Topic 4.7 Notes – Parametrization of Implicitly Defined Functions

What a Parametrization Is

An implicit equation describes a curve as all points $(x,y)$ that satisfy it. For example, $x^2+y^2=1$ means every point on the unit circle.

A parametrization writes that same curve as

$$(x(t),y(t))$$

where changing $t$ produces points on the curve.

Two checks matter every time:

• Substitution check
  Replace $x$ with $x(t)$ and $y$ with $y(t)$. The original equation should become true for every allowed $t$.

• Coverage check
  The $t$-interval has to trace the whole intended curve, or the intended part. An identity alone does not prove full coverage.

Parametrizations are not unique. The same curve can:

• start at different points
• move in different directions
• trace once, partly, or multiple times

If the curve is already a function $y=f(x)$, the fastest choice is usually just letting $x=t$.

Parametrizing Functions, Inverses, and Parabolas

Here are the basic constructions you need most often:

$$y=f(x)\quad \Rightarrow \quad (x(t),y(t))=(t,f(t))$$

$$x=f(y)\quad \Rightarrow \quad (x(t),y(t))=(f(t),t)$$

If $f$ is invertible, its inverse graph is the coordinates swapped:

$$(x(t),y(t))=(f(t),t)$$

Here $t$ runs through the original domain of $f$.

When a parametric curve is invertible

If $f(x)=x^3+2$, then

• graph of $f$
  $$(x(t),y(t))=(t,t^3+2)$$

• graph of $f^{-1}$
  $$(x(t),y(t))=(t^3+2,t)$$

That swap is a common quiz mistake.

Parabolas

A parabola works the same way. Solve for one variable and use the other as $t$.

• Vertical parabola
  $$y-k=a(x-h)^2$$
  can be written as
  $$x(t)=t,\qquad y(t)=k+a(t-h)^2$$
  or centered as
  $$x=h+t,\qquad y=k+at^2$$

• Horizontal parabola
  $$x-h=a(y-k)^2$$
  can be written as
  $$x(t)=h+a(t-k)^2,\qquad y(t)=t$$
  or centered as
  $$x=h+at^2,\qquad y=k+t$$

If solving gives $y=\pm\sqrt{g(x)}$, one sign only gives one branch.

Parametrizing Ellipses and Hyperbolas

Ellipse

$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$

Use

$$x=h+a\cos t,\qquad y=k+b\sin t,\qquad 0\le t\le 2\pi$$

This works because $\cos^2 t+\sin^2 t=1$.

What to know fast:

• starts at the rightmost point
• traces counterclockwise
• a circle is the special case $a=b$

Hyperbolas

These use

$$\sec^2 t-\tan^2 t=1$$

• Left-right hyperbola
  $$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
  $$x=h+a\sec t,\qquad y=k+b\tan t,\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}$$

• Up-down hyperbola
  $$\frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1$$
  $$x=h+a\tan t,\qquad y=k+b\sec t,\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}$$

The rule students forget most is this: secant goes with the variable in the positive term.

Recognize these six quick cases:

• ellipse
• left-right hyperbola
• up-down hyperbola
• parabola solved for $x$ or $y$
• function graph
• inverse function graph

How to Build and Check a Parametrization

1. Identify the curve type, or decide whether solving for $x$ or $y$ is easiest.
2. Match it to the right template.
3. Choose the correct parameter interval.
4. Substitute into the original equation and simplify to an identity.
5. Check what is actually traced.

That last check includes:

• full curve or only part
• one branch or both
• starting point and direction for ellipses
• excluded $t$-values where $\sec t$ and $\tan t$ are undefined

Common Mistakes and Fast Fixes

• Getting an identity and assuming you covered the whole curve.
• Using $x=t$ when the equation was solved for $x$ in terms of $y$.
• Forgetting inverse parametrizations swap coordinates.
• Mixing ellipse formulas with hyperbola formulas.
• Putting secant in the wrong coordinate of a hyperbola.
• Using $0\le t\le 2\pi$ for every conic.
• Ignoring undefined values for tangent and secant.
• Forgetting a restricted interval only traces part of a curve.