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. $$ x=\sqrt[4]{t}, \quad y=3-t $$

Short answer

The graph of the curve represents a function that starts at the point (0,3) and moves to the right and downwards. The corresponding rectangular equation for the given parametric equations is \(y = 3 - x^4\).

Step-by-step solution

Understanding the Parametric Equations

Given the pair of parametric equations \(x = t^{1/4}\) and \(y = 3 - t\), where 'x' and 'y' are both expressed in terms of the parameter 't'. 'x' represents the fourth root of 't', demonstrating that 't' must be non-negative to achieve real values of 'x'. The equation for 'y' demonstrates that as 't' increases, 'y' decreases, indicating downward orientation of the curve.

Sketching the Curve

Begin at the point where \(t = 0\), which gives us the point \(x = 0\) and \(y = 3\). As 't' increases, 'x' increases due to the relation \(x = t^{1/4}\), while 'y' decreases because 'y' depends on the term \(3 - t\). As such, sketch a curve that originates at (0,3) and decreases moving to the right.

Elimination of the Parameter t

To find the rectangular equation that corresponds to the parametric equations, get rid of the parameter 't'. In this case, it is easier to solve the equation \(x = t^{1/4}\) for 't', giving \(t = x^4\). Then, replace 't' in the equation \(y = 3 - t\) with the newly found value to get \(y = 3 - x^4\). This is the required rectangular equation.