Sketch the curve represented by the parametric equations $$x = e ^ { t } , y = e ^ { 3 t } + 2$$

Question

Quizplus · Accepted Answer

By eliminating the parameter $t$, we can obtain a Cartesian equation of the curve:
$$y = e^{3t} + 2 \implies e^{3t} = y - 2 \implies e^t = (y-2)^{1/3}$$
Substituting this into the equation for $x$, we get:
$$x = e^t = (y-2)^{1/3}$$
Taking both sides to the power of 3, we obtain:
$$x^3 = y-2$$
This is the equation of a curve in the $xy$-plane, which is a shifted cube root function. In particular, it is a curve that is symmetric about the point $(2,0)$ and approaches infinity as $|x|$ and $|y|$ increase. Therefore, the correct choice is (B).

Sketch the Curve Represented by the Parametric Equations x=et,y=e3t+2x = e ^ { t } , y = e ^ { 3 t } + 2x=et,y=e3t+2

Sketch the Curve Represented by the Parametric Equations $x = e ^ { t } , y = e ^ { 3 t } + 2$