1. 4.4 Conics
Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas.
Write the equation and find the focus of a
parabola.
Write the equation of a ellipse and find the
foci, vertices, the length of the major and minor
axis.

2. Parabolas
Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line.
Directrix

3. Parabolas Standard equation of the Parabola x2 = 4py p  0
Vertex (0, 0)
Directrix y = -p
Focus (0, p)
Line of sym x = 0
y2 = 4px p  0
Vertex (0,0)
Directrix x = -p
Focus (p, 0)
Line of sym y = 0

4. Parabola Examples Given Find the focus.
Since the squared variable is x, the parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down.
Focus (0, p)
Solve for x2 -p
Solve for p. • p
Focus

5. Parabola Examples
Write the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0).
Note that the focus is along the x axis, so the parabola is oriented
in the x axis direction, y2 = 4px.
Focus (2, 0) (p, 0)
p = 2
y2 = 4px = 4(2)x
y2 = 8x

6. Classwork
Page 370 problems 9 –14.
Page 371 problems 17 – 28.

7. Ellipse
Definition: An ellipse is the set of all points (x, y ) in a plane the sum of whose distances from two distinct fixed points (foci) is constant.
Major
Axis •
Focus
(x, y) d1 d2
d1 + d2 = constant •
Center
Vertex
Vertex
Minor
Axis

8. Ellipse
The standard form of the equation of an ellipse (Center at origin)
Major axis
along the x axis
Major axis
along the y axis
where 0 < b < a
c2 = a2 – b2
Major axis length = 2a Minor axis length = 2b

9. Ellipse Examples
Given 4x2 + y2 = Find the vertices, the end points of
the minor axis, the foci and center.
Change the equation to the standard form.
Major axis along the y-axis b2 a2
Center (0, 0) Vertices (0, 6) End points of minor axis (3, 0)
(0, a)
(b, 0)
To find the foci use c2 = a2 – b2
c2 = 36 – 9
c2 = 27
(0, c)

10. Classwork
Page 372 problems 35 – 40
45 – 55

11. Hyperbolas
Definition: a hyperbola is the set of all points (x, y) the difference
of whose distances from two distinct points (foci) is constant. d1
Focus
(x, y) • •
Vertex •
Branch d2
Transversal
Axis •
Center
Vertex •
Branch
Focus
d1 - d2 = constant