Unit 6 Introduction to Conic Sections 6...2015/04/14 · Unit 6 Lesson 1 An Introduction to...
Transcript of Unit 6 Introduction to Conic Sections 6...2015/04/14 · Unit 6 Lesson 1 An Introduction to...
Unit 6 Lesson 1 An Introduction to Conics.notebook
Unit 6 Introduction to Conic SectionsLEQ: What is a conic section and how do you represent one?
A conic section is a curve formed by the intersection of _________________________a plane and a double cone.
• Conic sections is one of the oldest math subject studied.
• The conics were discovered by Greek mathematician Menaechmus (c. 375325 BC)
Unit 6 Lesson 1 An Introduction to Conics.notebook
Facts: Circle Equation• Both variables are squared.
• Equation of a circle in centerradius form:
• What makes the circle different from the a line?
• What makes the circle different from the parabola?
Find the center and radius for each of the following circles.
1) (x 3)2 + (y + 1)2 = 16
2) (x + 5)2 + (y 2)2 = 15
3) x2 + (y 2)2 9 = 0
4. Write the equation of a circle centered at (2,7) and having a radius of 5.
5. Describe (x 2)2 + (y + 1)2 = 0
6. Describe (x + 1)2 + (y 3)2 = 1
.
Unit 6 Lesson 1 An Introduction to Conics.notebook
7. Write the equation of a circle whose diameter is the line segment joining A(3,4) and B(4,3).
What must you find first?
How can you find the center?
How can you find the radius?
Unit 6 Lesson 1 An Introduction to Conics.notebook
Unit 6 Lesson 1 An Introduction to Conics.notebook
ELLIPSES
Unit 6 Lesson 1 An Introduction to Conics.notebook
Facts: Ellipse Equation
• Both variables are squared.
• Equation:
• What makes the ellipse different from the circle?
Equation:
Equation Major Axis(length is 2a)
Minor Axis(length is 2b) Vertices Covertices
Horizontal Vertical (a,0) and (a,0) (0,b) and (0,b)
Vertical Horizontal (0,a) and (0,a) (b,0) and (b,0)
Standard Form for Elliptical
Equations
Unit 6 Lesson 1 An Introduction to Conics.notebook
where the center is at (h,k) and |2a| is the length of the horizontal axis and |2b| is the of the length of the vertical axis.
Procedure to graph:
1. Put in standard form (above): x squared term + y squared term = 1
2. Plot the center (h,k)
3. Plot the endpoints of the horizontal axis by moving “a” units left and right from the center.
4. Plot the endpoints of the vertical axis by moving “b” units up and down from the center.
Note: Steps 3 and 4 locate the endpoints of the major and minor axes.
5. Connect endpoint of axes with smooth curve.
6. Use the following formula to help locate the foci: c2 = a2 b2 if a>b or c2 = b2 – a2 if b>a
Move “c” units left and right form the center if the major axis is horizontal
OR Move “c” units up and down form the center if the major axis is vertical
Label the points f1 and f2 for the two foci.
7. Identify the length of the major and minor axes.
Unit 6 Lesson 1 An Introduction to Conics.notebook
Exp. 1: Graph
Unit 6 Lesson 1 An Introduction to Conics.notebook
Exp. 2: Graph 16x 2 + 9y2 = 144
Unit 6 Lesson 1 An Introduction to Conics.notebook
Challenge QuestionGiven the following information, write the equation of the ellipse. Sketch and find the foci.
Center is (4,3), the major axis is vertical and has a length of 12, and the minor axis has a length of 8.
Unit 6 Lesson 1 An Introduction to Conics.notebook
Unit 6 Lesson 1 An Introduction to Conics.notebook
EXAMPLES
What is the vertex? How does it open?
What is the vertex? How does it open?
Unit 6 Lesson 1 An Introduction to Conics.notebook
Hyperbolas
What I look like…two parabolas, back to back.
Standard Equations:
This equation opens left and right This equation opens up and down
Have you seen this before?
Center: (h , k)
EXAMPLE
Center:
Opens:
.
Unit 6 Lesson 1 An Introduction to Conics.notebook
Name the conic section and its center or vertex.
Unit 6 Lesson 1 An Introduction to Conics.notebook
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