Parabolas

Section 9-2: Parabolas

Objectives

  • Write equations of parabolas in standard form.
  • Graph parabolas.

Vocabulary

  • Parabola: A parabola is defined as the set of all points in a plane that are equidistant from a given point known as the focus and a given line known as the directrix.
  • Latus Rectum: The line segment through the focus of a parabola that is perpendicular to the axis of symmetry.

Basics of Parabolas

  • Parabolas can be oriented vertically or horizontally.
  • General form of a vertical parabola: (y = ax^2 + bx + c)
  • Standard form of a parabola: (y = a(x - h)^2 + k) where
    • $(h, k)$ is the vertex,
    • $a$ determines the direction and width of the parabola.
  • Vertex: The point $(h, k)$ is the vertex of the parabola and the axis of symmetry (AOS) is the vertical line $x = h$.
    • If $a > 0$, the parabola opens upward.
    • If $a < 0$, the parabola opens downward.

Key Features

  • Focus: The point from which distances to the parabola's points are measured.
  • Directrix: The line used to measure the distance to the parabola's points.
  • The width of the parabola is determined by the value of $|a|$:
    • Larger values of $|a|$ produce narrower parabolas.
    • Smaller values produce wider parabolas.

Steps to Write Equations in Standard/Vertex Form

  1. Isolate the squared variable.
  2. Complete the square for similar variables.
  3. Identify the vertex $(h, k)$ and determine the sign of $a$ for direction.

Example Transformations

  1. Transform (2y = x^2 - 12x + 6) into standard form:

    • Complete the square:
      • $(x^2 - 12x + 36 - 36 + 6)$
      • This results in vertex form where vertex is $(6, 3)$ and it opens upward.

  2. Identify the features:

    • Axis of symmetry: $x = 6$
    • Direction of opening: Upward.
    • Latus Rectum depending on $4p$ factors.

Graphing Parabolas

  • To graph the parabola based on the standard form:
    1. Plot the Vertex $(h, k)$ on the graph.
    2. Draw the Axis of Symmetry: this line divides the parabola into two symmetric parts.
    3. Plot Points based on the value of $a$ to establish the width and direction of the parabola.
    4. Identify the Focus and directrix to aid in drawing the parabola accurately.

Writing Equations from Given Features

  • To find the equation of a parabola given the vertex and directrix:
    • Use the vertex form (y - k) = rac{1}{4p}(x-h)^2 where $p$ is the distance from the vertex to the focus or directrix.
    • Given Vertex $(h, k)$ and the distance to the directrix, compute $p$ accordingly.

Application of Parabolas

  • Real-world Example: Parabolic mirrors harness solar energy by reflecting sun rays to a focus.
    • If the focus is $6.25$ feet above the vertex and the latus rectum is $25$ feet long, one can derive the equation of that parabola.

Homework

  • Complete exercises 1-11 odd from the provided worksheet to reinforce understanding of parabolas and their properties.