Curve in parametric form

Example: For curve given by x = e^2t, y = t + 1 (in figure below)

a. Eliminate the parameter, t, to find a Cartesian equation of the curve. i.e. y = y(x)

b. Sketch the curve and indicate with an arrow the direction in which the

curve is traced as the parameter, t, increases.

Part a: ln x = 2t so 2t = ln x Solve for t.

and t = (1/2) ln x = ln √ x, then substitute into the equation for y

Therefore y = ln √ x + 1 (result)

Part b: Plot y = ln √ x + 1

Note: The trajectory increases up and to the right as t increases.

Click here for a discussion of derivatives of parametric forms and of polar coordinates.

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