Parabolas: Upward and Downward

Parabolas Opening Upward and Downward

• Parabolas are U-shaped curves that open either upward or downward. Their equation can be represented in the form: y = ax² + bx + c. For parabolas that only open up or down, the "a" value determines the direction of the opening.

• If 'a' is positive, the parabola opens upward. If 'a' is negative, the parabola opens downward.

Focus of a Parabola

• The focus of a parabola is a fixed point inside the parabola. Every point on the parabola is equidistant from the focus and the directrix (a fixed line).
• The distance from the vertex (the turning point of the parabola) to the focus is 'p'.
• The parabola's equation is related to the focus and directrix.

Equations for Parabolas Opening Upward or Downward

• Standard Equation: Vertex at (h, k)

  • For a parabola opening upward: (x - h)² = 4p(y - k)
  • For a parabola opening downward: (x - h)² = -4p(y - k)

• Key element: The '4p' term is crucial in identifying the focus and the directrix of the parabola, from the equation. Its value directly relates to the distance from the vertex to the focus and the distance from the vertex to the directrix.

Determining the Focus

• Procedure

  1. Identify a.

  2. Determine p using the formula

     • 4p = ± a (from the standard equation)

  3. Identify the vertex coordinates (h, k) which is the point at the parabola's turning point.

  4. Calculate the coordinates of the focus.

     • Upward: Focus (h, k + p)
     • Downward: Focus (h, k - p)

• For instance: If x² = 8y means a = 8/1. Therefore, p= 2.If the parabola has vertex (0,0), the focus is (0, 2)

• Example Graph (Involves visualizing):

  • Draw a parabola that opens upward, with a vertex at (1, 2) and a focus at (1, 4) . Use the vertex and focus coordinates to help determine possible graph shapes (opening up or down). Then, draw the directrix.

  • Note how the focus is always inside the parabola. The distance from the vertex to the directrix and focus are exactly the same.

• Finding the Directrix: Given the focus, the directrix is an equally distant line, perpendicular to the axis of symmetry.

Important Considerations for Graphs

• Carefully consider the coefficients (values "a" and "4p") to determine the direction (up/down) and the rate of widening or narrowing of the parabola from the vertex.
• Verify your focus determination by plotting a few points on the parabola and calculating the distance from each point to the focus. All points on the parabola are equidistant from the focus and the directrix.