Conic Sections

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Conic Sections

• Parabola

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Conic Sections - Parabola
The parabola has the characteristic shape shown
above. A parabola is defined to be the set of
points the same distance from a point and a line.
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Conic Sections - Parabola
Focus
Directrix
The line is called the directrix and the point is
called the focus.
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Conic Sections - Parabola
Axis of Symmetry
Focus
Directrix
Vertex
The line perpendicular to the directrix passing
through the focus is the axis of symmetry. The
vertex is the point of intersection of the axis
of symmetry with the parabola.
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Conic Sections - Parabola
Focus
d1
Directrix
d2
The definition of the parabola is the set of
points the same distance from the focus and
directrix. Therefore, d1 d2 for any point (x,
y) on the parabola.
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Finding the Focus and Directrix

• Parabola

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Conic Sections - Parabola
Therefore, the distance p from the vertex to the
focus and the vertex to the directrix is given by
the formula
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Conic Sections - Parabola
Using transformations, we can shift the parabola
yax2 horizontally and vertically. If the
parabola is shifted h units right and k units up,
the equation would be
The vertex is shifted from (0, 0) to (h, k).
Recall that when a is positive, the graph opens
up. When a is negative, the graph reflects
about the x-axis and opens down.
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Example 1

• Graph a parabola.
• Find the vertex, focus and directrix.

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Parabola Example 1
Make a table of values. Graph the function.
Find the vertex, focus, and directrix.
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Parabola Example 1
The vertex is (-2, -3). Since the parabola opens
up and the axis of symmetry passes through the
vertex, the axis of symmetry is x -2.
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Parabola Example 1
x y -2 -1 0 1 2 3 4
-3
Make a table of values.
-1
Plot the points on the graph!Use the line of
symmetry to plot the other side of the graph.
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Parabola Example 1
Find the focus and directrix.
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Parabola Example 1
The focus and directrix are p units from the
vertex where
The focus and directrix are 2 units from the
vertex.
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Parabola Example 1
2 Units
Focus (-2, -1) Directrix y -5