Parametric Functions (old)

L3CALC

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Parametric functions are just ways to represent multiple values that don’t depend on each other, but both depend on the same independent variable. They often are used as a way of describing a curve using a third variable called a parameter (often '$t$').
Instead of writing $y$ directly as a function of $x$ like $y=f(x)$, we write BOTH as a function of $t$:

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

$t$ is the parameter, controlling the motion along the curve.
As $t$, changes the point $(X(t),Y(t))$ will change, and draw a curve.

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Most of the times, there will be two equations using the same independent variable $t$.
For example: $x=2t^2,y=6t$

To derive this, we first must differentiate each function separately:

$$\frac{dx}{dt}=4t$$ $$\frac{dy}{dt}=6$$

When differentiating, what we are looking for is still $\frac{dy}{dx}$, and a way to get it is:

$$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}$$

$dt$ ‘cancels out’, so we can let them be 1.

$$\frac{dy}{dx}=\frac{6}{1}\times \frac{1}{4t}$$ $$\frac{dy}{dx}=\frac{6}{4t}$$ $$\frac{dy}{dx}=\frac{3}{2t}$$

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Way to plot the parametric equation on Desmos is:

$$X(t)=2t^2$$ $$Y(t)=6t$$ $$(X(t),Y(t))$$