Question

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

Accepted Answer

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. 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 $$ y = 4 \pm \sqrt{\frac{x}{2}} . $$Step 3: For equation set (b), start with $$ x = 2t^{4} . $$Solve for $$ t^{4} = \frac{x}{2} . $$Then $$ y = 4 + t^{2} . $$Since $$ t^{2} = \sqrt{t^{4}} = \sqrt{\frac{x}{2}} $$, $$ y = 4 + \sqrt{\frac{x}{2}} . $$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, $$ y = 4 \pm \sqrt{\frac{x}{2}} . $$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.

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 video answer for a similar problem:

Key Concepts

Parametric Equations and Curves

Eliminating the Parameter

Domain and Range of Parameters

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

Verified video answer for a similar problem:

Key Concepts

Parametric Equations and Curves

Eliminating the Parameter

Domain and Range of Parameters

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