Parabola parametric form

Parabola in Parametric Form

Introduction to Parametric Form

A parabola can be represented using parametric equations.
The parametric form expresses the coordinates of the points of the curve as functions of a variable called the parameter.
This form is particularly useful when the relationship between x and y is more complicated, or where it simplifies the mathematical expressions involved.

Form of Parametric Equations of a Parabola

For a parabola whose axis of symmetry is vertical, the parametric equations are x = at² and y = 2at, where t is the parameter and a is the distance from the vertex to the focus.
For a parabola whose axis of symmetry is horizontal, the parametric equations are x = 2at and y = at².

Interpreting Parametric Form

The parameter t can be considered as the time, and x and y as the coordinates of the point moving along the parabola.
For a given t, x = at² and y = 2at give the x and y coordinates of a point on the parabola.
As t varies, the point moves along the parabola.

Converting Between Parametric and Cartesian Form

A parametric form can be converted to the Cartesian form by eliminating the parameter t.
To convert from parametric to Cartesian form, express t in terms of x from one equation, and substitute into the other parametric equation.

Properties of a Parabola from Its Parametric Form

The vertex of the parabola is the point (0,0) and the axis of symmetry is the y-axis for the vertical parabola, or the x-axis for the horizontal parabola.
The direction in which x or y increases or decreases as t changes can reveal the direction of opening of the parabola.
For a vertical parabola, as t increases, x and y both increase, indicating that the parabola opens upwards.

Applying and understanding the parabola in parametric form, can enrich your problem-solving skills in Pure Mathematics. A solid grasp of the principles behind the parametric form is essential.