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Parametric Equations |
The parametric equations of a
quadratic
polynomial,
parabola |
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The
parametric equations of the parabola, whose axis of symmetry is parallel
to the y-axis
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The parametric equations of the
parabola, whose axis of symmetry is
parallel to the x-axis |
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| The parametric equations of a
quadratic
polynomial,
parabola |
| The parametric equations of the parabola,
whose axis of symmetry is parallel
to the y-axis
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|
The quadratic polynomial y
=
a2x2
+ a1x + a0
or y
-
y0
= a2(x
- x0)2,
V(x0,
y0) |
 |
| are
the coordinates of translations of the source quadratic y
=
a2x2,
can be transformed to the parametric |
| form
by
substituting x
- x0
= t. |
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| are the
parametric equations of the
quadratic polynomial. |
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| Example: Given
are the parametric equations, x
=
t + 1
and y
=
- t2 + 4,
draw the graph of the curve. |
| Solution:
The equation x
=
t + 1
solve for t
and plug into y
=
- t2 + 4,
thus |
| t
=
x
- 1
=> y
=
- t2 + 4,
y
=
- (x
- 1)2 + 4 |
| i.e., y
-
4
=
- (x
- 1)2
or y
=
- x2 + 2x + 3 translated parabola with the vertex V(x0,
y0),
so V(1,
4). |
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| The parametric equations of the parabola, whose axis of symmetry is
parallel to the x-axis |
| The
quadratic expression y2
=
2px,
where p
is the distance between focus and directrix, |
| represents
the source or the
vertex form of the conic section called parabola
with the vertex at the origin |
| whose
axis
of symmetry coincide with the x-axis. |
| If
we rewrite the above equation into x
= ay2 then, |
| |
x
= ay2 + by + c
or
x
- x0
= a(y
- y0)2 |
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|
| represents
the translation of the parabola in the direction of the coordinate axes
by x0
and y0
i.e.,
V(x0,
y0). |
| Thus,
the parabola, whose axis of symmetry is parallel to the x-axis,
can be transformed to parametric form |
| by
substituting y
- y0
= t
into the above equation. |
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are the parametric equations
of the parabola. |
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| Example: Given
is the parabola x
=
- y2 + 2y + 3,
write its parametric equations and draw the graph. |
| Solution:
Rewrite given equation by calculating the coordinates of
translations x0
and y0
or using completing |
| the
square method, we get x
-
4
=
- (y
- 1)2,
where the vertex V(x0,
y0),
so V(4,
1). |
| By
substituting y
- 1
= t
obtained are |
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the parametric equations of the parabola. |
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| Functions
contents A |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
