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## Q. 2.3

Describe and sketch the curve given parametrically by $x=t+3, y=t^{2}-t+2$.

## Verified Solution

With problems of this type the easiest thing to do is to write t as a function of x or y, if possible. Since $x=t+3$, we immediately see that $t=x-3$ and $y=t^{2}-t+2=(x-3)^{2}-(x-3)+2=x^{2}-7 x+14$. This is the Cartesian equation of the curve and is the equation of a parabola. It is sketched in Figure 2. Note that in the Cartesian equation of the parabola, the parameter t does not appear.