Name: Class: Date: A2R 3.4 Problem Set - Major Characteristics of Polynomial Graphs

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1 Name: Class: Date: A2R 3.4 Problem Set - Major Characteristics of Polynomial Graphs Use the coordinate plane to sketch a graph with the given characteristics. If the graph is not possible to sketch, explain why. Circle the function(s) that could model each graph. Describe your reasoning for either eliminating or choosing each function. 1. Characteristics: 3. even degree positive a-value six x-intercepts absolute maximum at y = 1 relative minimum at y = 4 f(x) = x 4 2x 3 3x 2 f(x) = 2x 4 3x 2 x f(x) = 2(x 2)(x + 3)(x + 1) 2. Characteristics: always decreasing y-intercept at y = 2.5 x-intercept at x = 3 4. f(x) = 4x 6 + 2x 3 1 f(x) = (x + 2)(x 5)(x + 3) + 2 f(x) = 0.25(x + 2)(x 5)(x + 3) + 2 1

2 Name: 5. d. Determine the y-intercept of the graph of p(x). Then, describe the meaning of the y-intercept in the context of the problem. e. Determine the x-intercepts of the graph of p(x). Include only the x-intercepts that are in the domain of the problem situation. Then, describe the meaning of the x-intercepts in the context of the problem. f(x) = 3x x 4 10x 3 240x 2 250x f(x) = (2x 3)(x + 4)(x 10)(x + 14) + 20 f(x) = 3x x 6 20x x Julio has been tracking the stock price of Computer Dudes, Inc. over the 40 days since it started trading on the New York Stock Exchange. He models the stock price over the 40 days with the function p(x) = x x x 2 + x , where p(x) represents the stock price, in dollars, on the xth day since the stock started trading. a. Graph the function p(x) on your graphing calculator and view the graph using the following window settings: Xmin = 0, Xmax = 40, Xscl = 5, Ymin = 1, Ymax = 10, Yscl = 1, and Xres = 1. Is each of the extrema for the graph of p(x) visible in this window? Explain how you know. f. Determine the intervals over which the function is increasing and decreasing. Include only the intervals within the context of the problem. Explain how these intervals relate to the problem situation. g. At what point during the 40 days since the stock started trading did the price of the stock reach a maximum? What was the maximum price at that point? Explain your reasoning. h. On the 40th day since the stock started trading, Julio analyzes his model and tries to decide whether to buy stock in Computer Dudes, Inc. Based on his model, should Julio buy the stock? b. Determine the domain and range of p(x). i. Do you think the function p(x) will effectively model the stock price beyond the 40th day? c. Explain what the domain and range mean in the context of the problem. Then, determine the domain and range of p(x) in terms of the problem situation. 2

3 A2R 3.4 Problem Set - Major Characteristics of Polynomial Graphs Answer Section 1. ANS: This is not possible because an even degree function with a positive a-value will go towards positive infinity at both ends, so it cannot have an absolute maximum. 2. ANS: 3. ANS: f(x) = x 4 2x 3 3x 2 I chose this function because it represents an even degree polynomial with a positive a-value. I eliminated this function because the graph represents an even degree polynomial. This function has a negative a-value. I eliminated this function because the graph represents an even degree function and this function is an odd degree. 1

4 4. ANS: I eliminated this function because the graph represents an odd degree polynomial and this function is an even degree. I eliminated this function because the graph represents an odd degree polynomial with a negative a-value and this function has a positive a-value. I chose this function because the graph represents an odd degree polynomial with a negative a-value. 5. ANS: f(x) = 3x x 4 10x 3 240x 2 250x I chose this function because it represents an odd function with a positive a-value. I eliminated this function because the graph represents an odd function and this function is an even degree. I eliminated this function because the graph represents an odd function with a positive a-value and this function has a negative a-value. 2

5 6. ANS: a. Yes, each of the extrema is visible in this window. The function p(x) is a 4th degree polynomial function that has at most 3 extrema. b. The domain of p(x) is (, ) and the range is (, 9.03). c. The domain is the number of days since the stock started trading. The range is the stock price. In terms of the problem situation, the domain is [0, 40] and the range is [0, 9.03]. d. The y-intercept is (0, 3.70). The y-intercept indicates that the stock price 0 days since the stock started trading was $3.70. This means that the stock price was $3.70 when the stock started trading on the New York Stock Exchange. e. The x-intercepts are (16.93, 0) and (22.53, 0). The x-intercepts indicate the days since the start of trading when the stock price was $0. These x-intercepts indicate that the price of the stock was $0 approximately days and days since the start of trading. f. The function is increasing over the intervals [0, 3.94) and (19.76, 38.65). The function is decreasing over the intervals (3.94, 19.76) and (38.65, 40]. The intervals over which the function is increasing represent the span of days during which the stock price was increasing. The intervals over which the function is decreasing represent the span of days during which the stock price was decreasing. g. The function has an absolute maximum at (38.65, 9.03). The stock had a maximum price of $9.03 approximately days since the stock started trading. h. Julio should not buy stock in Computer Dudes, Inc. The stock reached a maximum price of $9.03 approximately days since the stock started trading. After that point, the stock price decreases as the number of days increases. i. The function will not likely model the stock price beyond the 40th day since the stock started trading. The value of the function will continue to decrease as the number of days since the stock started trading increases beyond 40. Based on the shape of the model over the first 40 days, the price of the stock will probably continue to rise and fall over various intervals beyond the 40th day. TOP: Assignment 3