This document defines and discusses parabolas. It begins by listing 4 learning outcomes related to understanding parabolas, their standard form equations, graphing them, and solving problems involving parabolas. It then defines a parabola as the set of all points that are the same distance from both a fixed focus point and directrix line. The standard form of the equation for a parabola is derived and explained to be x^2 = 4cy, where c is the distance between the focus and directrix. Key features of parabolas like the vertex, directrix, focus, and axis of symmetry are identified. Examples of determining standard equations and graphing parabolas are provided.

1. Parabolas
PRECALCULUS

2. Parabolas
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; and
(4) solve situational problems involving conic sections
(parabolas).

3. Introduction
A parabola is one of the conic sections. We have
already seen parabolas which open upward or
downward, as graphs of quadratic functions. Here,
we will see parabolas opening to the left or right.
Applications of parabolas are presented at the end.
Parabolas

4. Definition and Equation of a Parabola
Consider the point F(0,2) and the line ℓ having equation
y = -2, as shown in Figure 1.17. What are the distances of
A(4,2) from F and from ℓ?

5. Definition and Equation of a Parabola

6. Let F be a given point, and
ℓ a given line not
containing F. The set of all
points P such that its
distances from F and from
ℓ are the same, is called a
parabola. The point F is its
focus and the line ℓ its
directrix.
Definition and Equation of a Parabola

7. Consider a parabola with
focus F(0,c) and directrix ℓ
having equation y =-c. The
focus and directrix are c
units above and below,
respectively, the origin. Let
P(x,y) be a point on the
parabola so PF = PPℓ .
Definition and Equation of a Parabola

8. Consider a parabola with
focus F(0,c) and directrix ℓ
having equation y =-c. The
focus and directrix are c
units above and below,
respectively, the origin. Let
P(x,y) be a point on the
parabola so PF = PPℓ .
Definition and Equation of a Parabola

9. The vertex V is the point
midway between the focus
and the directrix. This
equation, x2 = 4cy, is then
the standard equation of a
parabola opening upward
with vertex V(0,0).
Definition and Equation of a Parabola

10. We collect here the features of
the graph of a parabola with
standard equation x2 = 4cy or
x2 = -4cy, where c > 0.
(1) vertex: originV (0,0)
• If the parabola opens
upward, the vertex is the
lowest point. If the parabola
opens downward, the vertex is
the highest point.
Equation of a Parabola

11. Equation of a Parabola
(2) directrix: the line y = - c or y = c
• The directrix is c units below or above the vertex.
(3) focus: F(0,c) orF(0, -c)
• The focus is c units above or below the vertex.
• Any point on the parabola has the same distance
from the focus as it has from the directrix.
(4) axis of symmetry: x = 0 (they-axis)
• This line divides the parabola into two parts
which are mirror images of each other.

12. Activity
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) x2 = 12y (2) x2 = -6y
(3) What is the standard
equation of the parabola

13. 4. Give the focus and directrix of the parabola with equation
x2 = 10y. Sketch the graph, and indicate the focus, directrix,
vertex, and axis of symmetry.
5. Find the standard equation of the parabola with focus
F(0,-3.5) and directrix y = 3.5.
Activity

14. THANK YOU
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