Tangent Lines to Parametric Curves

Trigonometry · Axiom Academy

LESSON Tangent Lines to Parametric Curves Discovering how to find the slope of a curve when position is described by two separate functions of time 1. What is a Parametric Curve? Instead of writing y as a function of x , parametric equations express both x and y as functions of a third variable, typically t (representing time or another parameter). As t varies, the point ( x , y ) traces out a curve in the plane. Think of it as a particle moving through space over time. To find the tangent line, we need to understand how the position changes. Both coordinates are changing with respect to t : dx/dt tells us the horizontal rate of change dy/dt tells us the vertical rate of change These two rates form a velocity vector that points in the direction the curve is moving at any instant. The slope of the tangent line is the ratio of vertical change to horizontal change. Using the chain rule concept: This makes intuitive sense: if y is changing twice as fast as x , then the slope should be 2. The formula captures this relationship precisely. Let's find the tangent line to the curve x = t ², y = t ³ at t = 1. Tangent line equation: y - 1 = (3/2)( x - 1)

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