Step 1

Let us find the plane curve by a vector-valued function

Given:

\(\displaystyle{y}={9}-{x}^{{{2}}}\)

The vector valued function for a plane curve:

\(\displaystyle{r}{\left({t}\right)}={x}{\left({t}\right)}{i}+{y}{\left({t}\right)}{j}\)

Step 2

\(\displaystyle{x}{\left({t}\right)}={t}\)

\(\displaystyle{y}{\left({t}\right)}={9}-{t}^{{{2}}}\)

\(\displaystyle{r}{\left({t}\right)}={t}{i}+{\left({9}-{t}^{{{2}}}\right)}{j}\)

Let us find the plane curve by a vector-valued function

Given:

\(\displaystyle{y}={9}-{x}^{{{2}}}\)

The vector valued function for a plane curve:

\(\displaystyle{r}{\left({t}\right)}={x}{\left({t}\right)}{i}+{y}{\left({t}\right)}{j}\)

Step 2

\(\displaystyle{x}{\left({t}\right)}={t}\)

\(\displaystyle{y}{\left({t}\right)}={9}-{t}^{{{2}}}\)

\(\displaystyle{r}{\left({t}\right)}={t}{i}+{\left({9}-{t}^{{{2}}}\right)}{j}\)