Lesson 3: PARABOLA
Precalculus / Basic Calculus
Science, Technology, Engineering, and Mathematics
Lesson 3: PARABOLA
Definition and Standard Equation of a
Parabola
Graphs of Parabola
Applications of Parabolas in Real-Life
Situations
Capstone
Precalculus
Project
Science, Technology, Engineering, and Mathematics
Lesson 3: PARABOLA
Definition and Standard Equation of a
Parabola
Capstone
Precalculus
Project
Science, Technology, Engineering, and Mathematics
Do you know how a
satellite dish works? A
satellite dish is
parabolic in shape so
that the signals
transmitted from the
satellite reflects onto
the receiver after it
bounces off the focus.
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Learning Competencies
At the end of the lesson, you should be able to do the following:
● Define a parabola.
● Determine the standard form of equation
of a parabola.
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Learning Objectives
At the end of the lesson, you should be able to do the following:
● Define a parabola.
● Identify the parts and properties of a parabola.
● Write the equation of a parabola.
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Parabola
Recall that conic sections are the
curves obtained when a plane
intersects a double-napped cone.
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Parabola
A parabola is generated when
the plane is parallel to one
generator and intersecting only
one of its nappes.
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Parabola
A parabola is the set of all
points in a plane that are
equidistant from a fixed line
called the directrix and a
fixed point not on the line
called focus (plural, foci).
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Parabola
The midpoint between the
focus and the directrix is
called the vertex, and the
line passing through the
focus and the vertex is
called the axis of symmetry
of the parabola.
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Parabola
The latus rectum of a
parabola is a line segment
that passes through the
focus, parallel to its directrix,
and with its endpoints on
the parabola.
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Standard Form of the Equation of a Parabola
Graph Axis of Standard
Vertex Directrix Focus
Opening Symmetry Form
upward -axis
downward -axis
to the right -axis
to the left -axis
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Standard Form of the Equation of a Parabola
• In the equation , if , the parabola opens upward and if
the parabola opens downward.
• In the equation , if , the parabola opens to the right and
if the parabola opens to the left.
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Standard Form of the Equation of a Parabola
Aside from the vertex at the origin, the graph of a parabola
can also have its vertex at any point in the Cartesian plane.
The horizontal and vertical translations are performed by
substituting with and with in the standard form of the
equation of the parabola.
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Standard Form of the Equation of a Parabola
Graph Axis of
Vertex Directrix Focus Standard Form
Opening Symmetry
upward
downward
to the right
to the left
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Tips
To easily determine the values of and
from the standard form of the equation
of the parabola, get the additive inverse
of the constants from the binomials
and .
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Tips
In the given equation
, the values of and are the additive
inverse of and . Thus, the vertex is at .
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Let’s Practice!
Determine the equation of the parabola in standard
form given its focus at and directrix .
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Let’s Practice!
Find the vertex, focus, and directrix of the parabola
given by the equation .
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Let’s Practice!
Find the standard form of the equation of the
parabola given that the vertex at and the focus at .
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Try It!
Find the standard form of the equation
of the parabola with focus at and
directrix .
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Try It!
Find the vertex, focus, and directrix of
the parabola given by the equation .
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Try It!
Find the standard form of the equation
of the parabola given that the vertex at
and the focus at .
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Let’s Sum It Up!
● A parabola is generated when the plane is parallel to
one generator and intersecting only one of its nappes.
● A parabola is the set of all points in a plane that are
equidistant from a fixed line called the directrix and a
fixed point not on the line called focus (plural, foci).
● The midpoint between the focus and the directrix is
called the vertex. The vertex of the parabola may be at
the origin or at any point on a Cartesian plane. 24
Let’s Sum It Up!
● The line passing through the focus and the vertex is
called the axis of symmetry of the parabola.
● If, a parabola with a horizontal axis of symmetry will
open to the left and the focus will lie on the left of the
directrix. If , a parabola with a vertical axis of symmetry
will open upwards and the focus will lie above the
directrix.
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Key Formulas
Concept Formula Description
Use these formulas
Standard Form of
when finding the
the Equation of a
equation of a parabola
Parabola with Vertex
when the vertex is at
at the Origin
the origin given its
focus and directrix.
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Key Formulas
Concept Formula Description
Use these formulas
Standard Form of
when finding the
the Equation of a
equation of a parabola
Parabola with Vertex
when the vertex is at
at
given its focus and
directrix.
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Lesson 3: PARABOLA
Graphs of Parabola
Capstone
Precalculus
Project
Science, Technology, Engineering, and Mathematics
Learning Competencies
At the end of the lesson, you should be able to do the following:
● Graph a parabola in a rectangular
coordinate system.
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Learning Objectives
At the end of the lesson, you should be able to do the following:
● Graph a parabola given its properties or equation.
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Recall: Definition of a Parabola
A parabola is the set of y
all points in a plane that
are equidistant from a
fixed line called the parabola
directrix and a fixed focus
point not on the line x
called the focus.
directrix
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Recall: Other Parts and Properties of a Parabola
line that divides a parabola
Axis of symmetry
into two equal parts
y
x
(h ,𝑘) Vertex
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Recall: Other Parts and Properties of a Parabola
distance from the vertex to
c focal distance the focus or to the directrix;
x also equal to
c
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Recall: Other Parts and Properties of a Parabola
y
• line segment
• connects two points of the parabola
• passes through the focus
latus rectum
• parallel to the directrix
• length:
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Graph and Properties of Parabolas
Equation:
Opening: upward
Focus:
Directrix:
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Graph and Properties of Parabolas
Equation:
Opening: downward
Focus:
Directrix:
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Graph and Properties of Parabolas
Equation:
Opening: to the right
Focus:
Directrix:
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Graph and Properties of Parabolas
Equation:
Opening: to the left
Focus:
Directrix:
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Graph and Properties of Parabolas
Here are other important parts and properties of a
parabola.
a. The vertex of a parabola is at .
b. The axis of symmetry or axis of the parabola is a line
that divides the parabola into two equal parts. It
passes through the focus and the vertex.
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Graph and Properties of Parabolas
c. The focal distance is the distance from the vertex to
the focus. It is also the distance from the vertex to
the directrix.
d. The latus rectum is a line segment connecting two
points of the parabola, passing through the focus
and parallel to the directrix. The length of the latus
rectum is .
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Graphing Parabolas
To graph parabolas, we can follow these steps:
● Identify its parts and properties.
● Plot the focus, directrix, vertex, and the
endpoints of the latus rectum on the coordinate
plane.
● Sketch the graph based on its parts and
properties.
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Tips
Graphing the Latus Rectum
The length of the latus rectum is .
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Tips
For a vertical parabola, the endpoints of
the latus rectum are units to the left and
the right of the focus.
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Tips
For a horizontal parabola, the endpoints
of the latus rectum are units above and
below the focus.
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Let’s Practice!
Where does the graph of the parabola given by the
equation open?
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Let’s Practice!
Identify the opening, vertex, focus, directrix, and
length of the latus rectum of the parabola given by
the equation .
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Let’s Practice!
Graph the parabola with focus at , and whose
directrix is given by the equation .
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Let’s Practice!
Graph the parabola that opens downward, and
whose endpoints of the latus rectum are at and .
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Try It!
Where does the graph of the parabola
given by the equation open?
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Try It!
Identify the opening, vertex, focus,
directrix, and length of the latus rectum
of the parabola given by the equation .
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Try It!
Graph the parabola with focus at and
whose directrix is given by the equation .
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Try It!
Graph the parabola that opens to the
right, and whose endpoints of the latus
rectum are at and .
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Let’s Sum It Up!
● A parabola is the set of all points in a plane that are
equidistant from a fixed line called the directrix and a
fixed point not on the line called the focus.
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Let’s Sum It Up!
● The standard form of the equation of the parabola with
vertex at depends on its opening.
Opening Standard Form of Equation
upward
downward
to the right
to the left
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Let’s Sum It Up!
● The other important parts and properties of the
parabola are as follows:
o The vertex of the parabola is at .
o The axis of symmetry or axis of the parabola is a
line that divides the parabola into two equal parts. It
passes through the focus and the vertex.
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Let’s Sum It Up!
o The focal distance is the distance from the
vertex to the focus. It is also the distance from
the vertex to the directrix.
o The latus rectum is a line segment connecting
tow points of the parabola, passing through the
focus and parallel to the directrix. The length of
the latus rectum is .
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Let’s Sum It Up!
● To sketch the graph of a parabola, we can follow
these steps:
o Determine its parts and properties.
o Graph the focus, vertex, directrix, and
endpoints of the latus rectum.
o Draw a curve that passes through the vertex
and the endpoints of the latus rectum.
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Lesson 3: PARABOLA
Applications of Parabolas in Real-Life
Situations
Capstone
Precalculus
Project
Science, Technology, Engineering, and Mathematics
Learning Competency
At the end of the lesson, you should be able to do the following:
Solve situational problems involving
parabolas.
59
Learning Objectives
At the end of the lesson, you should be able to do the following:
● Identify real-life applications of parabolas.
● Solve real-life problems involving parabolas.
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Applications of Parabolas
These radio signals or light
waves are very faint, but since
they are collected at the focus,
a stronger sound or image is
produced.
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Applications of Parabolas
Suspension bridges and
arches are also parabolic in
nature. This allows for more
support to the whole
structure.
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Let’s Practice!
A paraboloid light reflector is 12 inches wide and 6
inches deep. Where should the light source be placed
in order to produce a concentrated beam of light?
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Let’s Practice!
The path of the water from the spout of a fountain
on the ground travels a parabolic path and then
lands on the ground. The horizontal distance traveled
by the water is 8 feet and the maximum height
reached by the water is 8 feet. Write a function that
gives the height of the water given its horizontal
distance from the spout of the fountain assuming
that the spout is at the origin.
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Let’s Practice!
The cable of a suspension bridge forms a parabolic
arc such that the lowest point of the cable is at the
middle of two towers. The two towers are 210 meters
high and 1 000 meters apart. The lowest point of the
cable is 10 meters above the road. Find the height of
the cable 300 meters away from a tower.
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Try It!
A flashlight is shaped like a paraboloid.
If the paraboloid is 5 cm deep and 8 cm
wide on its surface, at what location
should the light source be placed?
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Try It!
An object is thrown upward from the
ground. Its trajectory is parabolic, its
maximum height is 6 meters, and the
horizontal distance it traveled is 12 meters
before it reached the ground. Write a
function that gives the height of the object
given its horizontal distance from the
starting point assuming that the starting
point is at the origin.
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Try It!
A cable is supported by two poles such
that the cable is parabolic in shape. The
lowest part of the cable is exactly
halfway between the two poles and is 8
feet above the ground. The two poles
are both 10 feet high and they are 16
feet apart. Find the height of the cable 2
feet away from a tower.
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Remember
The width of the parabolic object or the
diameter of the paraboloid is not
necessarily the latus rectum.
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Let’s Sum It Up!
● A parabola is the set of all points in a plane that are
equidistant from a fixed line called the directrix and a
fixed point not on the line called the focus.
● The features and parts of a parabola are the focus,
directrix, vertex, axis of symmetry, and latus rectum.
The exact locations of these features depend on the
equation of the parabola.
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Let’s Sum It Up!
● Parabolas have applications in different fields such as
physics and engineering. Real-life objects that are
parabolic in nature include satellite dishes, suspension
bridges, spotlights, headlights, and trajectory of objects.
● The reflective property of parabolas allow spotlights to
emit a focused beam and satellite dishes to receive
sound or light waves.
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Let’s Sum It Up!
● The trajectories of projectiles resemble parabolas.
● The parabolic structure of suspension bridges and
arches give them ample support to withstand
compression and tension.
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Let’s Sum It Up!
● In order to solve word problems on parabolas, it is
necessary to have knowledge on its graphs, equations,
and properties. Word problems on parabolas include
finding the focus, the equation of the parabola, and a
point on the parabola.
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