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)
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Precalculus
The PARABOLAConic Sections:
Von Christopher G. Chua
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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.
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.
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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
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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?
LETβS DISCUSS
To which direction does each of the following parabolas open to?
Determine the Orientation
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
YOUR TURN!
Determine the focus and directrix of the following parabola based from the given equation.
Focus & Directrix
WHAT IFβ¦
But what if the parabola does not have its vertex at the
point of origin?
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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
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 .
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v
focus
directrix
Axis of symmetry
v
focus
directrix
Axis of symmetry