1. 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.

2. 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.

3. 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.

4. 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.

5. 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.)

6. 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].

7. 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.

8. 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  (half the line y = x since t2  0).

9. 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. 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.

11. 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.