Standard Equation of a Parabola

Standard Equation of a Parabola:

1. General Form:

  • The standard equation of a parabola differs based on its orientation: vertical or horizontal.
  • A vertical parabola has the form: y=ax2+bx+c
  • A horizontal parabola has the form: x=ay2+by+c

2. Components of the Equation:

  • a:
    • Determines the shape and direction of the parabola.
    • If a>0, the parabola opens upward (for vertical) or rightward (for horizontal).
    • If a<0, the parabola opens downward (for vertical) or leftward (for horizontal).
  • b:
    • Influences the position of the axis of symmetry and the vertex.
    • The axis of symmetry is x=b2a for a vertical parabola and y=b2a for a horizontal parabola.
  • c:
    • Shifts the parabola vertically (for vertical) or horizontally (for horizontal).

3. Finding the Vertex:

  • For a vertical parabola (y=ax2+bx+c):
    • x=b2a gives the x-coordinate of the vertex. Substitute into the equation to find the y-coordinate.
    • Vertex: (b/2a,f(b/2a)), where f(x) is the function defining the parabola.
  • For a horizontal parabola (x=ay2+by+c):
    • y=b2a gives the y-coordinate of the vertex. Substitute into the equation to find the x-coordinate.
    • Vertex: (g(b/2a),b/2a), where g(x) is the function defining the parabola.

4. Axis of Symmetry:

  • Vertical Parabola: The axis of symmetry is a vertical line passing through the vertex, given by x=b2a.
  • Horizontal Parabola: The axis of symmetry is a horizontal line passing through the vertex, given by y=b2a.

5. Orientation and Shape:

  • The sign of a determines the orientation and direction of opening for the parabola.
  • Positive a opens upward (vertical) or rightward (horizontal).
  • Negative a opens downward (vertical) or leftward (horizontal).

Standard Equation of a Vertical Parabola:

1. Equation Form:

  • For a vertical parabola with its axis parallel to the y-axis, the standard form is: y=ax2+bx+c
  • This form represents a parabola that opens either upward or downward.

2. Example:

  • Consider the equation: y=2x24x+3
  • Identifying the Components:
    • Coefficients: a=2, b=4, c=3
    • Vertex Form: 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)
    • Axis of Symmetry: x=1 (line of symmetry for the parabola)
    • Direction: Upward (as a>0)

3. Plotting the Graph:

  • With the vertex at (1,1), use additional points or the axis of symmetry to draw the parabola.
  • Choose values for x to calculate y and plot points to sketch the parabolic curve.

Standard Equation of a Horizontal Parabola:

1. Equation Form:

  • For a horizontal parabola with its axis parallel to the x-axis, the standard form is: x=ay2+by+c
  • This form represents a parabola that opens either rightward or leftward.

2. Example:

  • Consider the equation: x=3y2+6y4
  • Identifying the Components:
    • Coefficients: a=3, b=6, c=4
    • Vertex Form: 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)
    • Axis of Symmetry: y=1 (line of symmetry for the parabola)
    • Direction: Leftward (as a<0)

3. Plotting the Graph:

  • With the vertex at (1,1), use additional points or the axis of symmetry to draw the parabola.
  • Choose values for y to calculate x and plot points to sketch the parabolic curve.