Parabola

1. PARABOLA

2. Parts of Parabola

3. Focus Equation
Parabolas
open
Directrix Axis of Symmetry
To the right
To the left
Upward
Downward
Equation of a Parabola with vertex at the origin

4.  The table above shows for equations of the parabola in standard
form with the vertex at . If the focus is at , the parabola opens
to the right and its equation is If the focus is at the parabola
opens to the left and its equation is .If, on the other hand the
focus is at , the parabola opens to the upward and its equation
is . The parabola that opens downward has focus and
equation .The table also shows the equations of the directrix
and the axis of symmetry of the parabola based on its opening.

5.  Example 1:
 Find the equation for the parabola with vertex and focus , and sketch its graph.
Since the parabola opens upward.
y=-2
F0,2
y
x
Solution: Since the focus is F0,2, you conclude that p=2 so the
directrix is y=-2. Thus, the equation of parabola is
=42y
=8y

6.  Example 2:
 Determine the focus and directrix of the parabola with the given equation. Sketch the
graph, and indicate the focus, directrix, vertex, and axis of symmetry.
1.
Solution:
The vertex and the parabola opens upward. From . The focus, units above the vertex, is .
The directrix, 3 units below the vertex, is. The axis off symmetry is .

7. 2.
The vertex is and the parabola opens downward. From . The focus, units below
the vertex, is . The directrix, units above the vertex, is . The axis of symmetry is .
 The parabolas you consider so far are “vertical” and have their vertices at the
origin. Some parabolas open instead horizontally (to the left or right),an d some
have vertices not at the origin.

8. Parabola opens Equation Focus Directrix
Upward
Downward
To the right
To the left
Equation of a Parabola with Vertex at
The table above summarizes the equations, opening, coordinates of the focus and
the equation of the directrix.

10.  example:
 Determine the vertex, focus, directrix, and the axis of symmetry of the parabola
with the given equation. Sketch the parabola, and include these points and lines.
1.
2.
 Solution:
(1) You complete the square on , and move to the other side.
 The parabola opens to the right. It has vertex . From, you get . The focus is units
to the left of .
 The (vertical) directrix is units to the left of . The (horizontal) axis is through

12. Solution:
You complete the square on , and move to the other side.
In the last equation, you divided by 5 for the squared part not to have any
coefficient. The parabola opens downward. It has vertex V-3,4.
From 4p=245, you get p=65=1.2. The focus is p=1.2 units below V:F-3,2.8.
The (horizontal) directrix is p=1.2 units above V:y=5.2. The (vertical) axis is
through V:x=-3.

13.  Example: A parabola has focus and directrix . Find its standard equation.
Solution: The directrix is horizontal, and the focus is above it. The parabola then
opens upward and its standard equation has the form . Since the distance from
the focus to the directrix is then . Thus, the vertex is , the point 3 units below .
The standard equation is then .

14. Determine the coordinates of the vertex, axis of symmetry, focus, focal
distance length of the latus rectum and endpoint of the latus rectum of the
parabola with the given equation. Sketch the graph. Put your answer in a
graphing paper. (5 points each item)
1.
2.
3.