When a curve is first introduced in the familiar form
y = f(x)
it is natural to wonder how it can later be written using parametric equations:
x = x(t)
y = y(t)
A particularly important question is:
If y = f(x) is already known, where does x(t) come from?
The answer reveals an important idea about parametric equations: the parameter t is an additional choice used to describe the curve.
1. What does y = f(x) tell us?
Consider the parabola:
y = x²
This equation tells us the relationship between x and y.
For example:
- x = 1 gives y = 1
- x = 2 gives y = 4
- x = 3 gives y = 9
The equation completely describes the curve, but there is no parameter t in it.
To introduce t, we have to make a choice.
2. The simplest choice: t = x
One possibility is to define:
t = x
Therefore:
x = t
Now substitute x = t into the original equation:
y = x²
giving:
y = t²
Thus the parametric equations become:
x(t) = t
y(t) = t²
Here, x(t) was not derived from the equation y = x². Instead, t was chosen to represent x.
3. Another choice: t = y
We could make a completely different choice.
Suppose:
t = y
Then:
y = t
But the original equation says:
y = x²
Therefore:
t = x²
and hence:
x = ±√t
If only the right-hand half of the parabola is required, we take:
x = √t
So the parametric equations become:
x(t) = √t
y(t) = t
This shows that the same curve can have different parameterizations.
4. Why can t be a combination of x and y?
There is actually no requirement that t must be x or y.
The parameter can be defined as a function of both coordinates:
t = g(x, y)
For example:
t = x + y
or:
t = x − y
or:
t = x² + y²
The important question is whether this choice allows the coordinates to be expressed in terms of t.
5. Example: t = x + y
Take the parabola:
y = x²
Suppose we choose:
t = x + y
Since y = x²:
t = x + x²
Therefore:
x² + x − t = 0
Solving this quadratic equation for x gives:
x = (−1 ± √(1 + 4t))/2
Once x(t) has been obtained, y(t) can be found from:
y(t) = [x(t)]²
Thus, even a combination of x and y can be used to construct a parameterization.
6. A particularly interesting example: a circle
Consider the unit circle:
x² + y² = 1
Instead of choosing t = x or t = y, suppose we choose:
t = x + y
Now we have:
x + y = t
and:
x² + y² = 1
The first equation gives:
y = t − x
Substituting into the circle equation gives:
x² + (t − x)² = 1
This equation can be solved for x in terms of t, after which y can be obtained from:
y = t − x
So t can successfully be constructed from both x and y.
7. But not every combination works
There is an important condition.
Consider the same unit circle:
x² + y² = 1
Suppose we choose:
t = x² + y²
But every point on the circle satisfies:
x² + y² = 1
Therefore:
t = 1
for every point on the circle.
That means t cannot distinguish one point from another.
Consequently, this is not a useful parameter for describing the entire circle.
A good parameter should change as the point moves along the curve, at least over the portion being described.
8. The deeper meaning of a parameter
A point on a curve has two coordinates:
(x, y)
But the equation of the curve creates a relationship between x and y. Therefore, only one independent quantity is needed to identify a point along a one-dimensional curve.
That quantity can be chosen in many different ways.
For example:
t = x
means that t identifies a point by its x-coordinate.
t = y
means that t identifies a point by its y-coordinate.
t = x + y
means that t identifies a point using a combination of its two coordinates.
Thus, the parameter is essentially a label for the position of a point on the curve.
9. Where does y(t) = f(x(t)) come from?
Once a parameterization has been chosen and x has been expressed as x(t), the original relationship
y = f(x)
can be used to find y.
Simply replace x by x(t):
y(t) = f(x(t))
For example, if:
y = x²
and:
x(t) = √t
then:
y(t) = (√t)²
so:
y(t) = t
Therefore, y(t) is often obtained from the original equation after x(t) has been established.
10. The complete picture
The process can therefore be understood as follows:
Original curve:
y = f(x)
↓
Choose a parameter:
t = g(x, y)
↓
Use the curve equation and parameter definition to find:
x = x(t)
↓
Substitute x(t) into the original equation:
y(t) = f(x(t))
This explains why x(t) and y(t) appear in parametric equations.
The key takeaway
The equation y = f(x) defines the curve, but it does not uniquely determine x(t).
The parameter t is introduced separately. It may be chosen as x, y, or even a suitable combination of x and y.
Once the choice of parameter has been made, the curve can be rewritten in terms of that parameter:
x = x(t)
y = y(t)
So a parametric equation is not necessarily a different curve. It is often simply a different way of describing the same curve, using a new variable to label the position of each point.
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