## Parametric Equations: Graphs

### Learning Outcomes

• 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 approximately $45^\circ$ 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.

Figure 1. Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)

## 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.

1. Construct a table with three columns: $t,x\left(t\right),\text{and}y\left(t\right)$.
2. Evaluate $x$ and $y$ for values of $t$ over the interval for which the functions are defined.
3. Plot the resulting pairs $\left(x,y\right)$.

### Example 1: Sketching the Graph of a Pair of Parametric Equations by Plotting Points

Sketch the graph of the parametric equations $x\left(t\right)={t}^{2}+1,y\left(t\right)=2+t$.

### Try It

Sketch the graph of the parametric equations $x=\sqrt{t},y=2t+3,0\le t\le 3$.

### Example 2: Sketching the Graph of Trigonometric Parametric Equations

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

\begin{align}&x=2\cos t \\ &y=4\sin t\end{align}

### Try It

Graph the parametric equations: $x=5\cos t,y=3\sin t$.

### Example 3: Graphing Parametric Equations and Rectangular Form Together

Graph the parametric equations $x=5\cos t$ and $y=2\sin t$. First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.

### Example 4: Graphing Parametric Equations and Rectangular Equations on the Coordinate System

Graph the parametric equations $x=t+1$ and $y=\sqrt{t},t\ge 0$, and the rectangular equivalent $y=\sqrt{x - 1}$ on the same coordinate system.

### Try It

Sketch the graph of the parametric equations $x=2\cos \theta \text{ and }y=4\sin \theta$, along with the rectangular equation on the same grid.

## Applications of 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 $\theta$ to the horizontal, with an initial speed of ${v}_{0}$, and at a height $h$ above the horizontal.

The path of an object propelled at an inclination of $\theta$ to the horizontal, with initial speed ${v}_{0}$, and at a height $h$ above the horizontal, is given by

\begin{align}x&=\left({v}_{0}\cos \theta \right)t \\ y&=-\frac{1}{2}g{t}^{2}+\left({v}_{0}\sin \theta \right)t+h \end{align}

where $g$ accounts for the effects of gravity and $h$ is the initial height of the object. Depending on the units involved in the problem, use $g=32\text{ft}\text{/}{\text{s}}^{2}$ or $g=9.8\text{m}\text{/}{\text{s}}^{2}$. The equation for $x$ gives horizontal distance, and the equation for $y$ gives the vertical distance.

### How To: Given a projectile motion problem, use parametric equations to solve.

1. The horizontal distance is given by $x=\left({v}_{0}\cos \theta \right)t$. Substitute the initial speed of the object for ${v}_{0}$.
2. The expression $\cos \theta$ indicates the angle at which the object is propelled. Substitute that angle in degrees for $\cos \theta$.
3. The vertical distance is given by the formula $y=-\frac{1}{2}g{t}^{2}+\left({v}_{0}\sin \theta \right)t+h$. The term $-\frac{1}{2}g{t}^{2}$ represents the effect of gravity. Depending on units involved, use $g=32{\text{ft/s}}^{2}$ or $g=9.8{\text{m/s}}^{2}$. Again, substitute the initial speed for ${v}_{0}$, and the height at which the object was propelled for $h$.
4. Proceed by calculating each term to solve for $t$.

### Example 5: Finding the Parametric Equations to Describe the Motion of a Baseball

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 $45^\circ$ to the horizontal, making contact 3 feet above the ground.

1. Find the parametric equations to model the path of the baseball.
2. Where is the ball after 2 seconds?
3. How long is the ball in the air?
4. Is it a home run?

## Key Concepts

• When there is a third variable, a third parameter on which $x$ and $y$ depend, parametric equations can be used.
• To graph parametric equations by plotting points, make a table with three columns labeled $t,x\left(t\right)$, and $y\left(t\right)$. Choose values for $t$ in increasing order. Plot the last two columns for $x$ and $y$.
• When graphing a parametric curve by plotting points, note the associated t-values and show arrows on the graph indicating the orientation of the curve.
• Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion.
• Projectile motion depends on two parametric equations: $x=\left({v}_{0}\cos \theta \right)t$ and $y=-16{t}^{2}+\left({v}_{0}\sin \theta \right)t+h$. Initial velocity is symbolized as ${v}_{0}$. $\theta$ represents the initial angle of the object when thrown, and $h$ represents the height at which the object is propelled.