If f and g are continuous functions of t on an interval I, then the set of ordered pairs (f(t),g(t)) is a C. The equations given by x=f(t) and y=g(t) are for C, and t is the .
Let's recall some facts about plane curves. A plane curve is a geometric object represented as a path in a coordinate plane. It is formed by a continuous sequence of points. Each point on the curve corresponds to a pair of coordinates (x, y), which is obtained by applying two continuous functions, f(t) and g(t), to a parameter t varying over a specific interval I.
In other words, as the parameter t varies within the interval I, the functions f(t) and g(t) generate a sequence of ordered pairs (x, y), where x = f(t) and y = g(t) are parametric equations for C. These ordered pairs collectively form the plane curve C. The parameter t serves as a way to traverse or parametrize the curve, allowing us to trace out its shape as t varies.
Now we can complete the sentence.
If f and g are continuous functions of t on an interval I, then the set of ordered pairs (f(t),g(t)) is a plane curve C. The equations given by x=f(t) and y=g(t) are parametric equations for C, and t is the parameter.
