The Parabola

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The Parabola

• Section 10.2

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To sketch a parabola, first write it in standard
form, then identify the vertex, (h,k) and the
distance from the vertex to the focus, p.
Example 1 Identify the vertex, focus, and
directrix of the parabola, then sketch it.
First, put the equation in standard form x2
2y This gives us our vertex (0,0). Plot this
point. Next solve for p 4p 2 p 0.5 Use
this value for p to identify the focus (h,kp)
(0,0.5). Then find the directrix y k-p ?
y -0.5. To aid in sketching, find two more
points on the graph. The easiest way to do this
is to move 2p units away from the focus in a
direction parallel to the directrix. Once you
have plotted these two additional points, sketch
the parabola through the three plotted points.
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Example 2 Find the vertex, focus, and directrix
of the parabola and sketch its graph.
First, put the equation in standard form y2 -x
This gives us our vertex (0,0). Plot this point.
Next solve for p 4p -1 p -0.25
Use this value for p to identify the focus
(hp,k) (-0.25,0).
Then find the directrix x h-p ? x 0.25.
To aid in sketching, find two more points on the
graph. The easiest way to do this is to move 2p
units away from the focus in a direction parallel
to the directrix.
Once you have plotted these two additional
points, sketch the parabola through the three
plotted points.
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Example 3 Find the vertex, focus, and directrix
of the parabola and sketch its graph.
First, put the equation in standard form (y-1)2
8(x5)
This gives us our vertex (-5,1). Plot this
point.
Next solve for p 4p 8 p 2
Use this value for p to identify the focus
(hp,k) (-3,1).
Then find the directrix x h-p ? x -7.
To aid in sketching, find two more points on the
graph. The easiest way to do this is to move 2p
units away from the focus in a direction parallel
to the directrix.
Once you have plotted these two additional
points, sketch the parabola through the three
plotted points.
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Example 4 Find the vertex, focus, and directrix
of the parabola and sketch its graph.
First, put the equation in standard form. This
requires completing the square. (x-1)2 -8(y1)
This gives us our vertex (1,-1). Plot this
point.
Next solve for p 4p -8 p -2
Use this value for p to identify the focus
(h,kp) (1,-3).
Then find the directrix y k-p ? y 1.
To aid in sketching, find two more points on the
graph. The easiest way to do this is to move 2p
units away from the focus in a direction parallel
to the directrix.
Once you have plotted these two additional
points, sketch the parabola through the three
plotted points.
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Example 5 Find the standard form of the equation
for the parabola with vertex at the origin and
the directrix x 2.
Start with the standard form for a
parabola Replace h and k each with zero because
the vertex is at the origin Since the directrix
is -p units from the vertex, p must equal -2.
Place this value in next Finally, simplify the
equation
(y-k)2 4p(x-h) (y-0)2 4p(x-0) (y-0)2
4(-2)(x-0) y2 -8x
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Example 6 Find the standard form of the equation
for the parabola with vertex (2,1) and focus
(2,4).
Start with the standard form for a
parabola Replace h and k each with 2 and 1
respectively Since the focus is p units from
the vertex, p must equal 3. Place this value in
next Finally, simplify the equation
(x-h)2 4p(y-k) (x-2)2 4p(y-1) (x-2)2
4(3)(y-1) (x-2)2 12(y-1)