Q 68. - Calculus

Question

Show that every function $y = f \left(x\right)$ can be written as parametric equations.

Step-by-Step Solution

Answer

For the function defined by $y=f\left(x\right)$ the way to obtain parametric equations is to let $x=t$ then $y=f\left(t\right)$ and $x\left(t\right)=t, y\left(t\right)=f\left(t\right)$ where $t$ is in the domain of $f$ are parametric equations of the curve.

Step 1: Given information

$y = f \left(x\right)$

Step 2: Calculation

Frequently, functions are described in terms of two variables that can be plotted as a single equation. Consider the function $y=f\left(x\right)$ where $y$ is explicitly expressed as a function of $x$

This is not a straightforward representation of every curve.

To get around this problem, each variable is stated as a function of a new variable called a parameter.

Every function $y=f\left(x\right)$ is rewritten in parametric form by letting $x=t$ then $y=f\left(t\right)$

Two equations define a parameterized curve: $x=f\left(t\right), y=g\left(t\right)$ All of the points of $(x, y)$ that may be generated by entering the values of $t$ from a certain domain into both equations are contained in the curve.

The set of points in the coordinate plane $\left\{\right(x\left(t\right)),y(t)\mid t\in I\}$ is known as a parametric curve.

Step 3: Calculation

For the function defined by $y=f\left(x\right)$ the way to obtain parametric equations is to let $x=t$ then $y=f\left(t\right)$ and $x\left(t\right)=t, y\left(t\right)=f\left(t\right)$ where $t$ is in the domain of $f$ are parametric equations of the curve.

Example:

If $y=x^2-4$ then the parametric representation of this curve is $y=t^2-4$ by letting $x=t,-\infty <t<\infty$

Hence the explanation.