QuestionJuly 15, 2025

Use the parameter to write the rectangular equation as a pair of parametric equations. x=t-1,y=-(t^2)/(3)+(2t)/(3)-(1)/(3) y=-(x^2)/(2) y=-(x^2)/(3) x=-(y^2)/(5) y=-(x^2)/(6)

Use the parameter to write the rectangular equation as a pair of parametric equations. x=t-1,y=-(t^2)/(3)+(2t)/(3)-(1)/(3) y=-(x^2)/(2) y=-(x^2)/(3) x=-(y^2)/(5) y=-(x^2)/(6)
Use the parameter to write the rectangular equation as a pair of parametric equations. x=t-1,y=-(t^2)/(3)+(2t)/(3)-(1)/(3) y=-(x^2)/(2) y=-(x^2)/(3) x=-(y^2)/(5) y=-(x^2)/(6)

Solution
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Answer

x = t - 1, \; y = -\frac{t^2 - 2t + 1}{3} Explanation 1. Identify the given rectangular equation The given rectangular equation is y = -\frac{x^2}{3}. 2. Express x in terms of parameter t Given x = t - 1, this is already a parametric equation for x. 3. Substitute x into the rectangular equation Substitute x = t - 1 into y = -\frac{x^2}{3} to find y in terms of t. 4. Calculate y in terms of t y = -\frac{(t-1)^2}{3} = -\frac{t^2 - 2t + 1}{3}.

Explanation

1. Identify the given rectangular equation<br /> The given rectangular equation is $y = -\frac{x^2}{3}$.<br /><br />2. Express $x$ in terms of parameter $t$<br /> Given $x = t - 1$, this is already a parametric equation for $x$.<br /><br />3. Substitute $x$ into the rectangular equation<br /> Substitute $x = t - 1$ into $y = -\frac{x^2}{3}$ to find $y$ in terms of $t$.<br /><br />4. Calculate $y$ in terms of $t$<br /> $y = -\frac{(t-1)^2}{3} = -\frac{t^2 - 2t + 1}{3}$.
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