How is y(t) Derived in Parametric Equations?

When we first encounter parametric equations, the transition from an ordinary equation such as y = f(x) to y(t) = f(x(t)) can seem confusing. However, the idea is actually very simple: it is just substitution.

Starting with y = f(x)

Suppose a curve is described by

y = f(x)

This tells us that the value of y depends on the value of x.

For example:

y = x²

Here, f(x) is x².

Introducing a parameter

In parametric equations, we describe the coordinates using another variable called the parameter, usually t.

Suppose we write:

x = x(t)

Now x is no longer being chosen directly. Instead, we choose a value of t, and that determines x.

For example:

x(t) = t + 1

If t = 2, then x = 3.

What happens to y?

We still have the original relationship:

y = f(x)

But we now know that x is equal to x(t). Therefore, wherever we previously had x, we substitute x(t):

y = f(x(t))

Since y is now also determined by t, we write it as y(t):

y(t) = f(x(t))

This is the entire reason behind the formula.

A simple example

Take the curve:

y = x²

Suppose we parameterize x as:

x(t) = t + 1

Substituting x(t) into the original equation gives:

y(t) = (t + 1)²

Therefore, the parametric equations are:

x(t) = t + 1

y(t) = (t + 1)²

The key idea

You can think of the process as:

t → x(t) → f(x(t)) → y(t)

So y(t) = f(x(t)) is not a new or mysterious formula. It simply says:

Once x is expressed in terms of t, substitute that expression for x in the original function to obtain y in terms of t.

This simple substitution is the foundation for many calculations involving parametric curves, including finding their area, slope, and surface area.