H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 136:39 AM
Good MorningPlease grab a warm up, but you don't know how to do it...so hang tight, I'm going to start teaching right away!
You will need the note sheet from last class and notebook for a little supplementary material!
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 2210:08 AM
SWBAT:1) Classify a conic section: parabola, ellipse, circle, hyperbola, line, and point
2) Identify important information in each conic section: center, radius, foci, eccentricity, vertices, and asymptotes.
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 2210:09 AM
What is a conic section?A conic section is the intersection of a plane and a cone.
The types of conic sections will be dependent
on where the plane intersects the cones
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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May 2210:14 AM
Circle Ellipse Parabola Hyperbola
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May 2210:22 AM
How do we classify a conic section by only looking at its equation?
Parabola: either x or y is squared (One variable is squared)
Example: x2 + 15y 12x + 82 = 0
Circle: sum of x2 and y2 AND their coefficients are EQUAL
Example: x2 + y2 40x 52y + 25 = 0
Ellipse: sum of x2 and y2 AND their coefficients are NOT EQUAL if # under x is bigger > Major axis is horizontal
if # under y is bigger > Major axis is vertical
Example: x2 + 4y2 + 14x 16y + 1 = 0
Hyperbola: difference of x2 and y2 If x2 y2 > transverse axis is horizontal
If y2 x2 > transverse axis is vertical
Example: x2 4y2 + 14x 16y + 1 = 0
** equation must be = to 1
** equation must be = to 1
** Please note none of these examples are in the standard equation
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 2312:11 PM
Classify the following Conic Sections:
x + 10x + y2 21 = 0
x2 + 2x + y 1 = 0
x2 y2 2x 8 = 0
9x2 + y2 72x 153 = 0
x2 + y2 + 6x 2y = 0
3x2 + 30x + y + 79 = 0
y2 + x2 + 8y 17 = 0
5x2 + 4x + 5y2 6y +14 = 0
Get a communicator!
parabola
circle
circle
parabola
hyperbola
parabola
hyperbola
hyperbola
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 142:33 PM
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 136:50 AM
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 2212:59 PM
Circles:
Standard Equation: (x h)2 + (y k)2 = r2 Where: (h, k) is the center
r is the radius
**Questions will ask for center and the radius.
Example 1: ( x 4)2 + (x + 3)2 = 25
What is the center?
What is the radius?
** be thinking about how you know
this is circle??
(h, k)r
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 221:07 PM
(Circles) Example 2:
6(x +6)2 + 6(y +10)2 = 24
What is different about this equation?
Why is it still a circle?
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Ellipses:
Major Axis (h + a, k)(h a, k)
vertices
Major Axis
vertices
(h, k + a)
(h, k a)
(x h)2 + (y k)2 a2 b2
= 1
"a" represents the bigger number!!!
**When the bigger number is under x major axis is horizontal
(x h)2 + (y k)2 b2 a2
**When the bigger number is under y major axis is vertical
(h, k) is the centerfoci
(h + c, k)(h c, k)
c = √ a2 b2
foci(h, k + c)
(h, k c)
(h, k)
= 1
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 136:54 AM
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 158:11 AM
Eccentricitycircle:
ellipse:parabolahyperbola:
O
0<e<11
e>1
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 221:37 PM
(Ellipses) Example 1:
To determine the center, vertices, and foci, the equation must be in standard form (Step 1)
(y 6)2 + 4(x 5)2 = 36Recall that ellipse equations must be = to 1
Find center, vertices, and foci
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April 17, 2015
May 221:41 PM
25(y +2)2 + 4(x 6)2 = 100
(Ellipses) Example 2: Find center, vertices, and foci
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April 17, 2015
Apr 176:49 AM
Good Morning, Please have your homework out!
Please grab a warm up and start working!
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
Apr 176:50 AM
Warm up!
Ellipse
Ellipse
hyperbola
Circle
hyperbola
parabola
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H PreCalc 9.19.3 Conic Sections Day 2.notebook
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May 221:54 PM
Hyperbolas:
(x h)2 (y k)2 a2 b2
= 1
(h, k) is the centerc = √ a2 + b2
(y k)2 (x h)2 a2 b2
= 1
(h, k)
vertices
(h + a, k)(h a, k)
foci
(h +c, k)(h c, k)
y = k + (x h)aby = k (x h)a
b
y = k + (x h)ba
y = k (x h)ba
(h, k)
(h, k+c)
(h, kc)
(h, k+a)
(h, ka)
verticesfoci
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 222:04 PM
(Hyperbolas) Example 1:
25(x 5)2 4(y + 3)2 = 100
Find center, vertices, foci, and asymptotes
H PreCalc 9.19.3 Conic Sections Day 2.notebook
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April 17, 2015
May 231:17 PM
(Hyperbolas) Example 2:
4(y 6)2 9(x + 2)2 = 36
Find center, vertices, foci, and asymptotes
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April 17, 2015
Apr 137:00 AM