In this section you will:

It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately

to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

Given a pair of parametric equations, sketch a graph by plotting points.

and

for values of

over the interval for which the functions are defined.

Sketch the graph of the parametric equations

Construct a table of values for

and

as in [link], and plot the points in a plane.

The graph is a parabola with vertex at the point

opening to the right. See [link].

As values for

progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As values of

become negative, they trace out the lower half of the parabola. There are no restrictions on the domain. The arrows indicate direction according to increasing values of

The graph does not represent a function, as it will fail the vertical line test. The graph is drawn in two parts: the positive values for

and the negative values for

Sketch the graph of the parametric equations

Construct a table of values for the given parametric equations and sketch the graph:

Construct a table like that in [link] using angle measure in radians as inputs for

and evaluating

and

Using angles with known sine and cosine values for

makes calculations easier.

 0

By the symmetry shown in the values of

and

we see that the parametric equations represent an ellipse. The ellipse is mapped in a counterclockwise direction as shown by the arrows indicating increasing

values.

We have seen that parametric equations can be graphed by plotting points. However, a graphing calculator will save some time and reveal nuances in a graph that may be too tedious to discover using only hand calculations.

Make sure to change the mode on the calculator to parametric (PAR). To confirm, the

window should show

Graph the parametric equations:

Graph the parametric equations

and

First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.

Construct a table of values like that in [link].

Plot the

values from the table. See [link].

Next, translate the parametric equations to rectangular form. To do this, we solve for

in either

or

and then substitute the expression for

in the other equation. The result will be a function

if solving for

as a function of

or

if solving for

as a function of

Then, use the Pythagorean Theorem.

In [link], the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.

Graph the parametric equations

and

and the rectangular equivalent

on the same coordinate system.

Construct a table of values for the parametric equations, as we did in the previous example, and graph

on the same grid, as in [link].

With the domain on

restricted, we only plot positive values of

The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.

Sketch the graph of the parametric equations

along with the rectangular equation on the same grid.

The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.

Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object’s path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.

A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of

to the horizontal, with an initial speed of

and at a height

above the horizontal.

The path of an object propelled at an inclination of

to the horizontal, with initial speed

and at a height

above the horizontal, is given by

where

accounts for the effects of gravity and

is the initial height of the object. Depending on the units involved in the problem, use

or

The equation for

gives horizontal distance, and the equation for

gives the vertical distance.

Given a projectile motion problem, use parametric equations to solve.

Substitute the initial speed of the object for

indicates the angle at which the object is propelled. Substitute that angle in degrees for

The term

represents the effect of gravity. Depending on units involved, use

or

Again, substitute the initial speed for

and the height at which the object was propelled for

Solve the problem presented at the beginning of this section. Does the batter hit the game-winning home run? Assume that the ball is hit with an initial velocity of 140 feet per second at an angle of

to the horizontal, making contact 3 feet above the ground.

Use the formulas to set up the equations. The horizontal position is found using the parametric equation for

Thus,

The vertical position is found using the parametric equation for

Thus,

Substitute 2 into the equations to find the horizontal and vertical positions of the ball.

After 2 seconds, the ball is 198 feet away from the batter’s box and 137 feet above the ground.

To calculate how long the ball is in the air, we have to find out when it will hit ground, or when

Thus,

When

seconds, the ball has hit the ground. (The quadratic equation can be solved in various ways, but this problem was solved using a computer math program.)

We cannot confirm that the hit was a home run without considering the size of the outfield, which varies from field to field. However, for simplicity’s sake, let’s assume that the outfield wall is 400 feet from home plate in the deepest part of the park. Let’s also assume that the wall is 10 feet high. In order to determine whether the ball clears the wall, we need to calculate how high the ball is when x = 400 feet. So we will set x = 400, solve for

and input

into

The ball is 141.8 feet in the air when it soars out of the ballpark. It was indeed a home run. See [link].

Access the following online resource for additional instruction and practice with graphs of parametric equations.

and

depend, parametric equations can be used.

and

Choose values for

in increasing order. Plot the last two columns for

and

and

Initial velocity is symbolized as

represents the initial angle of the object when thrown, and

represents the height at which the object is propelled.

What are two methods used to graph parametric equations?

plotting points with the orientation arrow and a graphing calculator

What is one difference in point-plotting parametric equations compared to Cartesian equations?

Why are some graphs drawn with arrows?

The arrows show the orientation, the direction of motion according to increasing values of

Name a few common types of graphs of parametric equations.

Why are parametric graphs important in understanding projectile motion?

The parametric equations show the different vertical and horizontal motions over time.

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

For the following exercises, sketch the curve and include the orientation.

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

For the following exercises, graph the equation and include the orientation.

For the following exercises, use the parametric equations for integers a and b:

Graph on the domain

where

and

and include the orientation.

Graph on the domain

where

and

, and include the orientation.

Graph on the domain

where

and

, and include the orientation.

Graph on the domain

where

and

, and include the orientation.

If

is 1 more than

describe the effect the values of

and

have on the graph of the parametric equations.

Describe the graph if

and

There will be 100 back-and-forth motions.

What happens if

is 1 more than

Describe the graph.

If the parametric equations

and

have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

Take the opposite of the

equation.

For the following exercises, describe the graph of the set of parametric equations.

and

is linear

and

is linear

The parabola opens up.

and

is linear

Write the parametric equations of a circle with center

radius 5, and a counterclockwise orientation.

Write the parametric equations of an ellipse with center

major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

For the following exercises, use a graphing utility to graph on the window

by

on the domain

for the following values of

and

, and include the orientation.

For the following exercises, look at the graphs that were created by parametric equations of the form

Use the parametric mode on the graphing calculator to find the values of

and

to achieve each graph.

For the following exercises, use a graphing utility to graph the given parametric equations.

Graph all three sets of parametric equations on the domain

Graph all three sets of parametric equations on the domain

Graph all three sets of parametric equations on the domain

The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

Explain the effect on the graph of the parametric equation when we switched

and

.

The

-intercept changes.

Explain the effect on the graph of the parametric equation when we changed the domain.

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation

, while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation

Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time

Neglect air resistance.

Find parametric equations that model the problem situation.

Find all possible values of

that represent the situation.

When will the dart hit the ground?

approximately 3.2 seconds

Find the maximum height of the dart.

At what time will the dart reach maximum height?

1.6 seconds

For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

An epicycloid:

on the domain

.

A hypocycloid:

on the domain

.

A hypotrochoid:

on the domain

.

A rose:

on the domain

.