Parametric Representation

Calculus 2 · Axiom Academy

LESSON Parametric Representation A curve isn't only a graph — it's a path traced over time. Let a parameter t drive the motion and watch curves that y=f(x) could never describe. 1. A Curve You Trace Over Time Give the parameter t to two functions, x=f(t) and y=g(t) . At every instant t you stand at the point ; as t increases, that point sweeps out a curve. Here x=t and y=t^2-1 : the input t slides the point left to right while y dips to a low of -1 and climbs back. A point in the plane, one for each value of t 2. Why Bother? Curves y=f(x) Can't Describe A function y=f(x) assigns exactly one y to each x — so a vertical line, a loop, or any path that revisits an x -value is off-limits. Watch the vertical line : as the parameter t runs, the point climbs straight up a single x -value, stacking many y 's over x=3 — impossible to write as y=f(x) , but trivial to parametrize. stacks infinitely many y 's over one x — never a function of x . A path can cross itself or close into a loop; that's many y 's per x , so not y=f(x) . The parameter records which way the curve is drawn — information a static graph throws away. Different parametrizations sweep the same shape at different rates — useful for motion in physics. 3. The Circle, and Eliminating the Parameter

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