Question

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

$x = 3e^{t}, y = 3 + e^{t}; t \geq 0$

choose the correct graph below.

determine the rectangular equation of the curve.

$y=\square,x\geq\square$

(simplify your answer. use integers or fractions for any numbers in the expression)

Explanation:

Step1: Solve for \(e^{t}\) from \(x = 3e^{t}\)

From \(x = 3e^{t}\), we can get \(e^{t}=\frac{x}{3}\) (since \(x = 3e^{t}\), divide both sides by 3).

Step2: Substitute \(e^{t}\) into the equation for \(y\)

We know that \(y = 3+e^{t}\). Substitute \(e^{t}=\frac{x}{3}\) into this equation. Then \(y=3+\frac{x}{3}\).

Step3: Find the domain of \(x\)

Since \(t\geq0\) and \(x = 3e^{t}\), and the function \(y = e^{t}\) is an increasing function. When \(t = 0\), \(x=3e^{0}=3\). As \(t\) increases (\(t\geq0\)), \(e^{t}\) increases, so \(x = 3e^{t}\geq3\).

Answer:

\(y = 3+\frac{x}{3},x\geq3\)