Graphing x = t² - 2t, y = t + 1

Calculus 2 · Axiom Academy

Graph a parametric curve from a table of values, then eliminate the parameter to find its Cartesian equation. Graph the parametric curve . Build a table of points (x, y) for several values of t , plot the curve, then eliminate the parameter to write the equation in terms of x and y alone. As t increases from −1 to 3, the point traces a parabola opening to the right. The leftmost point, the vertex, is reached at t = 1. Nice work! You graphed a parametric curve and converted it to Cartesian form. Here's what we used: Build a t-table: Pick several values of t (negative, zero, and positive) to sketch the curve's shape. Plot and connect: Plot the (x, y) points and connect them in order of increasing t — this also shows the direction of travel along the curve. Eliminate the parameter: Solve the simpler equation for t , then substitute into the other equation to relate x and y directly. Recognize the curve: x = y^2 - 4y + 3 is a parabola opening to the right, with vertex at (-1, 2) . What parametric adds: The Cartesian form shows the shape; the parametric form also encodes the direction of motion along the curve. Eliminating the parameter is a core Calculus 2 skill — you'll use it to analyze curves, find tangent lines, and compute arc length for parametric and polar equations.

This is the written version of the interactive lesson above. See the full Calculus 2 course.