10.4 Parametric Equations Parametric Equations of a Plane Curve A plane curve is a set of points (x, y) such that x = f (t), y = g(t), and f and g are both defined on an interval I. The equations x = f (t) and y = g(t) are parametric equations with parameter t.

10.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent Example For the plane curve defined by the parametric equations graph the curve and then find an equivalent rectangular equation. Analytic Solution Make a table of corresponding values of t, x, and y over the domain t and plot the points.

10.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent The arrow heads indicate the direction the curve takes as t increases.

10.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent To find the equivalent rectangular form, eliminate the parameter t. This is a horizontal parabola that opens to the right. Since t is in [–3, 3], x is in [0, 9] and y is in [–3, 9] . The rectangular equation is Use this equation because it leads to a unique solution.

10.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent Graphing Calculator Solution Set the calculator in parametric mode where the variable is t and let X1T = t2 and Y1T = 2t + 3. (We have been in rectangular mode using variable x.)

10.4 Example 2: Graph of a Parametric Equation and Its Rectangular Equivalent Example Graph the plane curve defined by Solution Get the equivalent rectangular form by substitution of t. Since t is in [–2, 2], x is in [1, 9].

10.4 Example 2: Graph of a Parametric Equation and Its Rectangular Equivalent This represents a complete ellipse. By definition, y  0. Therefore, the graph is the upper half of the ellipse only.

10.4 Graphing a Line Defined Parametrically Example Graph the plane curve defined by x = t2, y = t2, and then find an equivalent rectangular form. Solution x = t2 = y, so y = x. To be equivalent, however, the rectangular equation must be given as y = x, x  0 (half the line y = x since t2  0).

10.4 Alternative Forms of Parametric Equations Parametric representations of a curve are not always unique. One simple parametric representation for y = f (x), with domain X, is Example Give two parametric representations for the parabola Solution

10.4 Projectile Motion Application The path of a moving object with position (x, y) can be given by the functions where t represents time. Example The motion of a projectile moving in a direction at a 45º angle with the horizontal (neglecting air resistance) is given by where t is in seconds, 0 is the initial speed, x and y are in feet, and k > 0. Find the rectangular form of the equation.

10.4 Projectile Motion Application Solution Solve the first equation for t and substitute the result into the second equation. A vertical parabola that opens downward.