PARABOLAS IN ANALYTICAL GEOMETRY FOR SHS PRECALCULUS SUBJECY

1. WHAT DO YOU THINK???

2. Analytic Geometry
Lecture 2: Parabolas
Noven S. Villaber
Fatima College of Camiguin, Inc.
SHS Department
PRECALCUS
16

3. Learning Outcomes of the
Lesson
At the end of the lesson, the student is able to:
(1) define a parabola;
(2) determine the standard form of equation of a
parabola;
(3) graph a parabola in a rectangular coordinate
system

4. Parabolas
The parabola is the locus of all points in a plane that are
the same distance from a line in the plane, the directrix,
as from a fixed point in the plane, the focus.
| p |
| p |
 The parabola has one
axis of symmetry, which
intersects the parabola
at its vertex.
 The distance from the
vertex to the focus is | p |.
 The distance from the
directrix to the vertex is
also | p |.
Point Focus = Point Directrix
PF = PD

5. Parts of Parabola

6. Parts of Parabola
 Fixed line = Directrix
 Fixed point = Focus
 The line that passes through the focus and is
perpendicular to the directrix = axis of symmetry
 The point where the parabola intersects with its axis
of symmetry (it is point midway between the latus
rectum and the directrix = vertex
 Line segment that passes through the focus of a
parabola and is perpendicular to the axis of symmetry
= Latus Rectum/focal width
 Parabola has 4 openings (upward, downward, to the
right, and to the left).

7. Vertex
Focus
Directrix
Axis of Symmetry
x
y

8. The general form of the
equation is where D, E, and F
are constants

9. Examples
23

10. The Standard Form of the
Equation with Vertex (h, k)
24
Parabola
Axis of
Symmetry
Opening Focus
Latus
Rectum
Directrix
Horizontal
Right, if
p > 0
Left, if p<0
(h+p,k)
Length: 4p
Equation:
=ℎ+
Endpoints:
(ℎ+ , ±2 )
Equation:
=ℎ−
Vertical
Upward, if
p > 0
Downward
,
if p<0
(h,k+p)
Length: 4p
Equation:
= +
Endpoints:
(ℎ±2 , + )
Equation:
= −

11. General Form To Standard Form
Steps:
1.Group the terms with same variables.
2.Move the remaining terms to the right side of the
equation.
3.Create a perfect square trinomial by completing
the squares.
4.Simplify both sides of the equation.
5.Express the perfect square trinomials as square of
binomials to make the equation in the vertex
form.

12. Example
26

13. Solution
standard form/
vertex form

14. Standard Form To
General Form

15. Example
29

16. Solution
General
form

17. Standard form of a Parabola
with vertex at the origin (0,0)
31
Parabola
Axis of
Symmetry
Opening Focus
Latus
Rectum
Directrix
Horizontal
Right, if
p > 0
Left, if p<0
(p,0)
Length: 4p
Equation:
=±
Endpoints:
( ,±2 )
Equation:
=−
Vertical
Upward, if
p > 0
Downward,
if p<0
(0,p)
Length: 4p
Equation:
=±
Endpoints:
( ,±2 )
Equation:
=−

18. Example

19. Graph of Parabola

20. Graph of Parabola

21. Graph of Parabola

22. Graph of Parabola

23. Graph of Parabola

24. Graph of Parabola

25. Graph of Parabola

26. Graph of Parabola

27. Graph of Parabola

28. Graph of Parabola

29. Graph of Parabola
To graph at Vertex (h,k):
1. Locate the vertex of the parabola.
2. Determine its axis of symmetry by
considering the variable with the second
degree
3. Identify the opening of the parabola by
examining the value of its focus (p).
4. Locate the focus of the parabola.
5. Locate the endpoints of the latus rectum.
6. Plot the directrix.

30. Example

31. Solution
Step 1
Step 2
Step 3

32. Step 4 Locate the focus of the parabola.
Since p = 1, the focus of the parabola is 1 unit to the right
of the vertex. Thus, the focus is at (h+p,k) = (2+1, -3) =
(3, -3)
Step 5 Locate the endpoints of the latus rectum.
To locate the endpoints of the latus rectum
± ±
This means that from the focus (3,-3), we move 2 units
upward and another 2 units downward. Therefore, the
endpoints of the latus rectum are located at (3,-1) and
(3,-5).
Also, the length of the latus rectum is 4 units.

33. Step 6 Plot the directrix.

34. Graph of Parabola
To graph at Vertex (0,0):
1. Locate the vertex of the parabola.
2. Determine its axis of symmetry by
considering the variable with the second
degree
3. Identify the opening of the parabola by
examining the value of its focus (p).
4. Locate the focus of the parabola.
5. Locate the endpoints of the latus rectum.
6. Plot the directrix.

35. Example

36. Step 1
Step 2
Step 3
Step 4 Locate the focus of the parabola.
Since p = 5, the focus of the parabola is 5 units above the vertex.
Thus, the focus is at (0,5)
50

37. 51
Step 5 Locate the endpoints of the latus rectum.
To locate the endpoints of the latus rectum
±2 =±10.
This means that from the focus (0,5), we move 10 units to the right and
another 10 units to the left. Therefore, the endpoints of the latus rectum are
located at (-10,5) and (10,5).
Also, the length of the latus rectum is 20 units, which is actually the value
of 4p (the coefficients of y in the given example).
Step 6 Plot the directrix.