Vector-Valued Functions

AP Calculus · Axiom Academy

LESSON Vector-Valued Functions A new way to describe curves and motion Step 1: What is a Vector-Valued Function? A vector-valued function takes a number (usually t for time) and outputs a vector. Instead of a single output, we get a vector with components! This outputs a vector for each value of t As t varies, the tip of this vector traces out a curve in the plane. This is EXACTLY what parametric equations do! We can write vector-valued functions in several ways: They're all saying the same thing! Pick whichever notation feels most comfortable for you. Step 3: Three Important Examples As t increases, we move out along the x-axis while oscillating vertically Gives the parabola y = x² parametrically We can add, subtract, and scale vector-valued functions:

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