Alg2H Ch6: Investigating Exponential and Logarithmic Functions WK#14 Date:

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1 Alg2H Ch6: Investigating Exponential and Logarithmic Functions WK#14 Date: Purpose: To investigate the behavior of exponential and logarithmic functions Investigations For investigations 1 and 2, enter the function as Y 1 Complete Tables 1 and 2 using the ASK feature of the TABLE. 1. Given the exponential function: y = 2 x a. Complete Table 1: Directions to set table in Ask mode 1) Press 2 nd WINDOW to open TblSet 2) Arrow to Ask next to Indpnt and press Enter Now when you open the table, the calculator waits for you to enter a value for x. After doing so, press ENTER to set the corresponding output. x y = 2 x x y = 2 x b. As x, 2 x c. As x -, 2 x 2. Given the exponential function: y = (.5) x = (½) x a. Complete Table 1: x y = (0.5) x x y = (0.5) x b. As x, (.5) x c. As x -, (.5) x

2 3. Use the TI-83 calculator to graph y = 2 x and y = (0.5) x. Sketch each graph below. Graph of y = 2 x Graph of y = (0.5) x a. What is the domain and range of these functions in interval notation? Domain: Range: b. Which function is an increasing function and which is a decreasing function? Increasing Decreasing 4. Using the TI-83 calculator, investigate the tables and graphs associated with y = (1.1) x and y = (0.9) x a. What is the domain and range of these functions in interval notation? Domain: Range: b. Which function is an increasing function and which is a decreasing function? Increasing Decreasing 5. For the four functions graphed in investigations 3 and 4, what point do they all have in common?

3 6. Conclusions: Given the exponential function f(x) = b x, where b is a constant a. For any b such that b > 1, sketch the general shape and direction of the graph. Clearly label the y-intercept. (If necessary, use the y= key on the calculator to graph a few more graphs with b>1.) Write the domain of the function in interval notation: Write the range of the function in interval notation: b. For any b such that 0 < b < 1, sketch the general shape and direction of the graph. Clearly label the y-intercept. (If necessary, use the y= key on the calculator to graph a few more graphs with 0 < b < 1.) Write the domain of the function in interval notation: Write the range of the function in interval notation: c. What about other values of b? i. Set b = 1 and sketch the graph of f(x) = b x. What do you think? Is this the same type of function? ii. Try a value for b < 0. Use Tblset and change Tbl =.5 and Indpnt: AUTO Try to explain in your own words why the table reads as it does. Try graphing the problem. What happens? What restrictions do you think should be placed on the value of b so that f(x) = b x is a continuous function for all real values of x and maintain a general characteristic shape?

4 7. The function y = 4(1.1) x can be used to model the population of a city that is increasing at a rate of 10% per year. Assume that the population in 1995 was 4 hundred thousand. Let x represent the number of years since 1995 (x=0 on December 31, 1995) and y represent the population (measured in hundred thousands) after x years. a. What is the predicted population for the year 2000? Explain what you did. b. In what year will the population reach one million (10 hundred thousands)? Explain what you did. 8. The function y = 4(0.9) x can be used to model the population of a city that is decreasing at a rate of 10% per year. Assume that the population in 1995 was 4 hundred thousand. Let x represent the number of years since 1995 (x=0 on December 31, 1995) and y represent the population (measured in hundred thousands) after x years. a. What is the predicted population for the year 2000? Explain what you did. b. In what year will the population drop below one hundred thousand? Explain what you did.

5 9. Given the logarithmic function y = log 2 x a. Use the change of base property for logs first: Recall: log b a = log a which means log 2 x = log x log b log 2 You can use Y 1 = log x log 2 to complete Table 3 below. Table 3 x y = log 2 x x y = log 2 x x x x x10-16 b. As x, log 2 x c. As x 0, log 2 x d. Use the TI-83 calculator to graph the function. Sketch the graph here: e. Use the TI-83 calculator to graph together: Y 1 = log x and Y 2 = 2 x log 2 Sketch the results here: f. Compare the graphs and the tables (Table 1 and Table 3) of the 2 functions. Describe the relationship between them.