Question

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

x = t² + 4, y = t² - 2; -∞ < t < ∞

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 \(t^{2}=x - 4\), substituting \(t^{2}\) gives \(y=(x - 4)-2\).

Step3: Simplify the equation

Simplify \(y=(x - 4)-2\) to \(y=x-6\). Also, note that since \(t^{2}=x - 4\geq0\) (because \(t^{2}\geq0\) for all real \(t\)), then \(x\geq4\).

Answer:

\(y=x - 6,x\geq4\)