Multiple descriptions Which of the following parametric equations describe the same curve?
a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4
b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2
c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

Verified step by step guidance

1. Step 1: Understand the goal is to determine which parametric equations describe the same curve by eliminating the parameter \( t \) and expressing \( y \) as a function of \( x \) or vice versa for each set.
2. Step 2: For equation set (a), start with \( x = 2t^{2} \). Solve for \( t^{2} \) to get \( t^{2} = \frac{x}{2} \). Then express \( y \) in terms of \( t \) as \( y = 4 + t \). Since \( t = \pm \sqrt{\frac{x}{2}} \), \( y \) can be written as \( y = 4 \pm \sqrt{\frac{x}{2}} \).
3. Step 3: For equation set (b), start with \( x = 2t^{4} \). Solve for \( t^{4} = \frac{x}{2} \). Then \( y = 4 + t^{2} \). Since \( t^{2} = \sqrt{t^{4}} = \sqrt{\frac{x}{2}} \), \( y = 4 + \sqrt{\frac{x}{2}} \).
4. Step 4: For equation set (c), start with \( x = 2t^{\frac{2}{3}} \). Solve for \( t^{\frac{2}{3}} = \frac{x}{2} \). Then \( y = 4 + t^{\frac{1}{3}} \). To relate \( y \) and \( x \), express \( t^{\frac{1}{3}} \) in terms of \( x \) by noting that \( t^{\frac{1}{3}} = \pm \sqrt{\frac{x}{2}} \) because \( (t^{\frac{1}{3}})^2 = t^{\frac{2}{3}} = \frac{x}{2} \). Thus, \( y = 4 \pm \sqrt{\frac{x}{2}} \).
5. Step 5: Compare the expressions for \( y \) in terms of \( x \) from all three sets. Notice that sets (a) and (c) yield \( y = 4 \pm \sqrt{\frac{x}{2}} \), while set (b) yields \( y = 4 + \sqrt{\frac{x}{2}} \) only. Also, consider the domain restrictions on \( t \) to confirm the ranges of \( x \) and \( y \) for each curve.

Key Concepts

Parametric Equations and Curves

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually t. Understanding how x and y depend on t helps visualize the curve's shape and direction. Different parametric forms can represent the same curve if they trace the same set of points.

Eliminating the Parameter

To determine if two parametric equations describe the same curve, eliminate the parameter t to find a direct relationship between x and y. This often involves solving one equation for t and substituting into the other, revealing the underlying Cartesian equation of the curve.

Domain and Range of Parameters

The domain of the parameter t affects which portion of the curve is traced. Even if two parametric equations yield the same Cartesian equation, differences in parameter domains can result in different parts or directions of the curve being described.