Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )$$
A) $y = \frac { 3 } { 4 } ( x - 1 ) ^ { 2 } + 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
B) $y = \frac { 3 } { 4 } ( x + 1 ) ^ { 2 } - 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
C) $y = \frac { 3 } { 4 } ( x - 1 ) ^ { 2 } - 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$
D) $y = \frac { 3 } { 4 } ( x + 1 ) ^ { 2 } + 2 , \quad x \in ( - \infty , \infty ) , \quad \text { right to left }$

Question

Quizplus · Accepted Answer

To eliminate the parameter $t$, we solve the equation $x = -2t - 1$ for $t$: $t = -\frac{1}{2}x - \frac{1}{2}$. Substituting this into the equation for $y$, we get $y = 3\left(-\frac{1}{2}x - \frac{1}{2}\right)^2 + 2$, which simplifies to $y = \frac{3}{4}(x + 1)^2 + 2$. Since $x$ decreases as $t$ increases (given by the negative coefficient of $t$ in $x = -2t - 1$), the curve goes from right to left.

Find an Equation of Y as a Function of X x=−2t−1,y=3t2+2,t∈(−∞,∞)x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )x=−2t−1,y=3t2+2,t∈(−∞,∞)

Find an Equation of Y as a Function of X $x = - 2 t - 1 , \quad y = 3 t ^ { 2 } + 2 , \quad t \in ( - \infty , \infty )$