Write an equation of a translated parabola

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EXAMPLE 3
Write an equation of a translated parabola
Write an equation of the parabola whose vertex is
at (2, 3) and whose focus is at
(4, 3).
SOLUTION
STEP 1
Determine the form of the equation. Begin by
making a rough sketch of the parabola. Because
the focus is to the left of the vertex, the
parabola opens to the left, and its equation has
the form
(y k)2 4p(x h) where p lt 0.
2
EXAMPLE 3
Write an equation of a translated parabola
STEP 2
Identify h and k. The vertex is at (2,
3), so h 2 and k 3.
STEP 3
Find p. The vertex (2, 3) and focus (4, 3) both
lie on the line y 3, so the distance between
them is p 4
(2) 2, and thus p 2. Because p lt 0, it
follows that p 2, so 4p 8.
3
EXAMPLE 3
Write an equation of a translated parabola
ANSWER
The standard form of the equation is
(y 3)2 8(x 2).
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EXAMPLE 4
Write an equation of a translated ellipse
Write an equation of the ellipse with foci at (1,
2) and (7, 2) and co-vertices at (4,
0) and (4, 4).
SOLUTION
STEP 1
Determine the form of the equation. First sketch
the ellipse. The foci lie on the major axis, so
the axis is horizontal. The equation has this
form
5
EXAMPLE 4
Write an equation of a translated ellipse
STEP 2
Identify h and k by finding the center, which is
halfway between the foci (or the co-vertices)
(4, 2)
STEP 3
Find b, the distance between a co-vertex and the
center (4, 2), and c, the distance between a
focus and the center. Choose the co-vertex (4, 4)
and the focus (1, 2) b 4 2 2 and c
1 4 3.
6
EXAMPLE 4
Write an equation of a translated ellipse
STEP 4
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EXAMPLE 5
Identify symmetries of conic sections
Identify the line(s) of symmetry for each conic
section in Examples 1 4.
SOLUTION
For the hyperbola in Example 2, x 1 and y 3
are lines of symmetry
For the circle in Example 1, any line through the
center (2, 3) is a line of symmetry.
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EXAMPLE 5
Identify symmetries of conic sections
For the ellipse in Example 4, x
4 and y 2 are lines of symmetry.
For the parabola in Example 3, y 3 is a line of
symmetry.
9
for Examples 3, 4 and 5
GUIDED PRACTICE
5.
Write the equation of parabola with vertex at
(3, 1) and focus at (3, 2).
Write the equation of the hyperbola with vertices
at (7,3) and (1, 3) and foci at (9, 3) and
(1, 3).
6.
10
for Examples 3, 4 and 5
GUIDED PRACTICE
Identify the line(s) of symmetry for the conic
section.
7.

1
11
for Examples 3, 4 and 5
GUIDED PRACTICE
Identify the line(s) of symmetry for the conic
section.
(x 5)2 8(y 2).
8.
12
for Examples 3, 4 and 5
GUIDED PRACTICE
Identify the line(s) of symmetry for the conic
section.
9.
(y 2)2

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