3. Group Members
Adeel Iftikhar ID: 20397
Salman Haider ID:20357
Rahul Rai ID: 20400
Jahan Ban Hassan ID:
Abdul Latif ID:
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5. Agenda of today’s
presentation
Definition & types of parabola.
History of parabola.
How to solve parabola.
Quadratic Equations & function link with
parabola.
Examples in Engineering.
Examples in Business.
Examples regarding Einstein's theory.
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7. Parabola
A parabola is a two dimensional
curve.
Mirror symmetrical Curve.
Which is approximately U-Shaped.
This parabola represents Quadratic
equation, or Quadratic Functions
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9. Circle
The type of
parabola in :-
• Parabola
facing up
• Parabola
facing down
• Are equal
• & joined with
the same
centre.
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10. Ellipse
The case of uneven circle
It has uneven radius
Its more of an oval shaped parabola
In this the parabola facing & down are
joined
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14. All types of Parabola’s
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15. Edison Works Regarding
Parabola
Edison first invented the parabola
reflector.
To focus light on one point.
It uses used the principle on amplification
as well
In which parabolic reflector reflected the
light.
By amplifying light.
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16. Cont.
It is the men first step to use light.
This also gives him a giant leap in the field
of Mirror.
By knowing their focus property.
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18. Parabolic Microphones
In 21st century.
Engineers made use of the principle of
Echo sound & parabola.
To make a device known as Parabolic
Microphones.
Which helps in collecting the sound.
From long distances.
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20. Construction of Arches
Construction of arches can be done by
parabolas.
It construction they balance the forces at
any point.
So vertex plays a critical role in this.
Which determines the strength &
Also the maximum or minimum forces
The building can handle.
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23. History
Parabolas were first made by
Menaechmus in the fourth century BC.
While solving double cubes.
The area enclosed by a parabola and a
line segment is called "parabola
segment“.
Was made by Archimedes.
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24. History
The name "parabola" is due to Apollonius.
Who discovered many properties.
Conic sections
Focus property of the parabola and other
conics
Is due to Pappus.
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25. History
Galileo showed that the path of a
projectile
Follows a parabola.
& Situation of uniform acceleration
Due to gravity.
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26. Parabolas were 1st used in
Parabolic Reflectors were made due to
parabolas.
Which then produces Telescopes.
Isaac Newton built the First Reflecting
Telescope in 1668,
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27. Equation of Parabola
The last Equation of parabola is
y = ax2 + bx + c
Which include:-
X-intercept
Y-intercept
Concavity
Vertex
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28. X-Intercept
The point at which parabola touches X-
axis
Can be found by putting
y = 0
But this is optional dependent on
equation.
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29. Y-Intercept
The point at which parabola touches Y-
axis
Can be found by putting
x = 0
But this is optional dependent on
equation.
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30. Concavity
Determines the direction of
parabola
May be Concave up or Concave
down.
Which determines either graph is
Rising
or Falling.
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32. Parabola & 21st Century
Parabola have become a
great tool in 21st century.
The domestic devices are
1. Parabolic Solar Panels.
2. Parabolic solar Cookers.
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34. Parabola & Chemistry
Now a days, chemist are also using
parabolas.
Like to compare densities of different
chemicals.
As density = mass / volume
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36. Cont.
The chemical with the higher density
makes parabola.
The vertex of this chemical is the measure
Of density
Which shows how much denser is the
chemical.
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37. Projectile Motion
A parabolic motion of an object which
works under the constant force of gravity.
Vertex is measure of minimum
gravitational force at this height.
This is the result of Newtonian physics.
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41. How to solve parabola
General equation of parabola
f(x) = y = ax2 + bx + c
Concavity
X- intercept
Y- intercept
Vertex
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42. 1st Concavity
Of quadratic equation
f(x)= y = ax2 + bx + c
If a < 0 then parabola is concave down.
If a > 0 then parabola is concave up.
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43. 2nd X-intercept
Of quadratic equation
f(x)= y = ax2 + bx + c
Putting y = 0
We can find value
By simplifying or if may be
By using formulas like (a + b)2 ,(a - b)2 ,
(a + b)(a - b)
We can find the point at which it touches
x-axis
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44. 3rd Y-intercept
Of quadratic equation
f(x)= y = ax2 + bx + c
By putting x = 0
Quadratic Equation
We can find the point at which it touches
y-axis
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45. 4th Vertex
The maximum point of parabola concave
down (a > 0)
The minimum point of parabola concave
up (a < 0)
X = -b/2a
Y = f(x) = f(-b/2a) or (4ac – b2)/4a
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46. Beside that
The general equations of parabola is
Y = a(x - h)2 + k
Where h & k are vertex points
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47. Conversion from quadratic
equation to general equation
y = 2x2 – 4x + 5 into a(x-h)2 + k
Y = 2(x2 - 2x) + 5 now applying
completing the square method.
(+)ing & (-)ing by 1
Y = 2(x2 – 2x + 12 ) – 2 + 5
Y = 2(x - 1)2 – 2 + 5
Y = 2(x - 1)2 + 3
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48. Cont.
Y = 2(x - 1)2 + 3
Comparing with Y = a(x - h)2 + k
x – 1 = x – h
So h = 1
K = 3
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50. Equivalent Equations
The equation
Y = 2x2 - 4x + 5
& also the equation
Y = 2(x - 1)2 + 3
Are equivalent because they have same
graph.
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51. Example-1
Y = 4x2
Comparing with standard quadratic
equation :-
Y = ax2 + bx + c -> (A)
Y = 4x2 -> (B)
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52. Cont.
By comparing (A) & (B)
a = 4 , b = 0 , c = 0
Concavity :-
As a > 0, so parabola is concave up-ward
X-intercept :-
Putting Y = 0 in (B)
Y = 4x2
0 = 4x2
0 = x2
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53. Cont.
By taking square root
x = 0 (0,0)
Y-intercept
Putting x = 0 in (B)
Y = 4(0)2
Y = 4.(0)
Y = 0 (0,0)
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54. Cont.
Vertex :-
a = 4 , b = c = 0
As, x = -b/2a
x = -(0)/2.4
x = 0/8
x = 0
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55. Cont.
For Y = f(-b/2a) = f(0)
Y = f(0) = 4.(0)2
Y = 4.0
Y = 0 (0,0)
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57. Example-2
Y = -6x2
Comparing with standard quadratic
equation :-
Y = ax2 + bx + c -> (A)
Y = -6x2 -> (B)
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58. Cont.
By comparing (A) & (B)
a = 4 , b = 0 , c = 0
Concavity :-
As a < 0, so parabola is concave down-ward
X-intercept :-
Putting Y = 0 in (B)
Y = -6x2
0 = -6x2
0 = x2
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59. Cont.
By taking square root
x = 0 (0,0)
Y-intercept
Putting x = 0 in (B)
Y = -6(0)2
Y = -6.(0)
Y = 0 (0,0)
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60. Cont.
Vertex :-
a = -6 , b = c = 0
As, x = -b/2a
x = -(0)/2.-6
x = 0/-12
x = 0
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61. Cont.
For Y = f(-b/2a) = f(0)
Y = f(0) = -6.(0)2
Y = -6.0
Y = 0 (0,0)
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64. Applications of Parabola:
The trajectory of a thrown object is a parabolic
curve.
Compound Interest earned on a savings account is a
parabolic curve.
Acceleration and deceleration, if graphed, forms a
parabola.
The surface of water in a rotating container has a
parabolic curve.
parabolas can be found in lenses, lamps, flashlights,
and lighthouses. when a light source is placed at the
parabola's focus, light emanates in parallel rays,
optimizing its intensity.
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65. Applications in Business
Quadratic Equations are used in Business
to find Revenues , Quantity of a product.
In business it helps in predicting the
maximum or minimum revenue.
Example :- q = 6p -> (A)
q = Quantity demanded
p = Price of a unit
As, R = p*q
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66. Cont.
So (*)ing A by p
p.q = 4.p.p
R = 4p2
As this is now a quadratic equation.
R = ap2 + bx + c
By comparing
a = 4 , b = c = 0
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67. Cont.
Concavity :- As a > 0
So parabola is concave up-ward.
R-intercept :- putting p = 0 in Revenue
equation
R = 4.p2
R = 4.(0)2
R = 4.0
R = 0 (0,0)
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68. Cont.
P-intercept :- putting q = 0 in revenue
equation
q = 4p2
0 = 4.p2
0 = p2
p = 0 (0,0)
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69. Cont.
Vertex :-
As p = -b/2a
p = -0/2.4
p = 0/8
p = 0
As for R =f(-b/2a) = f(0) = 4.(0)2
R = 4.0
R = 0 (0,0)
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71. Solar Energy Farms
The solar panels which are used in the
generating electricity.
Have a certain angle.
Through which they direct there energy to
a central point.
By which the electricity is generated.
The parabola has a focus, bouncing all
waves onto a single receiving point
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73. Cont.
In construction of solar panels.
One must remember that the surface
area should be enough.
To collect maximum amount of solar
energy.
And one should place its sensor at a
optimum distance
To collect maximum amount of energy.
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74. Anatomy of the Solar penal
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75. Cont.
In the above figure solar penal are using
focusing property of parabola.
Under which they get solar energy &
Then reflect it back towards
evaporator(sensors).
To get solar energy.
This energy is then stored into condensors
for future use.
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76. Parabolas & Lenses
Parabolas play a vital in
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79. Applications in Construction
Parabola have been used in construction
during 19th century.
They are used in suspension bridges,
concrete bridges , dams etc.
The vertex point in construction is the
point.
At which maximum load can be
supported at a point.
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83. Examples Regarding Einstein’s
Theory
Before Einstein’s theory.
It was considered that Earth is
oval.
And sun is circling around the
earth.
So earth is considered the
centre of the universe.
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84. According to Einstein’s Theory
That all planets are lying on a same plane.
This plane is effected
By the force of Gravity of every planet.
Planets have their on gravitation to
attract meteor , satellites.
And also he concluded that sun is centre
of the whole universe.
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85. Forces Exerted by planet on a
plane
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87. Vertex of Einstein’s theory
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Point of
maximum
force of
earth
88. Examples in Dish Satellite
As parabola have a property to direct
force on a single point.
These principles are used in construction
of Satellite
So that they can direct their forces to their
central point.
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![Final Presentation
Instructor: Miss Rabia Javed
[ Lecturer Business
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