Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4$$
A) $y = - \frac { 1 } { 2 } x + 30$; for $x$ in $- 6 \leq x \leq 4$
B) $y = \frac { 1 } { 2 } x ^ { 2 } + 1 ;$ for $x$ in $- 6 \leq x \leq 4$
C) $\mathrm { y } = \mathrm { x } ^ { 2 } + 1 ;$ for $\mathrm { x }$ in $- 2 \leq \mathrm { x } \leq 2$
D) $y = \frac { 1 } { 4 } x ^ { 2 } + \frac { 1 } { 2 } x + \frac { 17 } { 4 } ;$ for $x$ in $- 9 \leq x \leq 7$

Question

Quizplus · Accepted Answer

To find a rectangular equation, we eliminate the parameter $t$. From $x = 2t - 1$, we solve for $t$ to get $t = \frac{x + 1}{2}$. Substituting this into $y = t^2 + 4$, we get $y = \left(\frac{x + 1}{2}\right)^2 + 4$, which simplifies to $y = \frac{1}{4}x^2 + \frac{1}{2}x + \frac{17}{4}$. The range for $x$ is found by substituting the limits of $t$ into the equation for $x$, giving $x$ values from $-9$ to $7$.

Find a Rectangular Equation for the Plane Curve Defined by the Parametric