (x – 1) (y + 2) = 1
(x – 1) (y + 2) = 1
Can you
graph this?

10. Standard Forms of Equations of Ellipses Centered at (h,k)
(h, k – a)
(h, k + a)
(h, k – c)
(h, k + c)
Parallel to the y-axis, vertical
(h, k)
b2 > a2 and a2 = b2 – c2
(h – a, k)
(h + a, k)
(h – c, k)
(h + c, k)
Parallel to the x-axis, horizontal
a2 > b2 and b2 = a2 – c2
Vertices
Foci
Major Axis
Center
Equation
(h, k)
Major axis x y
Focus (h – c, k)
Focus (h + c, k)
Vertex (h – a, k)
Vertex (h + a, k)

11. 9.4 Hyperbola
A hyperbola is the set of points in a plane the difference whose distances
from two fixed points (called foci) is constant x y
Transverse axis
Vertex
Focus
Center y x
Traverse Axis - line segment joining the vertices.
When you graph a hyperbola you must first locate the
center and direction of the traverse access
-- parallel or horizontal
Can you find the
traverse axis, center,
vertices and foci in
the hyperbola above?

12. Graphing Hyperbola
Hyperbola: (x – h)2 - (y – k)2 = or (y – k)2 - (x – h)2 = 1
a b a b2
transverse axis: Horizontal transverse axis: Vertical
center = (h, k) – BE Careful! (Foci-displacement)2 = a2 + b2
Verticies at center traverse-coordinate +a Asymptotes: y – k = m (x – h)
center traverse-coordinate –a m = +/- y-displacement
x-displacement
Example1 :
Center: (3, 1)
Traverse Axis: Horizontal
Foci-displacement
F2 = = 29
F = 29 = 5.4 C

13. Graphing Hyperbola (Cont.)
Hyperbola: (x – h)2 - (y – k)2 = or (y – k)2 - (x – h)2 = 1
a b a b2
transverse axis: Horizontal transverse axis: Vertical
center = (h, k) – BE Careful! (Foci-displacement)2 = a2 + b2
Verticies at center traverse-coordinate +a Asymptotes: y – k = m (x – h)
center traverse-coordinate –a m = +/- y-displacement
x-displacement
Example2 :
(y – 2)2 – (x + 1)2 = 1 C
Center: (-1, 2)
Traverse Axis: Vertical
Foci-displacement
F2 = = 25
F = 25 = 5

14. Graphing Hyperbola (Completing the Square)
Hyperbola: (x – h)2 - (y – k)2 = or (y – k)2 - (x – h)2 = 1
a b a b2
transverse axis: Horizontal transverse axis: Vertical
center = (h, k) – BE Careful! (Foci-displacement)2 = a2 + b2
Verticies at center traverse-coordinate +a Asymptotes: y – k = m (x – h)
center traverse-coordinate –a m = +/- y-displacement
x-displacement
Example3 : 4x2 – y2 + 32x + 6y +39 = 0

15. Standard Forms of Equations of Hyperbolas Centered at (h,k)
(h, k – a)
(h, k + a)
(h, k – c)
(h, k + c)
Parallel to the y-axis, vertical
(h, k)
b2 = c2 – a2
(h – a, k)
(h + a, k)
(h – c, k)
(h + c, k)
Parallel to the x-axis, horizontal
Vertices
Foci
Transverse Axis
Center
Equation
(h, k) x y
Focus (h – c, k)
Focus (h + c, k)
Vertex (h – a, k)
Vertex (h + a, k)