Which of the following best describes the graph of the parametric equations and as varies over all real numbers?
A
A parabola opening to the right
B
A circle centered at the origin
C
A straight line with negative slope
D
An ellipse elongated along the y-axis
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1
Start by writing down the parametric equations: \(x = t^{2} + 2t\) and \(y = -t\).
Express the parameter \(t\) in terms of \(y\) from the second equation: \(t = -y\).
Substitute \(t = -y\) into the equation for \(x\) to eliminate the parameter: \(x = (-y)^{2} + 2(-y)\).
Simplify the expression for \(x\): \(x = y^{2} - 2y\).
Recognize that this equation represents a parabola in the \(xy\)-plane, and since \(x\) is expressed as a quadratic function of \(y\), the parabola opens horizontally (to the right or left). Analyze the sign of the quadratic term to determine the direction.