Modelling Drug Concentration

A drug trial is studying the efficacy of a new drug for thalassemia, a genetic disorder in which a patient is unable to properly synthesize the hemoglobin molecule in the blood. In the trial, the rate at which the drug is metabolized by the body is studied. The data from the trial is then used to construct a model for the drug's metabolism. The concentration of a drug, ?(?) in parts per million, in a patient's blood ? hours after the drug is administered is given by the function

?(?) = ??5 +3?4 ?6?3 ?2?2 +60?

The goal of this tutorial is to graph this function and analyze and interpret the graph within the problem context.

Show the following steps on the blackboard or paper as your instructor directs:

a) State the coordinates of the y-intercept. Explain the meaning of this intercept within the problem context.

b) Use the procedure in Unit 2 Chapter 5 (p.211) to solve for all the zeros of f(t). Show clearly all steps of this procedure, identify the fully factored form of f(t) as part of the process.

c) Which of the zeros found in part b) are t-intercepts? Explain the meaning of the intercepts within the problem context.

d) Use the procedure for sketching polynomial functions (p. 189) to sketch the graph for f(t). Show clearly all steps of this procedure.

e) Identify the practical domain on your graph and state the practical domain in interval notation.

f) Using your graph, estimate when the drug attains maximum concentration. Check your answer by using graphing software (graphing calculator, wolframalpha phone app or website, desmos.com, etc).

II GRAPH, ANALYZE, AND INTERPRET HIGHER DEGREE POLYNOMIAL FUNCTIONS The objective in this section is to efficiently sketch the graphs of polynomial functions of degree 3 procedure of polynomial functions of higher degree: or larger. The Multiplicity Test and the Leading Coefficient Test are incorporated into the graphing Sketching Higher Degree Polynomial Functions: A. Inspect the function to see if the method of Transformations is applicable. B. If the method of Transformations does NOT apply, 1. Find the y-intercept of the function. 2. Find the x-intercepts of the function by factoring the function to linear and/or prime quadratic factors. We can then solve for the real zeros. 3. Do a Sign Chart to determine where the function is above or below the x-axis. 4. Use the Leading Coefficient Test and the Multiplicity Test to check your graph. Notate the end-behavior on your graph. The method of constructing a Sign Chart is illustrated in the following example. Note that if the function is not factorable and the x-intercepts cannot be found, a graphing utility (graphing calculator or computer software) would be required to sketch the function. Eg. 1 Modeling Drug Metabolism with a Polynomial Function A drug trial is studying the efficacy of a new drug for thalassemia, a genetic disorder in which a patient is unable to properly synthesize the hemoglobin molecule in the blood. In the trial, the rate at which the drug is metabolized by the body is studied. The data from the trial is then used to construct a model for the drug's metabolism. The concentration of a drug, f(x) in parts per million, in a patient's blood x hours after the drug is administered is given by the function f(x ) = -x*+ x3 + 2x2 Fourth degree polynomial function in Expanded Form a) According to the model, when is the drug is completely eliminated from the bloodstream? b) When does the drug attain maximum concentration? c) State the practical domain of this function. Solution: Sketch the function f(x) = - x4 + x3 + 2x on the mathematical domain, (-co, +co), following the steps above. 1. Find the y-intercept: (0, 0) . This point is plotted below. 2. Find the x-intercepts. Begin by factoring the function fully: f(x) = -x(x' - x - 2) f (x) = -x'(x - 2/x + 1) - Unit 2 Chapter 4: Page 208 -Ill FINDING ZEROS OF POLYNOMIAL FUNCTIONS In example I in the previous section, a logical process for finding the zeros of polynomial functions is used: expressed. Assuming that at least one rational zero exists, the following trial and error process can be *Finding Zeros of Polynomial Functions: 1. List all possible rational zeros using the Rational Zero Theorem. 2. Test each value in the list above until you find an actual rational zero. Test by using Synthetic Division to check the remainder. 3. Use the Division Algorithm and express the function in a fully factored form. 4. i) If the factors are linear and quadratic, solve for the remaining zeros. ii) If the factors are linear and higher degree (third degree or greater ), repeat the above steps for the higher degree factor. Eg. 1 For the third degree polynomial function, f(x) = 14x2 +15x3 -2-3x, a) List all possible rational zeros using the Rational Zero Theorem: Solution: Rewrite the function in descending order first: f(x) =15x +14x2 - 3x-2 All possible rational zeros: + 1,2 1,3,5,15 b) Test each value in the list above until you find an actual rational zero of the function. Solution: If a value in the list above is a zero, then the remainder of the division will be zero. The remainder can be found by Synthetic Division or by using the Remainder Theorem. It is usually easier to test by Synthetic Division: Test x = 1: Remainder Theorem: Synthetic Division: Remainder = f (1) -2 15 14 -3 Remainder = 15(1)3 + 14(1)2 -3(1) -2 Remainder = 24 X 29 26 - - 15 NO, x = 1 is NOT a zero! 24 X 29 26 15 Unit 2 Chapter 5: Page 237