Conics Instructional Unit

 

Day 3  - Parabolas

 

by

 

Mandy Stein

 

Parabola

 

The locus of all points P(x,y) equidistant from a fixed point, called the focus, and a fixed line, called the directrix.

 

Axis of symmetry – passes through the vertex and divides the parabola into two symmetric parts.

Vertex – the midpoint between the focus and directrix.

 

Standard Equation of a Parabola

Standard equation of a parabola with its vertex at (h,k)

Horizontal Directrix

y - k = ( x – h)2

 

p > 0 opens up

p < 0 opens down

Focus: (h, k + p)

Directrix: y = k – p

Axis of symmetry: x = h

 

Vertical Directrix

x - h = (y – k)2

 

p > 0 opens right

p < 0 opens left

Focus: (h + p, k)

Directrix: x = h - p

Axis of symmetry: y = k

 

 

Horizontal Directrix

 

Vertical Directrix

 

 

 

To graph a parabola:

  1. Determine if the parabola has a horizontal or vertical directrix
  2. Identify the vertex
  3. Identity the focus
  4. Identify the directrix
  5. Identify the axis of symmetry
  6. Graph a couple of points and use the axis of symmetry to plot other points

 

 

y - 1 = º (x – 1)2

º > 0 so the parabola opens up

Vertex: (1,1)

Focus: (1,2)

Directrix: y = 0

Axis of symmetry: x = 1

 

 

 

x – 3 = -  (y + 1) 2

- < 0 so the parabola opens down

Vertex: (3,-1)

Focus: (1,-1)

Directrix: x = 5

Axis of symmetry: y = -1

 

 

y + 3 = x2

 > 0 so the parabola opens up

Vertex: (0,-3)

Focus: (0,0)

Directrix: y = -6

Axis of symmetry: x = 0

 

 

To graph a parabola not in standard form:

  1. Write the equation in standard form by completing the square
  2. Determine if the parabola has a horizontal or vertical directrix
  3. Identify the vertex
  4. Identity the focus
  5. Identify the directrix
  6. Identify the axis of symmetry
  7. Graph a couple of points and use the axis of symmetry to plot other points

 

y2 – 8y + 8x + 8 = 0

 

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

y2 – 8y = -8x – 8

y2 – 8y + (-4) 2 = -8x – 8 +(-4)2

(y – 4)2 = -8x + 8

(y – 4)2 = -8(x – 1)

-(y – 4)2 = x – 1

x – 1 = -(y – 4)2

Vertical Directrix

- < 0 so the parabola opens down

Vertex: (1,4)

Focus: (-1,4)

Directrix: x = 3

Axis of symmetry: y = 4

 

 

To write the equation of a parabola:

  1. Determine if the parabola has a horizontal or vertical directrix and which formula to use
  2. Identify the vertex
  3. Identify the focus or the directrix
  4. Obtain the value of h, k, and p
  5. Substitute h, k, and p into the correct equation and simplify

 

 

Horizontal directrix

y - k = ( x – h)2

Vertex: (2, 3)

Focus: (2, 7)

Directrix: y = -1

h = 2, k = 3, p = 4

y – 3 = ( x – 2)2

 

 

Vertical directrix

x - h = (y – k)2

Vertex: (-4, 2)

Focus: (-1, 2)

Directrix: x = -7

h = -4, k = 2, p = 3

x + 4 = ( y – 2)2

 

Day 4 - Introduction to Ellipses

 

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