# 22 Parametric Equations: Graphs

### Learning Objectives

In this section you will:

- Graph plane curves described by parametric equations by plotting points.
- Graph parametric equations.

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 approximatelyto 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.

### Graphing Parametric Equations by Plotting Points

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.

### How To

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

- Construct a table with three columns:
- Evaluate and for values of over the interval for which the functions are defined.
- Plot the resulting pairs

### Sketching the Graph of a Pair of Parametric Equations by Plotting Points

Sketch the graph of the parametric equations

#### Analysis

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

### Try It

Sketch the graph of the parametric equations

## Show Solution

### Sketching the Graph of Trigonometric Parametric Equations

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

## Show Solution

Construct a table like that in (Figure) using angle measure in radians as inputs forand evaluatingandUsing angles with known sine and cosine values formakes calculations easier.

0 | ||

(Figure) shows the graph.

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 increasingvalues.

#### Analysis

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, thewindow should show

instead of

### Try It

Graph the parametric equations:

## Show Solution

### Graphing Parametric Equations and Rectangular Form Together

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

## Show Solution

Construct a table of values like that in (Figure).

Plot thevalues from the table. See (Figure).

Next, translate the parametric equations to rectangular form. To do this, we solve forin eitherorand then substitute the expression forin the other equation. The result will be a function if solving foras a function ofor if solving foras a function of

Then, use the Pythagorean Theorem.

#### Analysis

In (Figure), 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.

### Graphing Parametric Equations and Rectangular Equations on the Coordinate System

Graph the parametric equations andand the rectangular equivalent on the same coordinate system.

## Show Solution

Construct a table of values for the parametric equations, as we did in the previous example, and graphon the same grid, as in (Figure).

#### Analysis

With the domain onrestricted, we only plot positive values ofThe 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.

### Try It

Sketch the graph of the parametric equationsalong with the rectangular equation on the same grid.

## Show Solution

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.