
Skills
to Master
·
Recognize and
use the standard equation for a parabola that opens up or down
·
Recognize and
use the standard equation for a parabola that opens left or right
·
The
Parabola
You studied the
parabola as a function in chapter 3.
Parabolas are another kind of conic section. We are not concerned with examining the
parabola as a function but, like the circle before, as a set of points related
by a general form of an equation. We
will look at parabolas that open either up or down and also at parabolas
opening to the left or to the right.
The standard form of a
parabola opening up or down is
, where h and k are the x- and y-coordinates of the
vertex and the value of a will
determine whether the parabola opens up or down. As before, if
(positive), then the
parabola opens up; if
(negative), then the
parabola opens down.
The standard form of a
parabola that opens left or right is the same as above but with the variables x and y exchanged and the values of h
and k exchanged. That is,
. For this form, if
(positive), the parabola
opens to the right; if
(negative), the parabola
opens to the left.
Example
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Example
Find the equation of
the parabola with the given properties.
With vertex at (0, 2),
opening up and passing through (1, 4).
With vertex at (3, -4),
opening to the left, and passing through (-1, -6).
With vertex at (2, 1),
opening to the right, and passing through (4, 3).
With vertex at (-1,
-3), opening down, and passing through (1, -15).