Examples

Line in the plane from parametric equations

The parametric curve $(x(t),y(t))=(a+bt,c+dt)$ determines a line in ${\mathbb{R}}^2$.
Solve for $y$ in terms of $x$:

$$y=c+d\cdot \frac{1}{b}(x-a)=\frac{d}{b}x+(c-\frac{da}{b}).$$

The slope is $m=\frac{d}{b}$ and the $y$-intercept is $c-\frac{da}{b}$.

Observe that by setting $b=0,d\ne 0$, the image is a vertical line in ${\mathbb{R}}^2$ intersecting the $x$-axis at $x=a$.

Parametrization of a circle

Circle, radius $=r$, center $=(h,k)$
$(x(t),y(t))=(h+r\cos (t),k+r\sin (t)),t\in [0,2\pi ]$

Sine and cosine of 2t as a parametric curve

$(x,y)=(\sin (2t),\cos (2t)),t\in [0,2\pi ]$

Cubic curve with x equal to y cubed

$(x,y)=(-t^3,-t)$
The image curve satisfies $x=y^3$.