Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=e^{2 t}, \quad y=e^{t} $$

Short answer

The corresponding rectangular equation is \( x = y^2 \). The graph, a parabola, is oriented to the right.

Step-by-step solution

Understand the Parametric Equations

The parametric equations given are \( x = e^{2t} \) and \( y = e^t \). In Cartesian coordinate system, each point is represented by two coordinates (x, y). In the parametric form, x and y are both expressed in terms of a third variable, often time 't'. Here, both x and y are functions of 't'.

Eliminate the Parameter

To convert the parametric equations into rectangular form, the parameter needs to be eliminated. To do so, observe that the equation for \( x \) is simply the square of the equation for \( y \) (since squaring \( e^t \) gives \( e^{2t} \)). This lets us write \( x \) as a function of \( y \), giving us the rectangular equation \( x = y^2 \).

Graph the Curve

Using a graphing utility, put in the equation \( x = y^2 \) to graph the curve. This will give a parabola opening to the right. The parametric equations indicate that as \( t \) increases, both \( x \) and \( y \) increase, indicating that the curve is oriented to the right.