use the desmos graph linked here, where you can graph parametric equations to match the graph with its…

Explanation:

Step1: Analyze option a

For $\begin{cases}x(t)=t^{2}\y(t)=t^{3}\end{cases}$, when $t = 0$, $x = 0$ and $y = 0$. As $t$ varies, the curve has a self - intersecting and non - circular shape.

Step2: Analyze option b

For $\begin{cases}x(t)=t\cos(t)\y(t)=t\sin(t)\end{cases}$, this is a form of a spiral. The distance from the origin $r=\sqrt{x^{2}+y^{2}}=\sqrt{t^{2}\cos^{2}(t)+t^{2}\sin^{2}(t)} = |t|$. As $t$ increases from $- 5$ to $5$, it will form a spiral - like curve.

Step3: Analyze option c

For $\begin{cases}x(t)=2+\cos(t)\y(t)=\sin(t)\end{cases}$, we know that $(x - 2)^{2}+y^{2}=\cos^{2}(t)+\sin^{2}(t)=1$. This is a circle centered at $(2,0)$ with radius $r = 1$.

Step4: Analyze option d

For $\begin{cases}x(t)=t+\cos(t)\y(t)=t+\sin(t)\end{cases}$, as $t$ varies, the curve has a linear - like trend with some wiggles due to the trigonometric terms.

Step5: Analyze option e

For $\begin{cases}x(t)=e^{t}\y(t)=t^{2}\end{cases}$, when $t=0$, $x = 1$ and $y = 0$. As $t$ increases, $x=e^{t}$ grows exponentially and $y=t^{2}$ is a parabola - like function in terms of $t$.

We need to match the given graphs with these parametric equations. Without actually graphing them on Desmos (since we are doing a step - by - step analysis here), if we assume the first graph is a non - circular self - intersecting curve, it may match with $\begin{cases}x(t)=t^{2}\y(t)=t^{3}\end{cases}$; if the second graph is a circle, it may match with $\begin{cases}x(t)=2+\cos(t)\y(t)=\sin(t)\end{cases}$; if the third graph is a spiral, it may match with $\begin{cases}x(t)=t\cos(t)\y(t)=t\sin(t)\end{cases}$ etc. But since the graphs are not labeled clearly which is which, we assume we need to know the general shapes of these parametric curves.

If we assume we are matching the top - most graph which has a non - circular, self - intersecting shape, the answer is a. If we assume the middle graph which looks like a circle centered at $(2,0)$ the answer is c. If we assume the bottom graph which looks like a spiral, the answer is b.

Let's assume we are matching the top - most graph:

Answer:

a. $\begin{cases}x(t)=t^{2}\y(t)=t^{3}\end{cases}$