parabola

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).

• Conic Sections in Polar Coordinates

  • Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse.
  • The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas.
  • As in the figure, for $e = 0$, we have a circle, for $0 < e < 1$ we obtain an ellipse, for $e = 1$ a parabola, and for $e > 1$ a hyperbola.