parabolas; hyperbolas

1. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
Hyperbolas
(3, -1)
(7, -1)(-1, -1) 4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1
When we use completing the square to get to the standard
form of the hyperbolas, depending on the signs,
we add a number or subtract a number from both sides.

2. 9(x + 1)24(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
Group the x’s and the y’s:
4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients
4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square
4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9
16 –9
Hyperbolas
(y – 2)2 (x + 1)2
32 22
– = 1
Center: (-1, 2), x-rad = 2, y-rad = 3
The hyperbola opens up and down.
9 4

3. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2)
Draw. Get y-int:
–y2 + 2y + 15 = 0
y2 – 2y – 15 = 0
(y – 5) (y + 3) = 0
y = 5, -3
(15, 0)
(15, 2)
More Graphs of Parabolas
(16, 1)
(0, -3)