Standard Forms of a Parabola

Standard Forms of a Parabola:

1. Vertical Parabola:

  • General Equation: y=ax2+bx+c
  • Standard Form: y=a(xh)2+k or (xh)2=4p(yk)
  • Key Components:
    • a: Determines the direction and width of the parabola's opening.
    • h: Represents the horizontal shift of the vertex.
    • k: Indicates the vertical shift of the vertex.
    • p=14a: Distance between the vertex and the focus (if a>0) or directrix (if a<0).
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2. Horizontal Parabola:

  • General Equation: x=ay2+by+c
  • Standard Form: x=a(yk)2+h or (yk)2=4p(xh)
  • Key Components:
    • a: Determines the direction and width of the parabola's opening.
    • h: Represents the horizontal shift of the vertex.
    • k: Indicates the vertical shift of the vertex.
    • p=14a: Distance between the vertex and the focus (if a>0) or directrix (if a<0).
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3. Vertex Form:

  • Vertical Parabola: y=a(xh)2+k
  • Horizontal Parabola: x=a(yk)2+h
  • Key Points:
    • Represents the parabola with its vertex at (h,k).
    • Describes the parabola's shape and vertex without explicitly showing the focus and directrix.

4. Finding the Focus and Directrix:

  • Vertical Parabola: Focus is at (h,k+14a) and directrix is y=k14a.
  • Horizontal Parabola: Focus is at (h+14a,k) and directrix is x=h14a.

5. Identifying the Axis of Symmetry:

  • For both vertical and horizontal parabolas, the axis of symmetry is a line passing through the vertex and perpendicular to the directrix/focus.

Example: Consider the equation y=2x24x+3

  • Identifying Components:
    • a=2, b=4, c=3
    • To find the vertex, use x=b2a:
      • x=42×2=1
      • Substitute x=1 into the equation to find y:
      • y=2×124×1+3=24+3=1
    • Vertex: (1,1)
    • p=14a=18
    • Focus: (1,1+18)=(1,1.125)
    • Directrix: y=118=78

Example: Consider the equation x=3y2+6y4

  • Identifying Components:
    • a=3, b=6, c=4
    • To find the vertex, use y=b2a:
      • y=62×(3)=1
      • Substitute y=1 into the equation to find x:
      • x=3×12+6×14=3+64=1
    • Vertex: (1,1)
    • p=14a=112
    • Focus: (1+112,1)=(1112,1)
    • Directrix: x=1112=1312

Example :

Given the equation: y=2(x3)2+4

This equation is in the vertex form of a parabola with the vertex at (3,4).

Key Components:

  • a=2 determines the direction and width of the parabola.
  • h=3 represents the horizontal shift of the vertex.
  • k=4 indicates the vertical shift of the vertex.

Finding Additional Components:

  • Axis of Symmetry: x=h=3 (vertical line passing through the vertex).
  • Focus and Directrix: The focus is at (h,k+14a)=(3,4+18)=(3,4.125). The directrix is y=k14a=418=3.875.

Graphical Representation:

  • The vertex at (3,4) is the lowest point on the parabola.
  • The parabola opens upward because a>0.
  • The axis of symmetry is a vertical line passing through x=3.
  • Use the vertex and the direction of opening to sketch the parabola accurately on a graph.