This document discusses the translation of conic sections, including parabolas, ellipses, and hyperbolas. It provides the standard equations for each type of conic section and explains how to translate them by shifting the origin to a new point (h, k). Several examples are worked through step-by-step, showing how to identify the type of conic, its standard form, and its translated form with the new center (h, k) given. Homework problems are assigned.

1. 10.4 Translation of Conics
Matthew 7:21-23 "“Not everyone who says to me, ‘Lord,
Lord,’ will enter the kingdom of heaven, but the one who
does the will of my Father who is in heaven. On that day
many will say to me, ‘Lord, Lord, did we not prophesy in
your name, and cast out demons in your name, and do many
mighty works in your name?’ And then will I declare to
them, ‘I never knew you; depart from me, you workers of
lawlessness.’"

2. 10.4 Translation of Conics
Parabola: ( x − h ) 2
= 4 p( y − k)

3. 10.4 Translation of Conics
Parabola: ( x − h ) 2
= 4 p( y − k)
center is at ( h, k ) rather than the origin
so everything is shifted ...
horizontally h units
vertically k units

4. 10.4 Translation of Conics
Parabola: ( x − h ) 2
= 4 p( y − k)
center is at ( h, k ) rather than the origin
so everything is shifted ...
horizontally h units
vertically k units
2
Same for: (y − k) = 4 p ( x − h)

5. 10.4 Translation of Conics
2 2
Ellipse: ( x − h) +
(y − k) =1
2 2
a b
(major axis is horizontal)
Translations work the same here as well.

6. 10.4 Translation of Conics
2 2
Ellipse: ( x − h) +
(y − k) =1
2 2
a b
(major axis is horizontal)
2 2
( x − h) +
(y − k) =1
2 2
b a
(major axis is vertical)
Translations work the same here as well.

7. 10.4 Translation of Conics
2 2
( x − h) (y − k)
Hyperbola: a 2
−
b 2
=1
(opens left-right)
h is always ‘tied’ to x ... k is always ‘tied’ to y

8. 10.4 Translation of Conics
2 2
( x − h) (y − k)
Hyperbola: a 2
−
b 2
=1
(opens left-right)
2 2
(y − k) −
( x − h) =1
2 2
a b
(opens up-down)
h is always ‘tied’ to x ... k is always ‘tied’ to y

9. 10.4 Translation of Conics
page 782 # 24 (read directions)

10. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1

11. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1
2x + ( y − 2y + 1) = 2
2 2

12. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1
2x + ( y − 2y + 1) = 2
2 2
2 2
2x + ( y − 1) = 2

13. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1
2x + ( y − 2y + 1) = 2
2 2
2 2
2x + ( y − 1) = 2
2
x ( y − 1)
2
+ =1
1 2

14. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1
2x + ( y − 2y + 1) = 2
2 2
2 2
2x + ( y − 1) = 2
2
x ( y − 1)
2
+ =1
1 2
h=0 a= 2
k =1 b =1
c =1

15. 10.4 Translation of Conics
page 782 # 24 (read directions)
2 2
2x + y = 2y + 1
2x + ( y − 2y + 1) = 2
2 2
ellipse
2
2x + ( y − 1) = 2
2 center : ( 0, 1)
2
(
vertices : 0, 1 ± 2 )
x ( y − 1)
2
+ =1 foci : ( 0, 0 ), ( 0, 2 )
1 2
major : 2 2
h=0 a= 2 minor : 2
k =1 b =1
c =1

16. 10.4 Translation of Conics
page 782 # 28

17. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1

18. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1
2 2
x − y = 10x − 10y + 1

19. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1
2 2
x − y = 10x − 10y + 1
(x 2
− 10x ) − ( y − 10y ) = 1
2

20. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1
2 2
x − y = 10x − 10y + 1
(x 2
− 10x ) − ( y − 10y ) = 1
2
(x 2
− 10x + 25 ) − ( y − 10y + 25 ) = 1
2

21. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1
2 2
x − y = 10x − 10y + 1
(x 2
− 10x ) − ( y − 10y ) = 1
2
(x 2
− 10x + 25 ) − ( y − 10y + 25 ) = 1
2
2 2
( x − 5) −
( y − 5) =1
1 1

22. 10.4 Translation of Conics
page 782 # 28
2 2
x − y = 10 ( x − y ) + 1
2 2
x − y = 10x − 10y + 1
(x 2
− 10x ) − ( y − 10y ) = 1
2
(x 2
− 10x + 25 ) − ( y − 10y + 25 ) = 1
2
2 2
( x − 5) −
( y − 5) =1
1 1
hyperbola opening left − right

23. 10.4 Translation of Conics
page 782 # 28 h = 5 a =1
2 2
x − y = 10 ( x − y ) + 1 k = 5 b =1
2 2 c= 2
x − y = 10x − 10y + 1
(x 2
− 10x ) − ( y − 10y ) = 1
2
(x 2
− 10x + 25 ) − ( y − 10y + 25 ) = 1
2
2 2
( x − 5) −
( y − 5) =1
1 1
hyperbola opening left − right

24. 10.4 Translation of Conics
page 782 # 28 h = 5 a =1
2 2
x − y = 10 ( x − y ) + 1 k = 5 b =1
2 2 c= 2
x − y = 10x − 10y + 1
hyperbola
(x 2
− 10x ) − ( y − 10y ) = 1
2
center : ( 5, 5 )
(x 2
− 10x + 25 ) − ( y − 10y + 25 ) = 1
2
vertices : ( 6, 5 ) , ( 4, 5 )
2
( x − 5) − ( y − 5) = 1 2 foci : (5 ± 2, 5 )
1 1 asymptotes :
y = x; y = −x + 10
hyperbola opening left − right

25. 10.4 Translation of Conics
Video: Why do we have to learn all this stuff???
Meeting “The Wizard” ...

26. 10.4 Translation of Conics
Video: Why do we have to learn all this stuff???
Meeting “The Wizard” ...

27. 10.4 Translation of Conics
HW #8
“What happens is not as important as how you react
to what happens.” Thaddeus Golas