# At a Glance - Graphing Parametric Equations

Graphing parametric equations is similar to graphing vector functions.

One way to graph parametric equations is to find points for certain values of t, graph those points, then play connect-the-dots. The points will usually give an idea of the shape of the graph. Then we know whether to connect the dots with straight lines or with curves.

We can label the points with their t-values and put arrows on the connecting lines or curves to show the order in which the points are drawn.

It's also possible to graph parametric functions when given graphs instead of formulas for the component functions f(t) and g(t).

The graph of x = f(t) has t- and x-axes.
The graph of g has t- and y-axes.

We take information from these graphs and build a new graph with x- and y-axes.

One way to collect the information we need for the new graph is to build a table of values like we did before.

Instead of using formulas to calculate the values of x and y at particular t values, we read the values of x and y off their respective graphs.

How do we know the lines are straight?

If the functions x = f(t) and y = g(t) are both made up of straight line segments, then the graph of the parametric function will also be made up of straight line segments.

Take our word on that for now. We'll justify it when we talk about slopes of parametric functions.

Making a table of values can help keep the t and the x and the y straight. However, it's not necessary.

As long as we're careful, we can go straight from the graphs of f(t) and g(t) to the final graph of <f(t), g(t)>.

#### Example 1

 Graph the function defined by the parametric equationsx(t) = -cos ty(t) = -sin tfor 0 ≤ t ≤ 2π.

#### Example 2

 Graph the function described by the parametric equations x(t) = 2t, y(t) = t2 for -4≤ t ≤ 4.

#### Example 3

 Graph the function described by x = f(t) and y = g(t) where the functions f(t) and g(t) are as shown below.

#### Example 4

 Graph the function where the functions f(t) and g(t) are as shown below.

#### Exercise 1

Graph the function defined by the parametric equations , and .

#### Exercise 2

Graph the function described by the given parametric equations.

• x(t) = t, y(t) = cos t for -π≤ t ≤ π

#### Exercise 3

Exercises. Graph the function described by the given parametric equations.

• x(t) = 4 + 2t, y(t) = 1 + t

#### Exercise 4

Exercises. Graph the function described by the given parametric equations.

• x(t) = cos t, y(t) = sin t for -π ≤ t ≤ π

#### Exercise 5

Gillian was asked to graph the function <f(t), g(t)> with f(t) and g(t) given by the following graphs:

She came up with this:

What did Gillian do wrong?

#### Exercise 6

Graph the function < f(t), g(t)> where x = f(t) and y = g(t) are given by the graphs.

#### Exercise 7

Graph the function < f(t), g(t)> where xf(t) and yg(t) are given by the graphs.

#### Exercise 8

Graph the function < f(t), g(t)> where xf(t) and yg(t) are given by the graphs.