# Algebra 2

Name _________________________________ I.D. Number _______________________ Project 1 Evaluation 31 Second Year Algebra 2 (MTHH 040 059) Be sure to include ALL pages of this project (including the directions and the assignment) when you send the project to your teacher for grading. Don’t forget to put your name and I.D. number at the top of this page! This project will count for 8% of your overall grade for this course and contains a possible 100 points total. Be sure to read all the instructions and assemble all the necessary materials before you begin. You will need to print this document and complete it on paper. Feel free to attach extra pages if you need them. When you have completed this project you may submit it electronically through the online course management system by scanning the pages into either .pdf (Portable Document Format), or .doc (Microsoft Word document) format. If you scan your project as images, embed them in a Word document in .gif image format. Using .gif images that are smaller than 8 x 10 inches, or 600 x 800 pixels, will help ensure that the project is small enough to upload. Remember that a file that is larger than 5,000 K will NOT go through the online system. Make sure your pages are legible before you upload them. Check the instructions in the online course for more information. Part A – Fitting Exponential Curves to Data (possible 16 points) Cooling Tea Activity: You can use your graphing calculator to ﬁt an exponential curve to data and ﬁnd the exponential function. The table at the right shows the number of degrees above room temperature for a cup of tea after x minutes of cooling. Graph the data. Find the best-ﬁtting exponential function. Step 1 Press STAT ENTER on your calculator to enter the data in lists. Step 2 Set up STAT PLOT for Plot 1 to draw the scatter plot. Press 2ND Y=, select On, point graph, L1 for XList, and L2 for YList. Project 1 113 MTHH 040 Step 3 Find the equation for the best-fitting exponential function. Press STAT CALC Exp Reg. The line of x best fit is estimated by f(x) = 133.458(0.942) . Step 4 Graph f(x). Press Y= Clear VARS 5, arrow over twice and press ENTER to display the ExpReg results. Press GRAPH to display both the function and scatter plot together. To Zoom results, Press ZOOM 9. For additional information on exponential curves and line of best fit, please visit http://www.phschool.com/webcodes10/index.cfm?wcprefix=age&wcsuffix=2123&area=view and http://mathbits.com/MathBits/TISection/Statistics1/LineFit.htm. Use a graphing calculator to ﬁnd the exponential function that best ﬁts each set of data. Graph each function. Sketch your graph. 1. (3 pts) 2. (3 pts) 3. (3 pts) 4. (3 pts) Project 1 114 MTHH 040 5. In the Cooling Tea Activity example about at the beginning of this section, the function appears to level off. Explain why this happens. (1 pt) 6. a. Find a linear function that models the data in Question 3 from above. (2 pts) b. Which is a better ﬁt, the linear function or the exponential function? Explain. (1 pt) Part B – The pH Concept (possible 14 points) If you have taken Chemistry, you may recall molarity or molar concentration. Molarity is the number of moles of solute dissolved in 1 liter of solution. Expressing hydrogen-ion concentration [H+] in molarity is not practical, therefore another system called the pH scale is used. The pH scale ranges from 0 to 14. A solution with a pH less than 7 is acidic. A solution with a pH greater than 7 is basic. The pH of a solution is the negative logarithm of the hydrogen-ion concentration. In pure water or a neutral solution, the [H+] = 1 x 10 −7 M, and the pH is 7. pH = −log [H+] −7 = −log (1 x 10 ) = −(log 1 + (−7) log 10) = −(0 + (−7)) = 7 Activity: Use the information in the table to find the pH or hydrogen-ion concentration of each solution (or food item) listed. Tell whether the solution (or food item) is basic or acidic. Round to the nearest hundredth. Fill in the chart completely. (14 pts) Project 1 115 MTHH 040 Part C – Linear and Exponential Models (possible 15 points) You can transform an exponential function into a linear function by taking the logarithm of each side. Since linear models are easy to recognize, you can then determine whether an exponential function is a good model for a set of values. y = ab x Write the general form of an exponential function. x log y = log ab Take the logarithm of each side. log y = log a + x log b Product property and power property. log y = (log b)x + log a Rewrite. If log b and log a are constants, then log y = (log b)x + log a is a linear equation in slope-intercept form. To conﬁrm that log b is a constant, check that the graph of log y = (log b)x + log a is a line. Activity: Determine whether an exponential function is a good model for the values in the table. Step 1 Enter the values into STAT lists L1 and L2. To enter the values for log y, place cursor in the heading of L3 and press LOG 2ND 2 ENTER. Step 2 To graph log y, set-up STAT PLOT feature and press 1. Then enter L3 next to YList. To Zoom results, Press ZOOM 9. Since the graph of log y = (log b)x + log a is linear, the slope log b is constant, and b also is constant. An exponential function therefore is a suitable model. Step 3 x Press STAT arrow over right 0 ENTER to find the exponential function; y = 3(1.3) . For each set of values, determine whether an exponential function is a good model. If so, find the function. If not, explain why. 1. Project 1 (2 pts) 116 MTHH 040 2. (2 pts) 3. (2 pts) 4. (2 pts) 5. (2 pts) 6. The value of a car depreciates over time. The table shows the value of a car over a 4 year period. (5 pts) a. Determine which kind of function best models the data. b. Write an equation for the function that models the data. c. Use your equation to determine how much the car is worth after 7 years. Project 1 117 MTHH 040 Part D – Exponential and Logarithmic Inequalities (possible 20 points) Legend tells of a Roman general who was so successful in his campaigns that the emperor offered the general his choice of reward. The general asked the emperor for all the silver that he could carry out of the imperial treasury in one month (31 days). The emperor agreed, but offered two conditions. On the ﬁrst day, the general would receive 1 silver denarius. On the second day, he would turn in the previous day’s coin and receive a coin valued at 2 denarii. Each day, the general was to come to the treasury turn in the previous day’s coin and the treasurer would mint a special coin that would be twice as heavy and worth twice as much as the coin the general got a day earlier. One denarius coin weighed about 0.006 kg. The largest object the general could carry or roll without help could weigh no more than 300 kg. Activity 1: 1. a. What function V(x) gives the value of the coin that the general received on day x? (2 pts) b. What is the domain of this function? (1 pt) 2. Write a function M(x) that gives the mass of each coin in kilograms as a function of day x. (2 pts) 3. a. Use M(x) from Question 2 to write an inequality that describes the mass of a coin that the general could carry or roll out of the treasury. (2 pts) b. Make a table of values of this function. (3 pts) c. Use the table to solve the inequality. (2 pts) 4. What is the total value of the coins that the general would receive? (1 pt) Project 1 118 MTHH 040 Activity 2: Scientists are growing bacteria in a laboratory. They start with a known population of bacteria and measure how long it takes this population to double. 5. Sample A starts with 200,000 bacteria. The population doubles every hour. Write an exponential function that models the population in Sample A as a function of time in hours. (1 pt) 6. Sample B starts with 50,000 bacteria. The population doubles every half hour. Write an exponential function that models the population growth in Sample B as a function of time in hours. (2 pts) 7. a. Write an inequality that models the population in Sample B overtaking the population in Sample A. (2 pts) b. Use a graphing calculator to solve the inequality. (2 pts) Part E – Rational Inequalities (possible 35 points) When you solve rational inequalities you can’t necessarily solve them exactly like rational equations. If you multiply both sides of a rational inequality by the same algebraic expression just as you have done with equations, you can introduce extraneous solutions or lose solutions! For additional information on rational inequalities and how to solve rational inequalities by graphing. Use your Internet browser and please visit: https://www.khanacademy.org/math/algebra-home/algrational-expr-eq-func/alg-rational-inequalities/v/rational-inequalities and https://www.youtube.com/watch?v=wbCC51lHQig. Activity 1: 1. Here is Neely’s solution of the rational inequality Project 1 119 N < 3. Read it carefully. 4N MTHH 040 N

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