If you have used a drawing program on a computer, you have probably worked with a cu...

If you have used a drawing program on a computer, you have probably worked with a curve known as a curve. ² Such a curve is defined parametrically by using several control points in the plane to shape the curve. In Exercises 7–12, we discuss various aspects of quadratic curves. These curves are defined by using three fixed control points (x₁, y₁), (x₂, y₂), and (x₃, y₃) and a nonnegative constant w. The curve defined by this information is given by x: [0, 1] → R², x(t) = (x(t), y(t)), where

Determine the parametrization for the portion of the parabola y = x² between the points (−2, 4) and (2,4) as follows:

(a) Two of the three control points must be (−2,4) and (2,4). Find the third control point using the result of Exercise 10.

(b) Using part (a) and Exercise 9, we must have that x(1/2) lies on the y-axis and, hence, at the point (0,0). Use the result of Exercise 11 to determine the constant w.

(c) Now write the parametrization. You should be able to check that your answer is correct.