Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$

Short answer

The curve represented by the parametric equations has left to right orientation. The corresponding rectangular equation after eliminating the parameter 't' is \(y = x^3 + 1\).

Step-by-step solution

Understand the parametric equations

Defining \(x=e^{t}\) and \(y=e^{3t} + 1\), the value of 't' leads to a unique (x, y) coordinate. This aids in plotting the curve for given 't' values. Moreover, the parameter 't' can be eliminated to obtain the rectangular equation.

Sketch the curve

To sketch the curve, plot a few points by selecting different 't' values. Notably, as 't' increases, x and y increase exponentially. The orientation of the curve is from left to right, which is obvious since 'x' and 'y' both increase as 't' increases.

Convert to rectangular form

To eliminate the parameter 't', first express 't' in terms of 'x' from the first equation by taking natural logarithm (ln) of both sides. This yields \(t=ln(x)\). Substituting \(t = ln(x)\) into the second equation, the rectangular form of the equation becomes \(y = e^{3ln(x)} + 1\). Simplifying this equation using the property \(a^{ln(b)} = b^a\), we obtain: \(y = x^3 + 1\).