1. 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 < 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 < 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).

4. 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:
(x – h)2 a2 +
(y – k)2 b2
= 1

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)
(h, k) = 2 ) ( ,
= (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
Find a. For an ellipse, a2 = b2 + c2 = = 13, so a = 13
ANSWER
The standard form of the equation is
(x – 4)2 13 +
(y – 2)2 4
= 1

7. EXAMPLE 5
Identify symmetries of conic sections
Identify the line(s) of symmetry for each conic section in Examples 1 – 4.
SOLUTION
For the circle in Example 1, any line through the center (2, – 3) is a line of symmetry.
For the hyperbola in Example 2 x = – 1 and y = 3 are lines of symmetry

8. 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. GUIDED PRACTICE
for Examples 3, 4 and 5 5.
parabola with vertex at (3, – 1) and focus at (3, 2).
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 (x – h)2 = 4p(y – k) where p > 0.

10. GUIDED PRACTICE
for Examples 3, 4 and 5
STEP 2
Identify h and k. The vertex is at (3,– 1), so h = 3 and k = –1.
STEP 3
Find p. The vertex (3, – 1) and focus (3, 2) both lie on the line x = 3, so the distance between them is | p | = | – 2 – (– 1) | = 3, and thus p = + 3. Because p > 0, it follows that p = 3, so 4p = 12.
The standard form of the equation is (x – 3)2 = 12(y + 1).
ANSWER

11. GUIDED PRACTICE
for Examples 3, 4 and 5 6.
Write an equation of the hyperbola with vertices at (– 7,3) and (– 1, 3) and foci at (– 9, 3) and (1, 3).
SOLUTION
STEP 1
Determine the form of the equation. First sketch the horizontal. The foci lie on the x- axis with a horizontal traverse axis. The equation has this form:
(x – h)2 a2
(y – k)2 b2
= 1 –

12. ) ( GUIDED PRACTICE for Examples 3, 4 and 5 STEP 2
Identify h and k by finding the center, which is halfway between the foci (or the co-vertices) 2
(h, k) = – ) ( ,
= (– 4, 3)
STEP 3
Find b, the distance between a vertex and the center (– 4, 3), and c, the distance between a focus and the center. Choose the co-vertex (–7, 3) and the focus (–9, 3): b = | – 7 –(– 4) | = 3 and c = | –9 – (– 4) | = 5.

13. GUIDED PRACTICE
for Examples 3, 4 and 5
STEP 4
Find a. For an ellipse, b2 = c2 + a2 = = 16, so b = 4
ANSWER
The standard form of the equation is
(x + 4)2 9 –
(y – 3)2 16
= 1

14. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section.
(x – 5)2 64
(y)2 16 7. +
= 1
For the ellipse the lines of symmetry are x = 5 and y = 0.
ANSWER

15. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section. 8.
(x + 5)2 = 8(y – 2).
For parabola the lie of symmetry are x = – 5
ANSWER

16. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section. 9.
(x – 1)2 49
(y – 2)2 –
= 1
121
For horizontal lines of symmetry are x = 1 and y = 2.
ANSWER