Question

graph the curve whose parametric equations are given, and show its orientation. find the rectangular equation of the curve.

$x = t^{2}+4$, $y = t^{2}-2$; $-\infty

choose the correct graph below.

the rectangular equation of the curve is

Explanation:

Step1: Express \(t^{2}\) from the first equation

From \(x = t^{2}+4\), we can get \(t^{2}=x - 4\).

Step2: Substitute \(t^{2}\) into the second equation

Since \(y=t^{2}-2\) and \(t^{2}=x - 4\), substitute \(t^{2}\) into \(y=t^{2}-2\). Then \(y=(x - 4)-2\).

Step3: Simplify the equation

Simplify \(y=(x - 4)-2\) to get \(y=x-6\). Also, because \(t^{2}=x - 4\geqslant0\) (since \(t^{2}\geqslant0\) for all real \(t\)), then \(x\geqslant4\).

Answer:

The rectangular equation of the curve is \(y=x - 6,x\geqslant4\)