Parabolas Centered at the Origin

Parabolas can be constructed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, a parabola is formed. In this section, you will explore the characteristics of parabolas and use them to construct equations of parabolas. Note that these are not the parabolas we studied before because they are not functions.

The Parabola

Learning Objectives

In this section, you will:

  • Graph parabolas with vertices at the origin.
  • Write equations of parabolas in standard form.
  • Graph parabolas with vertices not at the origin.
  • Solve applied problems involving parabolas.

Figure 1 The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony.

Did you know that the Olympic torch is lit several months before the start of the games? The ceremonial method for lighting the flame is the same as in ancient times. The ceremony takes place at the Temple of Hera in Olympia, Greece, and is rooted in Greek mythology, paying tribute to Prometheus, who stole fire from Zeus to give to all humans. One of eleven acting priestesses places the torch at the focus of a parabolic mirror (see Figure 1), which focuses light rays from the sun to ignite the flame.

Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

Source: Rice University, https://openstax.org/books/college-algebra/pages/8-3-the-parabola
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