![Page 1: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/1.jpg)
1
Consider three points: and
Determine the distance between each point and .
Determine the distance between each point and the line,
πͺ(β1.5 ,2.25)
π©(2 ,4 )
π΄(1 ,1)
![Page 2: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/2.jpg)
2
Precalculus
The PARABOLAConic Sections:
Von Christopher G. Chua
![Page 3: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/3.jpg)
3
PARABOLAS:Equations and Graphs
In the next two sessions, you are expected to develop the ability toβ¦
OUR LEARNING
GOALS
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 parabolas.
This slideshow presentation will be made available through the course website: mathbychua.weebly.com.Download the document to use it as reference.
![Page 4: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/4.jpg)
DEFINITION PARABOLA
Let be a given point, and β a given line not containing . The set of all points such that its distances from and from β are equal, is called a parabola. The point is its focus and the line β its
directrix.
![Page 5: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/5.jpg)
55
EQUATION of a PARABOLA
π (π₯ , π¦)
π·(π₯ ,βπ)
πΉ (0 ,π)
The standard form of the equation of a parabola
with vertex is at the point of origin and opens
upward or downward is
If a parabola with its vertex at the opens
sideways, the standard form of its equation is verte
x
axis of symmetry
![Page 6: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/6.jpg)
6
π₯2=4ππ¦ ,π>0
When the parabola opens
upward
π₯2=4ππ¦ ,π<0
When the parabola opens
downward
π¦ 2=4ππ₯ ,π>0
When the parabola opens
to the right
π¦ 2=4ππ₯ ,π<0
When the parabola opens
to the left
TYPES of PARABOLA4
1 2
3 4
SIDE QUESTION:
What do you notice about the position of the
focus with respect to the
graph?
![Page 7: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/7.jpg)
LETβS DISCUSS
To which direction does each of the following parabolas open to?
Determine the Orientation
![Page 8: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/8.jpg)
EXAMPLE
How do we determine the focus and directrix of a parabola with vertex at the origin?
Since the quadratic variable is and the coefficient of is positive, the parabola opens upward.
The vertex is at and the axis of symmetry is the -axis or
Compared to , we can determine that The focus of the parabola is therefore at and its directrix
is the line
Focus & Directrix
![Page 9: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/9.jpg)
YOUR TURN!
Determine the focus and directrix of the following parabola based from the given equation.
Focus & Directrix
![Page 10: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/10.jpg)
WHAT IFβ¦
But what if the parabola does not have its vertex at the
point of origin?
!
![Page 11: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/11.jpg)
PARABOLAS with Vertex NOT on
WHAT IFβ¦
The standard form of the equation of a parabola that opens upward or
downward is
If a parabola opens sideways, the standard form of its equation is
(π₯βh)2=4π (π¦βπ)
(π¦βπ)2=4 π(π₯βh)
v
![Page 12: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/12.jpg)
EXAMPLE
Describe the parabola, .What is its graphβs orientation?What are the coordinates of its vertex?What is the value of in the equation?So, if , what are the coordinates of the focus?What is the equation of the line that is the directrix?
The graph opens upward.Its vertex is at If , then The coordinates of the focus are The directrix is .
![Page 13: [PPT]The PROBLEM and its BACKGROUND · Web viewEQUATION of a PARABOLA π(π₯, π¦) π·(π₯, −π) πΉ(0, π) The standard form of the equation of a parabola with vertex](https://reader031.vdocuments.us/reader031/viewer/2022021818/5aa485cb7f8b9a1d728bfda4/html5/thumbnails/13.jpg)
13
v
focus
directrix
Axis of symmetry
v
focus
directrix
Axis of symmetry