Exercise · ZW-PARAMETRIC-CURVE-009 · ★★★ Advanced
Let \(\mathbf r_1(t)=\langle t,t^2\rangle\) and \(\mathbf r_2(t)=\langle t^2,t^4\rangle\), with \(t\in\mathbb R\). Which is true?
Choose one option.
A Both trace the whole parabola \(y=x^2\) B \(\mathbf r_2\) traces only the half \(x\ge0\) of \(y=x^2\) C \(\mathbf r_2\) traces a line D \(\mathbf r_1\) traces only the half \(x\ge0\) Show answer B. \(\mathbf r_2\) traces only the half \(x\ge0\) of \(y=x^2\)← Back to the question
Explanation Both satisfy \(y=x^2\), but in \(\mathbf r_2\) the abscissa \(t^2\) is never negative. The parametrization decides which part of the graph is traced.
A parametric curve is the path traced by a point in space when its coordinates depend on a parameter \(t\) through a vector-valued function \(\mathbf r\).
\[\mathbf r(t)=\langle x(t),y(t),z(t)\rangle,\quad a\le t\le b\]
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