Conics Instructional Unit

 

Day 8  - Hyperbolas

 

by

 

Mandy Stein

 

Hyperbola

 

The locus of all points P(x,y) such that the difference of the distance from P to two fixed points, called foci, are constant

 

 

Transverse axis – contains the vertices as endpoints

Conjugate axis – contains the co-vertices as endpoints

 

 

Standard Equation of a Hyperbola

Standard equation of an hyperbola centered at (h , k)

 

Horizontal Transverse Axis

 -  = 1

 

Vertices: (h – a, k) & (h + a, k)

Co-Vertices: (h, k + b) & (h, k – b)

Foci: (h – c, k) & (h + c, k)

Slope of asymptote = + and -

Transverse axis = 2a

Conjugate axis = 2b

a2 + b2 = c2

Vertical Transverse Axis

 -  = 1

 

Vertices: (h, k – a) & (h, k + a)

Co-Vertices: (h – b, k) & (h + b, k)

Foci: (h, k – c) & (h, k + c)

Slope of asymptote = + and -

Transverse axis = 2a

Conjugate axis = 2b

a2 + b2 = c2

 

 

 

To graph a hyperbola:

  1. Determine if the transverse axis is vertical or horizontal
  2. Identify the center of the hyperbola
  3. Identify the vertices
  4. Identify the co-vertices
  5. Sketch a rectangle using those four points and sketch the asymptotes through the center and the corners of the rectangle
  6. Sketch the hyperbola using the vertices and the asymptotes

 

 -  = 1

Horizontal transverse axis

Center: (2,1)

Vertices: (0,1) & (4,1)

Co-Vertices: (2,1 + ) & (2,1 - )

Foci: (-1,1) & (5,1)

 

 

 -  = 1

Vertical transverse axis

Center: (-4,3)

Vertices: (-4,7) & (-4,-1)

Co-Vertices: (1,3) & (7,3)

Foci: (-4,-2) & (-4,8)

 

To graph an equation not in standard form:

  1. Write the equation in standard form by completing the square
  2. Determine if the transverse axis is vertical or horizontal
  3. Identify the center of the hyperbola
  4. Identify the vertices
  5. Identify the co-vertices
  6. Sketch a rectangle using those four points and sketch the asymptotes through the center and the corners of the rectangle
  7. Sketch the hyperbola using the vertices and the asymptotes

 

 

-2x2 + y2 +4x + 6y + 3 = 0

 

First, we put the equation in standard form by completing the square

-2x2 + y2 + 4x + 6y = -3

(y2 + 6y + 9) - 2(x2 - 2x + 1) = -3 + 9 – 2

(y + 3) 2 + -2(x – 1) 2 = 4

 -  = 1

Vertical transverse axis

Center: (1,-3)

Vertices: (1,-5) & (1,-1)

Co-Vertices: (1 - ,-3) & (1 + ,-3)

Foci: (1,-3 - ) & (1,-3 + )

 

To write the equation of a hyperbola

  1. Determine if the hyperbola has a horizontal or vertical transverse axis
  2. Identify the center
  3. Identify the vertices
  4. Identify the co-vertices
  5. Identify the foci
  6. Obtain a, b, and c
  7. Substitute a and b into the correct equation and simplify

 

Horizontal transverse axis

Center: (0,0)

Vertices: (0,-3) & (0,3)

Co-vertices: (-5,0) & (5,0)

Foci: (0,- ) & (0, )

Asymptotes: y = - &  y =

Day 9 - Parametric Equations

 

Return to my EMAT 6990