B)The volume of the box, V(x), is given by the cubic function: ?(?) = ?(20 ? 2?)(16 ? 2?) Explain how this function has been derived. Fully expand this cubic function. c) Sketch the graph of this function, without technology, and estimate the value of x that will maximize the volume of the box. d) What is the practical domain of this polynomial function? Explain your answer. 3. Graph and Interpret a Cubic Function The following function, ?(?) = (? ? 2.6)(? ? 4.5)2 models the annual profit in $ millions of a mid-size marketing firm with t in years from start-up in 2012 (t = 0 is the year 2012). Sketch the graph of this function, without technology, and then answer the questions that follow . a) Is the profit increasing or decreasing after 2017 (t = 5)? _______________________ Explain how you know this from the equation of the function? b) For what values of t is the profit exactly zero? t = _____________________. c) The practical domain is t ? [0,8]. State the interval for t in which the profit is negative in the practical domain: t ? _____________ How do you know this from the equation of the function?

Math 0132 Tutorial 4 Worksheet: Higher Degree Polynomial Functions 1. Find a Regression Model for Temperature Variation The following information describes the outdoor temperature change in Calgary on a day in mid-February from midnight to midnight. The temperature at midnight was 0 degrees Celsius. The temperature was below zero through the night and warmed back to zero at 9 am. The temperature increased throughout the day and cooled again to 0 degrees at 7 pm. The temperature was below zero and decreasing between 7 pm and midnight. Time, r Temperature, T(!) (hours from midnight) degrees Celsius) 4 -6 9 0 12 15 19 0 a) Perform Cubic Regression using the data in the table above. Write the model found using Cubic Regression: How good is the correlation? b) From the graph, estimate the maximum high temperature for that day and the time when this occurs. ) What does the model predict the temperature will be at midnight the next day (at hour 24)? Is this an unreasonable answer? Could you continue to use the model to predict the temperature beyond the practical domain given? Explain. 2. Graph and Interpret a Volume Function A rectangular piece of cardboard of size 16 inches by 20 inches is to be used in a factory to create boxes to transport apricots to market. The cardboard is formed into a box by cutting squares of dimensions x by x from each of the four corners and then folding up and taping the sides. The goal is to estimate the value of x that will maximize the enclosed volume. a) Draw both a plan view of the cardboard sheet and a three dimensional sketch of the taped box. Label the dimensions of the cutout square "x"