Let's explore the equation y equals 2 times the quantity 3 minus x cubed. This is a cubic function that transforms the basic cubic function y equals x cubed. We can see the original blue curve and our target red curve. Notice how the red curve has its inflection point at 3 comma 0, shifted from the origin.
Let's break down the transformations step by step. We can rewrite the equation as negative 2 times the quantity x minus 3 cubed. This reveals three key transformations: first, a horizontal shift 3 units to the right, then a reflection across the x-axis, and finally a vertical stretch by a factor of 2. Watch how each transformation affects the graph.
Now let's examine the key features of this function. The inflection point is at 3 comma 0, which is the center of the S-shaped curve. The function is decreasing everywhere - as x increases, y decreases. Notice the behavior at the extremes: as x approaches infinity, y approaches negative infinity, and as x approaches negative infinity, y approaches positive infinity. The function is also steeper than the basic cubic due to the vertical stretch factor of 2.
Let's calculate specific values to verify our understanding. When x equals 0, we get y equals 54. When x equals 1, y equals 16. At x equals 2, y equals 2. The inflection point occurs at x equals 3 where y equals 0. Then as we continue, x equals 4 gives y equals negative 2, x equals 5 gives negative 16, and x equals 6 gives negative 54. Notice the symmetry around the inflection point.
In summary, the equation y equals 2 times the quantity 3 minus x cubed represents a transformed cubic function. It combines a horizontal shift 3 units right, a vertical stretch by factor 2, and a reflection across the x-axis. The resulting function is decreasing everywhere with an inflection point at 3 comma 0. Such cubic functions appear in many real-world applications including physics for motion equations, economics for cost functions, and engineering for design curves. Understanding these transformations helps us analyze and predict the behavior of complex mathematical models.