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 parabola opening to the right
• B. A straight line passing through the origin
• C. A circle centered at the origin
• D. An ellipse centered at 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 \(t\) in terms of \(y\) since \(y = t\), so \(t = y\).
3. Substitute \(t = y\) into the equation for \(x\) to eliminate the parameter: \(x = (y)^{2} = y^{2}\).
4. Recognize that the equation \(x = y^{2}\) represents a parabola that opens to the right because \(x\) is expressed as a square of \(y\).
5. Note the domain of \(t\) restricts \(y\) to values between \(-3\) and \$3\(, so the graph is the portion of the parabola \)x = y^{2}\( for \)y\( in \)[-3, 3]$.