Determine whether each point is on the graph of the parametric equations
x(t) = t3 y(t) = t4
(8, 16)
(8, -16)
(-27, 81)
Answer
Since the equation x(t) = t3 can have only one solution while the equation y(t) = t4 can have two, we'll start with the x equation.
When x = 8 we must have t = 2. Then y = (2)4 = 16.The point (8, 16) is on the graph and occurs when t = 2.
When x = 8 we have t = 2. Then y = 16 not -16. Since the only value of t that makes x = 8 doesn't make y = -16, the point(8, -16) isn't on the graph.
When x = -27 we have t = -3. Theny = (-3)4 = 81.The point (-27, 81) is on the graph when t = -3.
Example 2
Show that the point (5, -6) is on the graph of the parametric equations
x(t) = t2 + 1 y(t) = 3t
Answer
We'll use the y-equation first since we'll find only one t-value that way.
When y = -6 we must have t = -2. Then
x = (-2)2 + 1 = 5.
The point (5, -6) occurs on the graph when t = -2.
Example 3
Show that the point (11, 9) is not on the graph of the parametric equations
x(t)= t2 + 1 y(t) = 3t
Answer
Using the y-equation first, if y = 9 then t = 3. Then
x = (3)2 + 1 = 10.
Since x ≠ 11 at the only t-value that makes y = 9, the point (11, 9) is not on the graph.
It's also possible to determine the points at which parametric equations intersect. This is how we'd know if Mario ran into a red koopa flying loftily over the canyon during his jump.