Calculus 2 Parametric Curve

Question

A parametric curve is defined by

x = 2cos^2t - 1 and y = sint + 1 for 0<=t<=3pi/2

(a) Determine the domain and range of the curve on the given interval.

(b) Eliminate the parameter t to find the relation between x and y.

(c) Sketch the curve and indicate its orientation by choosing and labeling at least four points on the curve.

Answer 1

For question a): For t on [0, 3π/2]:

Looking at x, cos(t) varies between 1 and -1 so cos²(t) varies between 0 and 1. Multiplying by 2 makes it vary between 0 and 2. Subtracting 1 means the domain varies between -1 and 1

Looking at y, sin(t) varies between 1 and -1. Adding 1 means the range varies between 0 and 2.

For question b:

Start with the equation x = 2cos²(t) - 1 and add 1 to both sides:

x + 1 = 2cos²(t) [now divide both sides by 2

1/2(x + 1) = cos²(t)

Now work with the other equation y = sin(t) + 1 and subtract 1 from both sides:

y - 1 = sin(t) [now square both sides:

(y - 1)² = sin²(t) [now add it to the modified first equation:

(y - 1)² = sin²(t)

1/2(x + 1) = cos²(t)

----------------------------

(y - 1)² + 1/2(x + 1) = sin²(t) + cos²(t)

Using the Pythagorean Identity: sin²(t) + cos²(t) = 1 so the equation becomes:

(y - 1)² + 1/2(x + 1) = 1

1/2(x + 1) = 1 - (y - 1)²

(x + 1) = 2 - 2(y - 1)²

x = -2(y - 1)² + 1

For question c) This is a sideways parabola opening left with the vertex at (1, 1) and a horizontal stretch factor of 2.: