Question

graph the curve whose parametric equations are given and show its orientation. find the rectangular equation of the curve.
x = 2e^{t}, y = 5 + e^{t}, t ≥ 0
choose the correct graph below.

Explanation:

Step1: Eliminate the parameter \( t \)

Given \( x = 2e^{t}\), then \( e^{t}=\frac{x}{2}\).
Since \( y = 5+e^{t}\), substitute \( e^{t}=\frac{x}{2}\) into the equation for \( y\).
We get \( y = 5+\frac{x}{2}\), which is a linear function.

Step2: Analyze the orientation

When \( t = 0\), \( x=2e^{0}=2\), \( y = 5+e^{0}=6\).
As \( t\) increases (\(t\geq0\)), since \(x = 2e^{t}\) and \(y = 5+e^{t}\) are both exponential - increasing functions of \(t\).
For \(y = 5+\frac{x}{2}\), as \(x\) increases (because \(x = 2e^{t}\) and \(t\) increases), \(y\) also increases.

Answer:

C.