What does the graph of the parametric equations x(t)=4−t and y(t)=(t−1)^2, where t is on the interval [−2,2], look like?
The parametric equations graph as a portion of a parabola. The initial point is _____, and the terminal point is _____. The vertex of the parabola is ____. Arrows are drawn along the parabola to indicate motion ____.
x = 4 - t ⇒ t = 4 - x (1)
y = (t - 1)^2 (2)
Put (1) into (2)
y = ( 4 - x - 1)^2
y = ( 3 - x)^2
y = ( x - 3)^2
y = a ( x -3)^2 + 0
In the form y = a(x - h)^2 + k
a = 1 and the vertex is (h,k) = (3,0)
The initial point is when t = -2
x = (4 - -2) = 6 y = (-2 - 1)^2 = (-3)^2 = 9 ⇒ (6,9)
When t = 2
x = (4 - 2) = 2 y = (2 - 1)^2 = 1^2 = 1 ⇒ (2, 1)
Since (6.9) is to the right of (2,1) the motion is right to left