1. U5D5 Have out: Bellwork: Answer the following for the equation:
red pen, highlighter, GP notebook, calculator, ruler
U5D5
Have out:
Bellwork:
Answer the following for the equation:
(a) Write the equation for the ellipse in standard from.
(b) Identify:
center
a, b, & c
foci
(c) Sketch a graph of the ellipse and the foci.
total:

2. Bellwork: 4 4
100 16 +1 +1 +1 +1 1 1 1 +1 1 25 4 +3

3. Bellwork: (2, 3) Center: (2, –2) (0, -2) (4,-2) a = 5 b = 2 c = 4.5810 y x
–10
Bellwork:
(2, 3) +1
Center:
(2, –2) +1
(0, -2)
(4,-2)
a = 5 +1 +1 +1
b = 2
c =
4.58
(2, -7)
foci:
(2, –2 ± 4.58) +1
(2, 2.58)
and
(2, –6.58)
c2 = a2 – b2 +1 +1 +1
c2 = 25 – 4
+1 graphed foci
c2 = 21
total:
Which denominator is bigger?
Y, so this is a vertical ellipse
c ≈ 4.58

4. Hyperbolas Let’s review the four types of conic sections: Parabola
Circle
Ellipse
Hyperbola
Hyperbolas
There is only one left to study:

5. Hyperbola: The set of all points in a plane such that the ________________ of the ___________ of the distances from two fixed points, called the _____, is constant.
absolute value
difference
foci
positive
d2 – d1 is a _________ constant.
Recall:
 For an ellipse, the sum of the distances is a positive constant.
 But for the hyperbola, the difference of the distances is a positive constant.

6. A hyperbola has some similarities to an ellipse.
 The distance from the center to a vertex is ___ units. a
 The distance from the center to a focus is ___ units. c y x
vertices
(0, b)
focus
focus
(-c, 0)
(-a, 0)
(a, 0)
(c, 0)
(0, -b)

7. 2 symmetry transverse 2a vertices conjugate 2b perpendicular
 There are _____ axes of ___________: 2
symmetry
transverse 2a
The ___________ axis is a line segment of length ____ units whose endpoints are _________ of the hyperbola.
vertices
The ___________ axis is a line segment of length ____units that is ________________ to the ____________ axis at the _________.
conjugate 2b
perpendicular y x
conjugate axis
transverse
center
(0, b)
transverse axis
(-c, 0)
(-a, 0)
(a, 0)
(c, 0)
(0, -b)

8. c2 = a2 + b2 asymptote asymptote center branches asymptotes
The values of a, b, and c relate differently for a hyperbola than for an ellipse. For a hyperbola, ___________.
c2 = a2 + b2
Hmm… the ellipse is a “minus” (c2 = a2 – b2) and the hyperbola is a “plus” (c2 = a2 + b2).
That looks like the Pythagorean Theorem!?! y
asymptote
(0, b)
asymptote
As a hyperbola recedes from its _______, the ___________ approach the lines called ______________.
center
branches x
asymptotes
(-c, 0)
(-a, 0)
(a, 0)
(c, 0)
(0, -b)

9. Summary Chart of Hyperbolas with Center (0, 0)
Standard Form of Equation
Direction of Transverse Axis
Foci
Vertices
Length of Transverse Axis
Length of Conjugate Axis
Equations of Asymptotes
horizontal
vertical
(±c, 0)
(0, ±c)
(±a, 0)
(0, ±a) 2a 2a 2b 2b

10. Note: The equations for hyperbolas and ellipses are identical except for a _______ sign. In graphing either, the __________ axis variable, x or y, is the axis which contains the _________ and _______.
minus
positive
vertices
foci
Okay, so what does all of this mean?
With a couple of exceptions that we will talk about, graph hyperbolas like you would an ellipse. But instead of drawing an ellipse, we will draw a rectangle and some lines.

11. Which variable is positive? x So, it’s a horizontal hyperbola a)
I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes.
Which variable is positive? x
So, it’s a horizontal hyperbola a)
–10 x y 10
Center:
(0, 0)
c2 = a2 + b2
a =
(0,3) 5
c2 =
b = 3
c2 = 34
c =
5.83
(-5, 0)
(5, 0)
foci:
(± 5.83, 0)
c ≈ 5.83
vertices:
(±5, 0)
(0,-3)
asymptotes:
vertices

12. Which variable is positive? y So, it’s a vertical hyperbola b)
I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes.
Which variable is positive? y
So, it’s a vertical hyperbola b)
–10 x y 10
Center:
(0, 0)
c2 = a2 + b2
a = 4
c2 =
(0, 4)
b = 7
c2 = 65
(–7,0)
(7,0)
c =
8.06
foci:
(0, ± 8.06)
c ≈ 8.06
(0,–4)
vertices:
(0, ±4)
asymptotes:

13. Take a few minutes to complete Part I.

14. Which variable is positive? x So, it’s a horizontal hyperbola c)
I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes.
Which variable is positive? x
So, it’s a horizontal hyperbola c)
–10 x y 10
Center:
(0, 0)
c2 = a2 + b2
(0, 4)
a = 2
c2 =
b = 4
c2 = 20
c =
4.47
(–2, 0)
(2, 0)
foci:
(± 4.47, 0)
c ≈ 4.47
vertices:
(±2, 0)
(0,–4)
asymptotes:

15. Which variable is positive? So, it’s a vertical hyperbola
I. Graph each hyperbola. Label the vertices and foci. Identify the asymptotes. y
Which variable is positive?
So, it’s a vertical hyperbola
–10 x y 10
Center:
(0, 0)
(0,6)
a = 6
b = 8
(–8,0)
(8,0)
c = 10
c2 = a2 + b2
foci:
(0, ±10)
c2 =
vertices:
(0, ±6)
asymptotes:
c2 = 100
(0,–6)
c = 10

16. a = 3 b = 2 c = equation: x2 y2 32 22 general form
II. Write the standard form equation for each hyperbola. a) y x -7 7
The hyperbola opens left and right.
Therefore, x is positive.
“a” will be under x. b a
a = 3
b = 2
c =
equation: x2 y2 32 22
general form

17. a = 1 equation: b = 2 y2 x2 c = 12 22 general form
II. Write the standard form equation for each hyperbola. b)
The hyperbola opens up and down. y x -5 5
Therefore, y is positive.
“a” will be under y.
To determine “a” and “b”, we need to draw in the rectangle. a b
Look at the places where the asymptotes cross the grid.
Draw a rectangle that goes through the vertices and intersections on the grid.
a = 1
equation:
b = 2 y2 x2
c = 12 22
general form

18. a c a = 2 b = equation: x2 y2 c = 4 22 general form
II. Write the standard form equation for each hyperbola. 5 -5 x y d)
The hyperbola opens left and right.
Therefore, x is positive.
“a” will be under x.
We don’t know b, and drawing a rectangle won’t help since the asymptotes are not given. a c
However, we know the foci.
a = 2
b =
equation: x2 y2
c = 4 22
general form

19. Finish today’s worksheets.