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May 06, 2015
Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus, and directrix.
x2 + 4x +6y - 2 = 0
Vertex: _________
Focus: ___________
directrix: __________
focal width: ________
standard form: _______________
Name: _________________
May 06, 2015
Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus, and directrix.
x2 + 4x +6y - 2 = 0
Vertex: _________
Focus: ___________
directrix: __________
focal width: ________
standard form: _______________
Name: _________________
x2 + 4x + 4 = -6y + 2 + 4
(x + 2)2 = -6y + 6
(x + 2)2 = -6(y - 1)
(x + 2)2 = -6(y - 1)
(-2, 1)
4p = -6
p = -3/2 = -1.5(opens down)
(-2, -0.5)
y = 2.5
6
May 06, 2015
Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus, and directrix.
y2 - 6y + 12x + 21 = 0
Vertex: _________
Focus: ___________
directrix: __________
focal width: ________
standard form: _______________
Name: _________________
May 06, 2015
Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus, and directrix.
y2 - 6y + 12x + 21 = 0
Vertex: _________
Focus: ___________
directrix: __________
focal width: ________
standard form: _______________
Name: _________________
y2 - 6y + 9 = -12x - 21 + 9(y - 3)2 = -12x - 12(y-3)2 = -12(x + 1)
(y-3)2 = -12(x + 1)
4p = -12
p = -3
(-1, 3)
(-4, 3)
x = 2
12
May 06, 2015
(8.2) Circles and Ellipses
Objective: To learn about the equation and parts of ellipses (and circles) and write their equations.
Why: Ellipses are the paths of planets and comets around the Sun, or of moons around planets.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Circles
C(h,k)
P(x,y)r
What would happen if you stretched the circle horizontally by the factor a and vertically by the factor b?
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Ellipse: the set of all points in a plane whose distances from two fixed points (foci) have a constant sum.
Defn.
d1d2
F1 F2
P(x,y) d1 + d2 = constant
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
+
+
(x-h)2 (y-k)2
a2 b2
(x-h)2
a2
(y-k)2
b2
= 1
= 1
major axis horizontal
major axis vertical
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Find the center, vertices, foci, and eccentricity. Draw an accurate sketch.
(x-2)2
16
(y+1)2
1= 1+
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
2. 4x2 + y2 = 36
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
3. 4x2 + y2 - 8x + 4y - 8 = 0
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
HW:
(REG) (8.2) Pg.590: 7-10, 13, 15, 37, 39, 45
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Write the equation of the ellipse in standard form.
1. Major axis 14 on x-axis, minor axis length 10.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
2. Major axis endpoints are (-5,2) and (3,2), minor axis length 6.
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
3. Foci (-2,1) and (-2,5) and major axis endpoints (-2,-1) and (-2,7).
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
(2,3)
(2,6)
(6,3)
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
HW:
(REG) (8.2) Pg.591: 21-35 odd, 47
HW:
(REG) (8.2) Pg.590: 7-10, 13, 15, 37, 39, 45
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Kepler's First Law of Planetary Motion (1609) states that the path of a planet's orbit is an ellipse with the Sun at one of the foci.
(closest point to sun)
(farthest point to sun)
May 06, 2015
Obj: To learn about the equation and parts of ellipses (and circles) and write their equations.
Calculate the aphelion distance and perhelion distance of Earth.
