Graphing Parametric Equations Examples

Graph the function defined by the parametric equations

x(t) = -cos t
y(t) = -sin t

for 0 ≤ t ≤ 2π.

Start by making a table of values with points for certain values of t:

Graph these points and label them with their t-values:

When we connect the points in the order they were drawn, we find a quarter-circle:

If we graph 0 and 2π, we get 

We've come (wait for it) full circle.

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

We make a table, then we graph the points and play connect the dots.

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

Make a table of values. When t = 0 we have x = f(0) = 1 and y = g(0) = 0. 

 When t = 1 we have x = f(1) = 1 and y = g(1) = 1.

Continuing like this, we make the following table: 

 We plot the points and connect the dots as we did when we were given equations. 

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

When t = 0 we have x = f(0) = 1 and y = g(0) = 0. That means we need to plot the point (x, y) = (1, 0) on our new graph: 

When t = 1 we have x = f(1) = 1 and y = g(1) = -1. 

When t = 2 we have x = f(2) = -1 and y = g(2) = -1. 

When t = 3 we have x = f(3) = -1 and y = g(3) = 1. 

When t = 4 we have x = f(4) = 0 and y = g(4) = 1. 

The final picture is