parabola

(noun)

The shape formed by the graph of a quadratic function.

Related Terms

  • quadratic equation
  • zeros
  • dependent variable
  • independent variable
  • quadratic function
  • axis of symmetry
  • hyperbola
  • asymptote
  • zero
  • vertex
  • quadratic

(noun)

The conic section formed by the intersection of a cone with a plane parallel to a plane tangent to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix).

Related Terms

  • quadratic equation
  • zeros
  • dependent variable
  • independent variable
  • quadratic function
  • axis of symmetry
  • hyperbola
  • asymptote
  • zero
  • vertex
  • quadratic

Examples of parabola in the following topics:

  • Parts of a Parabola

    • The graph of a quadratic function is a U-shaped curve called a parabola. 
    • Parabolas also have an axis of symmetry, which is parallel to the y-axis.
    • The y-intercept is the point at which the parabola crosses the y-axis.
    • The x-intercepts are the points at which the parabola crosses the x-axis.
    • A parabola can have no x-intercepts, one x-intercept, or two x-intercepts.
  • Converting the Conic Equation of a Parabola to Standard Form

  • Parabolas As Conic Sections

    • All parabolas have the same set of basic features.
    • It forms the rounded end of the parabola.
    • All parabolas have a directrix.
    • Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are similar.
    • Describe the parts of a parabola as parts of a conic section
  • Graphing Quadratic Equations In Standard Form

    • If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.
    • The axis of symmetry for a parabola is given by:
    • For example, consider the parabola $y=2x^2-4x+4 $ shown below.
    • The coefficient $c$ controls the height of the parabola.
    • The point $(0,c)$ is the $y$ intercept of the parabola.
  • Applications of the Parabola

    • The parabola has many important applications, from the design of automobile headlight reflectors to calculating the paths of ballistic missiles.
    • This is the exact mathematical relationship we know as a parabola.
    • As in all cases in the physical world, using the equation of a parabola to model a projectile's trajectory is an approximation.
    • So, at low speeds the parabola shape can be a very good approximation.
    • The parameters $a$, $b$, and $c$ determine the direction as well as the exact shape and position of the parabola.
  • Standard Form and Completing the Square

    • In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as:
    • In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates of x, x2, x3,...xn.
    • The standard parabola is the case n = 2.
    • This form of quadratic equation is known as the "standard form" for graphing parabolas in algebra; from this equation, it is simple to determine the x-intercepts (y = 0) of the parabola, a process known as "solving" the quadratic equation.
    • The graph of this quadratic equation is a parabola with x-intercepts at -1 and -5.
  • Eccentricity

    • Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix.
    • From the definition of a parabola, the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix.
    • Therefore, by definition, the eccentricity of a parabola must be $1$.
  • What Are Conic Sections?

    • The three types of conic sections are the hyperbola, the parabola, and the ellipse.
    • If the plane is parallel to the generating line, the conic section is a parabola.
    • As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.
    • In the next figure, four parabolas are graphed as they appear on the coordinate plane.
    • They could follow ellipses, parabolas, or hyperbolas, depending on their properties.
  • Types of Conic Sections

    • Every parabola has certain features:
    • All parabolas possess an eccentricity value $e=1$.
    • As a direct result of having the same eccentricity, all parabolas are similar, meaning that any parabola can be transformed into any other with a change of position and scaling.
    • Thus, like the parabola, all circles are similar and can be transformed into one another.
    • Notice that the value $0$ is included (a circle), but the value $1$ is not included (that would be a parabola).
  • Conics in Polar Coordinates

    • Previously, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line).
    • Consider the parabola $x=2+y^2$.
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