# Represent the parabola y=x^{2}+1 by a vector-valued function

Represent the parabola $$\displaystyle{y}={x}^{{{2}}}+{1}$$ by a vector-valued function

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Jeffrey Parrish

Step 1
Given Parabola
$$\displaystyle{y}={x}^{{{2}}}+{1}$$
The Vector valued function is
$$\overrightarrow{r}(t)=x(t)\hat{i}+y(t)\hat{j}$$
Step 2
In Parabola eqn, y is explicity defined in terms of x
$$\displaystyle\therefore{x}={t}$$
$$\displaystyle{y}={t}^{{{2}}}+{1}$$
Ans: $$\overrightarrow{r}(t)=t\hat{i}+(t^{2}+1)\hat{j}$$