# Cubic Functions

NOTE: Figure not drawn to scale. In the figure above, ABCD is a rectangle. Points A and C lie on the graph of y = px3, where p is a constant. If the area of ABCD is 4, what is the value of p ?

p = 16 Level of Difficulty E M H There are several things to make note of before beginning work on this problem: ABCD is a rectangle whose sides are parallel to the x and y axes; the cubic graph goes through opposite corners of the rectangle; the given cubic polynomial is symmetric about the origin (it is an odd function). Given the observations above, we can conclude that the rectangle is centered about the origin and that the magnitude of a, b, c, and d are all the same. One way to express the height of the rectangle, therefore, is 2a. Notice the base of the rectangle has length = 1. Area = base x height; 4 = (1)(2a); a = 2. Therefore, 2 is the distance of the rectangle above and below the x axis. Let's label point A with its coordinates: (-1/2, -2). This point must satisfy the cubic equation because it lies on the graph of that function. After plugging in: -2 = p(-1/2)3 Solving for p, you should get p = 16.