NAME: DATE: Algebra 2: Lesson 11-8 Exponential and Logarithmic Regression Learning Goals How do we write an equation that models an exponential or logarithmic function Warm Up Answer the following question to prepare for today s lesson: Show that the set of ordered pairs (x, y) in the table below satisfied a quadratic relationship. Find the equation of the form that all of the ordered pairs satisfy. Think Pair Share Using the warm-up to help you, attempt the following question. You will take 2 minutes to try this individually and then you will share your thinking with a partner. 1) A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial. Write an exponential regression equation, rounding the calculated values to the nearest tenthousandth.
LOCATION OF REGRESSIONS IN THE CALCULATOR These non-linear regressions are also found using the graphing calculator. All types of regressions on the calculator are prepared in a similar manner. Your regression options can be found under STAT CALC (scroll for more choices) Choose if you want to write an exponential regression equation. Choose if you want to write a logarithmic regression equation. EXPONENTIAL y = ab x LOGARITHMIC (Natural Log) y = a + blnx Practice Using the Calculator to Find a Model 2) An object at a temperaute of 160 0 C is removed from a furnace and placed in a room at 20 0 C. The table shows the temperatures d at selected times t (in hours) after the object was removed from the furnace. d 1 0.55 0.25 0.12 0.06 0.02 t 0 5 10 15 20 25 Write a logarithmic regression equation for this set of data, rounding coefficients to 3 decimal places.
Practice Using the Model to Estimate Values 3) Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Write the exponential regression equation for this set of data, rounding all values to two decimal places. Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase. Using this equation, also find how many years, to the nearest tenth, it will take for her investment to grow to reach $900 4) The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10ºF. Write the logarithmic regression equation for this set of data, rounding coefficients to the nearest ten thousandth. Based on your equation, if the wind chill factor is 0, what is the wind speed, to the nearest mile per hour?
Combining concepts 5) The following set of data shows U.S. gas prices in recent years. The table below represents that U.S. price of gas from 2005 to 2014, where t = 1 represents the year 2005. a) Based on the table what was the average rate of change in the price of gasoline from 2005 to 2014, to the nearest thousandth? b) What is the exponential regression for the data in the table, rounding coefficients to the nearest thousandth? Year Price ($) 1 1.78 2 2.24 3 2.33 4 3.11 5 1.68 6 2.67 7 3.07 8 3.29 9 3.29 10 3.33 c) Based upon your regression, what is the average rate of change in the price of gasoline from 2005 to 2014, to the nearest thousandth? d) Why is there a difference between your answers using the table and using the regression equation?
Extend your thinking 6) The price of a postage stamp in the years since the end of World War I is shown in the scatterplot below. The equation that best models the price, in cents, of a postage stamp based on these data is 1) 3) 2) 4) 7) Data for the number of users at two different social media companies is given below. Assuming an exponential growth rate, which company is adding users at a faster annual rate? Explain how you know. Social Media Company A Number of Years since Users 2009 (Millions) Social Media Company B Number of Years since Users 2009 (Millions)