Multiple Choice
Graph the plane curve formed by the parametric equations and indicate its orientation.
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16. Parametric Equations
Graphing Parametric Equations
Multiple Choice
Which of the following best describes the graph of the parametric equations , for ?
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
Verified step by step guidance1
Identify the parametric equations given: \(x = t^{2}\) and \(y = t\), with the parameter \(t\) ranging from \(-3\) to \(3\).
Express \(t\) in terms of \(y\) since \(y = t\), so \(t = y\).
Substitute \(t = y\) into the equation for \(x\) to eliminate the parameter: \(x = (y)^{2} = y^{2}\).
Recognize that the equation \(x = y^{2}\) represents a parabola that opens to the right because \(x\) is expressed as a square of \(y\).
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 \([-3, 3]\).
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