Parametric Equation of a Parabola

Parametric Equation of a Parabola:

1. Parametric Equations:

  • Parametric equations define curves using parameterized variables x and y as functions of a third parameter, often denoted as t.
  • For a parabola, parametric equations are expressed as:
    • Vertical Parabola: x=at2+bt+c y=t
    • Horizontal Parabola: x=t y=at2+bt+c

2. Key Components:

  • Parameter t: Serves as the independent variable that traces the curve.
  • Coefficients a, b, and c: Determine the shape, position, and orientation of the parabola.

3. Deriving the Parametric Equations:

  • Vertical Parabola: x=at2+bt+c and y=t
    • The parameter t represents the x-coordinate, while y remains linear, producing a vertical parabola.
  • Horizontal Parabola: x=t and y=at2+bt+c
    • The parameter t represents the y-coordinate, while x remains linear, producing a horizontal parabola.

4. Parameter t Interpretation:

  • As the parameter t varies, it traces points on the parabolic curve, generating the shape of the parabola.

5. Advantages of Parametric Equations:

  • Simplifies tracing complex curves by introducing a single parameter that defines multiple coordinates.
  • Enables easy manipulation and animation of the curve by altering the parameter values.

6. Graphical Representation:

  • Plotting the parametric equations involves varying t within a specified range to trace points on the parabola.
  • Plotting multiple values of t produces a series of points that, when connected, form the parabolic curve.

Example: 

Vertical Parabola: Let's consider a vertical parabola defined by the equation y=2x2. We'll express this parabola in parametric form.

Parametric Equations for Vertical Parabola: For a vertical parabola, the parametric equations are: x=t y=2t2

Here, the parameter t allows us to trace points on the parabolic curve. As t varies, the corresponding values of x and y generate coordinates that form the parabola.

Horizontal Parabola: Now, let's consider a horizontal parabola defined by the equation x=3y2. We'll express this parabola in parametric form as well.

Parametric Equations for Horizontal Parabola: For a horizontal parabola, the parametric equations are: x=3t2 y=t

Similar to the vertical parabola, the parameter t allows us to trace points on the curve. Varying t generates different coordinates along the parabolic curve.

Graphical Representation:

  • For the vertical parabola, by choosing various values of t within a specific range, we can plot points (t,2t2) to trace the parabolic curve.
  • For the horizontal parabola, by choosing values of t and plotting points (3t2,t), we can trace the parabolic curve.

Visualizing the Curves:

  • Varying the parameter t within a suitable range will generate points on the parabolas.
  • By connecting these points, we can visualize and plot the corresponding vertical and horizontal parabolic curves on a graph.

Vertical Parabola:

Given the parametric equations: x=2t2+3t1

 y=t

Key Components:

  • Parameter t represents the independent variable.
  • Coefficients 2, 3, and 1 determine the shape and position of the parabola.

Interpretation:

  • As t varies, the values of x and y change, tracing points on the parabolic curve.

Graphical Representation:

  • Plotting values of x and y for different t within a specified range will generate points on the parabola.
  • As t varies, these points form the parabolic curve.

Horizontal Parabola:

Given the parametric equations: x=t

 y=t2+4t+2

Key Components:

  • Parameter t serves as the independent variable.
  • Coefficients 1, 4, and 2 determine the shape and position of the parabola.

Interpretation:

  • t controls the values of x and y, generating points on the parabolic curve.

Graphical Representation:

  • Varying t within a specified range will produce different x and y values, tracing points on the parabola.
  • Connecting these points obtained for various t values will reveal the shape of the parabolic curve.