Problem 46 - Precalculus with Limits

Question

In Exercises 41-48, find a set of parametric equations for the rectangular equation using (a) \(t=x\) and (b) \(t=2-x\). \(y=1-2x^2\)

Step-by-Step Solution

Answer

The sets of parametric equations are (a) \(x=t, y=1-2t^2\) and (b) \(x=2-t, y = -7 + 8t - 2t^2\).

Step 1: Parameterize with t=x

Substitute \(t = x\) into the given equation. This gives the parametric equations \(x=t\) and \(y=1-2t^2\).

Step 2: Parameterize with t=2-x

Substitute \(t = 2 - x\) into the given equation. This will involve a bit more algebra as the equations will be more complex. Rearrange to give \(x = 2 - t\), and then substitute \(x\) into the equation for y to solve for y: \(y = 1 - 2x^2 = 1 - 2(2 - t)^2 = 1 - 2(4 - 4t + t^2) = 1 - 8 + 8t - 2t^2 = -7 + 8t - 2t^2\). Therefore the parametric equations are \(x=2-t\) and \(y=-7+8t-2t^2\).

Key Concepts

Parametrization of Equations

Parametrization is a valuable tool that provides a unique perspective on equations and is critical when dealing with various mathematical applications. By introducing a parameter, complex relationships between variables can become more manageable, which is a fundamental concept in algebra and calculus.