 # Mathematics Practices

Exploration 15-6a: Rehearsal #1 for Test

Objective: Analyze polynomial functions and their rates of change.

 Answer all of these questions on separate paper. You may draw graphs on this sheet where specified. 1. Let Plot the graph of f. Sketch the result here. 2. By synthetic substitution , show that 2 is a zero of . 3. Write as the product of a linear binomial and a cubic polynomial . By synthetic substitution, show that -1 is a zero of the cubic polynomial. 4. Write as the product of two linear binomials and a quadratic trinomial . Show that the quadratic trinomial has no real zeros . 5. Find the two nonreal complex zeros of 6. Explain how the graph in Problem 1 agrees with the zeros you found algebraically in Problems 2–5. 7. Let Plot the graph of g on the same screen as in Problem 1. Use the result to find the two integer zeros of 8. To five decimal places, find the other two real zeros of g(x). For Problems 9–14, a freight train backs up, stops, and then goes forward. The displacement, kilometers, of its engine from a railroad crossing is given by where t is time, in minutes (see graph). 9. What is the average velocity of the train from to minutes? What is the average velocity from to minutes? Based on these numerical answers , what would you conjecture is the instantaneous velocity at ? 10. Write an expression for the average velocity of the train from 3 minutes to t minutes. By simplifying the resulting fraction , calculate algebraically the instantaneous velocity at minutes. How does the answer compare with your conjecture in Problem 9? 11. Plot a line through the point where on the previous graph , with slope equal to the instantaneous velocity at time How does the line relate to the graph? 12. At time minute, is the train approaching the railroad crossing or moving away from it? At what instantaneous rate? 13. Let Find and .Show that these values agree with the instantaneous velocity of the train at and ? 14. in Problem 13 is the velocity function. At what two times is the velocity equal to zero? At the positive value of t, is the displacement positive or negative? Give a real-world interpretation of your answer. 15. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-6b: Rehearsal #2 for Test

Objective: Analyze polynomial functions and their rates of change

 1. Mr. X, the math teacher, is making up a test for his students. He wants them to find the particular equation of a cubic function, but all he will reveal about it is that for its three zeros, and , this is true: Find the particular equation for . Plot the graph, and sketch the result here. 2. One of the zeros of is a real number. Are the other two zeros real numbers or nonreal complex numbers? How can you tell this from the graph? 3. Confirm your conclusion in Problem 2 by calculating the three zeros. 4. Use the answers to Problem 3 to show that the zeros have the sum and product properties specified in Problem 1. 5. Function has a double zero (a zero of multiplicity 2) because two of the factors are zero for the same value of x, namely, . Plot the graph and sketch the result here. What happens to the graph when there is a real zero of multiplicity 2? 6. Let . By synthetic substitution, show that -1 is a zero of .Then find the other zeros of . 7. A quartic function such as is supposed to have exactly four zeros. How do you explain the answer to Problem 6 in regard to this property? 8. On this set of axes, sketch the graph of h. 9. At a double zero, the graph of a polynomial function just touches the x-axis but does not cross it. Is the same thing true of a triple zero? How can you tell from the graph that a polynomial function might have a triple zero? For Problems 10–13, a bumblebee flies past a flower. It decides to go back for another look. Just before it reaches the flower, it turns and flies off. The bumblebee’s displacement from the flower, feet, is given by where t is time, in seconds (see graph). 10. What is its average velocity from to seconds? 11. Write an expression for the bee’s average velocity of the train from 6 seconds to t seconds. By simplifying the resulting fraction, calculate algebraically the bee’s exact instantaneous velocity at seconds. 12. Plot a line through the point where on the given graph, with slope equal to the instantaneous velocity at time . How does the line relate to the graph? 13. At what (positive) time was the bee closest to the flower? How close? 14. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-6c: Rehearsal #3 for Test, Date:
Annotated List

Objective: Analyze polynomial functions and their rates of change.

 1. Find the particular equation of a polynomial function. a. From points, e.g., a cubic function containing the points (2, 7), (3, 34), (4, 91), and (5, 190). b. From zeros, e.g., a cubic function with leading coefficient 1 and zeros -5, 3, and 6. Write in factored form and then multiply and simplify . Confirm by graphing. c. From sums and products of zeros, e.g., sum= 1, product = -15, sum of pairwise products = -7. 2. Find zeros of a polynomial function. a. By graphing, e.g., Explain how you know that two of the zeros of are complex numbers. b. By synthetic substitution, e.g., Find the complex zeros using the quadratic formula. 3. Find discontinuities in a rational function . a. Vertical asymptotes, e.g., Sketch the graph. Explain how the graph is a transformation of b. Removable discontinuities, e.g., Find the x- and y-coordinates algebraically. Confirm graphically using a friendly window that includes the discontinuity as a grid point. 4. Find the rate of change of a polynomial function. a. Average rate numerically, e.g., the displacement of a moving object, , in feet, is given by where x is in seconds. Find the average rate of change of for the time intervals [3, 3.1], [3, 3.01], and [3, 3.001]. Give the units. b. Instantaneous rate numerically, e.g., use the results of part a to make a conjecture about the instantaneous rate of change of at . Explain the basis for your conjecture. c. Instantaneous rate algebraically, e.g., for in part a, write the equation for a rational algebraic function that gives the average velocity of the moving object for the time interval [3, x]. Simplify the equation by doing the calculations needed to remove the removable discontinuity. Use the simplified equation to calculate the instantaneous velocity at exactly. d. Instantaneous rate by shortcut, e.g., let . Show that equals the instantaneous rate of change of at Explain how you can find the equation for from the equation for . e. Instantaneous rate graphically, e.g., the figure here shows the graph of from part a. Plot a line on the graph at the point with slope equal to the instantaneous rate of change of at that point. How is the line related to the graph? The next figure shows the graph of a function. Estimate graphically the instantaneous rate of change of the function at . f. Instantaneous rate verbally, e.g., “The instantaneous rate at is the —?— of the average rates as —?—” and “The instantaneous rate of change of at a particular point is called the —?— of ” 5. What did you learn as a result of doing this Exploration that you did not know before?
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