Unit 6 Introduction to Conic Sections 6...2015/04/14 · Unit 6 Lesson 1 An Introduction to Conics.notebook Unit 6 Introduction to Conic Sections LEQ: What is a conic section and

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)


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

.


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?



ELLIPSES


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


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.


Exp. 1: Graph


Exp. 2: Graph 16x 2 + 9y2 = 144


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.



EXAMPLES

What is the vertex? How does it open?

What is the vertex? How does it open?


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:

.


Name the conic section and its center or vertex.


.

Documents

Unit 6 Introduction to Conic Sections 6...2015/04/14 · Unit 6 Lesson 1 An Introduction to Conics.notebook Unit 6 Introduction to Conic Sections LEQ: What is a conic section and