Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 1. f (x) = e x + 7 Evaluate the function for several x-values in its domain. x 7 5 4 3 y 1 7.4 20.1 54.6 Then use a smooth curve to connect each of these ordered pairs. Use the graph of f (x) to describe the transformation that results in the graph of g(x). Then sketch the graphs of f (x) and g(x). 3. ; This function is of the form. Therefore, the graph of g(x) is the graph of f (x) translated 3 units to the right and 4 units up. The translation to the right is indicated by the subtraction of 3 in the exponent. The translation up is indicated by the addition of 4. Use the graphs of the functions to confirm this transformation. List the domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. D = (, ); R = (, 0); y-intercept: e 7 ; asymptote: x-axis; decreasing for (, ) esolutions Manual - Powered by Cognero Page 1
5. MULTIPLE CHOICE For which function is A f (x) = 2 3 x B C f (x) = log 8 (x 5) D f (x) = log 3 ( x) 6 For option A, as x approaches infinity, 3 x approaches 0, so the function approaches 0. For Sketch the graph of each function. 11. g(x) = log ( x) + 5 Evaluate the function for several x-values in its domain. x 3 2 1 0 y 5.48 5.30 5 undef. Then use a smooth curve to connect each of these ordered pairs. option B, as x approaches infinity, approaches 0, so the function approaches 0. For option D, as x approaches infinity, log 3 ( x) is undefined. For option C, as x approaches infinity, log 8 (x 5) approaches infinity and log 8 (x 5) approaches negative infinity. The correct choice is C. Evaluate each expression. 7. log 32 2 Expand each expression. 13. log 6 36xy 2 15. GEOLOGY Richter scale magnitude of an earthquake can be calculated using where E is the energy produced 9. and E 0 is a constant. a. An earthquake with a magnitude of 7.1 hit San Francisco in 1989. Find the scale of an earthquake that produces 10 times the energy of the 1989 earthquake. b. In 1906, San Francisco had an earthquake registering 8.25. How many times as much energy did the 1906 earthquake produce as the 1989 earthquake? a. If E is the energy produced by the 1989 earthquake, then 10 times this energy is 10E. Use the 1989 data to solve for E 0, then replace E with 10E to find the magnitude of an earthquake with 10 esolutions Manual - Powered by Cognero Page 2
times the energy. Now, solve for R. The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake. Condense each expression. 17. 1 + ln 3 4 ln x The magnitude is about 7.8. b. For this problem, we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E. Let the 1906 earthquake = ne. Solve each equation. 19. e 2x 3e x + 2 = 0 x = ln 2 or 0 esolutions Manual - Powered by Cognero Page 3
21. log 2 (x 1) + 1 = log 2 (x + 3) 23. For Exercises 23 and 24, complete each step. a. Find an exponential or logarithmic function to model the data. b. Find the value of each model at x = 20. a. An exponential regression cannot be completed with negative y-values. Calculate the logarithmic regression. a. Linearize the data, assuming a quadratic model. Graph the data and write an equation for a line of best fit. b. Use the linear model to find a model for the original data. Is a quadratic model a good representation of population growth? Explain. a. What linearizing data according to a quadratic model, use (x, ). The rounded logarithmic regression equation is f (x) = 8.20 5.11ln x. Plot the data. b. Using the full regression equation, f (20) 7.11. Find the regression. 25. CENSUS The table gives the U.S. population between 1790 and 1940. Let 1780 = 0. The rounded regression equation is y = 0.07x. esolutions Manual - Powered by Cognero Page 4
b. Substitute for and solve for y. The rounded regression equation is y = 0.0041x 2 + 0.1331x + 1.0816. Population growth generally does not grow without bounds. Typically, it will start to level off and plateau. For these reasons, a logistic model would be more appropriate. esolutions Manual - Powered by Cognero Page 5