Chapter 10.7, Problem 13AYU

To determine

The graph of the curve of a parametric equation $x=3t^2,y=t+1;-\infty <t<\infty$, show its orientation. Find the rectangular equation of curve.

Answer to Problem 13AYU

Solution:

The graph of the parametric equations $x=3t^2,y=t+1;-\infty <t<\infty$ is

The rectangular equation of the parametric equation $x=3t^2,y=t+1;-\infty <t<\infty$ is $x=3{\left(y-1\right)}^2$.

Explanation of Solution

Given information:

The parametric equation $x=3t^2,y=t+1;-\infty <t<\infty$

Explanation:

To draw the given parametric equation, plug some values of $t$ in the given equation and find few points on the curve.

The interval for $t$ is $-\infty <t<\infty$

$\begin{array}{cccc}t& x\left(t\right)=3t^2& y\left(t\right)=t+1& \left(x,y\right)\\ -2& 12& -1& \left(12,-1\right)\\ -1& 3& 0& \left(3,0\right)\\ 0& 0& 1& \left(0,1\right)\\ 1& 3& 2& \left(3,2\right)\\ 3& 27& 4& \left(27,4\right)\end{array}$

The orientation is curve traced out in a certain direction by the corresponding succession of points $\left(x,y\right)$.

The arrow shows the orientation along the curve as $t$ varies from $-\infty <t<\infty$

As $t$ takes values from $-\infty <t<\infty$, the corresponding orientation is from $\left(12,-1\right)$ to $\left(27,4\right)$

Use points in the table to sketch the curve.

The graph of the parametric equation $x=3t^2,y=t+1;-\infty <t<\infty$ is

Now, let $x=3t^2$ ——- (1)

$y=t+1$ ——- (2)

Solve equation (2) for variable $t$,

$\Rightarrow t=y-1$

Substitute this result for $t$ into equation (1),

$\Rightarrow x=3{\left(y-1\right)}^2$

The rectangular equation of the parametric equation $x=3t^2,y=t+1;-\infty <t<\infty$ is $x=3{\left(y-1\right)}^2$.