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 < t < infty )

choose the correct graph below.

the rectangular equation of the curve is (square).

Explanation:

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

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

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

Since \(y=t^{2}-2\) and \(t^{2}=x - 4\), substitute \(t^{2}\) in the \(y\) - equation. We get \(y=(x - 4)-2\).

Step3: Simplify the equation

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

Answer:

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