Parametric Equations and Curves — Question 9

Question 9 – Solution

Solution

1. To trace the parabola from right to left, choose the $x$-coordinate so that it decreases as $t$ increases. A simple choice is $$x=-t.$$

2. The curve must satisfy $y=x^2$. Substituting $x=-t$ gives $$y=(-t)^2=t^2.$$ Hence one suitable parametrization is $$\boxed{x=-t,\qquad y=t^2,\qquad -2\le t\le2}.$$

3. Verify that the curve is correct by eliminating the parameter: $$t=-x \quad\Longrightarrow\quad y=t^2=(-x)^2=x^2.$$

4. Verify the direction. At $t=-2$, the point is $(2,4)$; at $t=0$, it is $(0,0)$; and at $t=2$, it is $(-2,4)$.

5. Also, $dx/dt=-1<0$, so $x$ decreases continuously. Therefore, the parabola is traced from right to left as required.