Which of the following best describes the graph of the parametric equations $x=t^2$, $y=t$ for $-3\le t\le 3$?

• A. A right-opening parabola traced from bottom to top as $t$ increases
• B. A circle centered at the origin traced counterclockwise
• C. An ellipse centered at the origin
• D. A straight line passing through the origin

Answer: (A)

Verified step by step guidance

1. Identify the parametric equations given: \(x = t^{2}\) and \(y = t\), with the parameter \(t\) ranging from \(-3\) to \$3$.
2. Express \(x\) in terms of \(y\) by substituting \(t = y\) into \(x = t^{2}\), which gives \(x = y^{2}\).
3. Recognize that the equation \(x = y^{2}\) represents a parabola that opens to the right along the positive \(x\)-axis.
4. Consider the direction in which the curve is traced as \(t\) increases from \(-3\) to \$3\(: since \)y = t\(, \)y\( increases from \)-3\( to \)3\(, so the graph is traced from bottom (negative \)y\() to top (positive \)y$).
5. Conclude that the graph is a right-opening parabola traced from bottom to top as \(t\) increases.