How do we model data by using exponential and logarithmic functions? Curve Fitting with Exponential and Logarithmic Models Essential Questions How do we model data by using exponential and logarithmic functions? How do we use exponential and logarithmic models to analyze and predict? Holt McDougal Algebra 2 Holt Algebra 2

In previous chapters, you used a graphing calculator to perform linear progressions and quadratic regressions to make predictions. You can also use an exponential model, which is an exponential function that represents a real data set. Once you know that data are exponential, you can use ExpReg (exponential regression) on your calculator to find a function that fits. This method of using data to find an exponential model is called an exponential regression. The calculator fits exponential functions to abx, so translations cannot be modeled.

If you do not see r2 and r when you calculate regression, and turn these on by selecting DiagnosticOn. Remember!

Tuition of the University of Texas College Application Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. Tuition of the University of Texas Year Tuition 1999–00 $3128 2000–01 $3585 2001–02 $3776 2002–03 $3950 2003–04 $4188 Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature. An exponential model is f(x) ≈ 3236(1.07x), where f(x) represents the tuition and x is the number of years after the 1999–2000 year.

Tuition of the University of Texas College Application Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. Tuition of the University of Texas Year Tuition 1999–00 $3128 2000–01 $3585 2001–02 $3776 2002–03 $3950 2003–04 $4188 Step 2 Graph the data and the function model to verify that it fits the data. Graph the line y = 6000 on Y2. 7500 15 Zoom Out, then use 2nd Trace (Calc) and hit Intersect The tuition will be about $6000 when t = 9 or 2008–09.

College Application Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000? Time (min) 1 2 3 4 5 Bacteria 200 248 312 390 489 610 Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature. An exponential model is f(x) ≈ 199(1.25x), where f(x) represents the number of bacteria and x is the number of minutes.

Zoom Out, then use 2nd Trace (Calc) and hit Intersect College Application Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000? Time (min) 1 2 3 4 5 Bacteria 200 248 312 390 489 610 Step 2 Graph the data and the function model to verify that it fits the data. Graph the line y = 2000 on Y2. 2500 15 Zoom Out, then use 2nd Trace (Calc) and hit Intersect The bacteria will be about 2000 when x = 10.3 min.

Many natural phenomena can be modeled by natural log functions Many natural phenomena can be modeled by natural log functions. You can use a logarithmic regression to find a function Most calculators that perform logarithmic regression use ln rather than log. Helpful Hint

Global Population Growth Population (billions) Application Global Population Growth Population (billions) Year 1 1800 2 1927 3 1960 4 1974 5 1987 6 1999 Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000? Step 1 Enter data into two lists in a graphing calculator. Use the logarithmic regression feature. An logarithmic model is f(x) ≈ 1824 + 106ln x, where f(x) is the year and x is the population in billions.

Global Population Growth Population (billions) Application Global Population Growth Population (billions) Year 1 1800 2 1927 3 1960 4 1974 5 1987 6 1999 Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000? Step 2 Graph the data and the function model to verify that it fits the data. Because we are looking for the y-value when x = 9, you can use the table and scroll down to x = 9. 2500 15 The population will exceed 9,000,000,000 in the year 2057.

Application Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s? Time (min) 1 2 3 4 5 6 7 Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6 Step 1 Enter data into two lists in a graphing calculator. Use the logarithmic regression feature. An logarithmic model is f(x) ≈ 0.59 + 2.64 ln x, where f(x) is the time and x is the speed.

Zoom Out, then use 2nd Trace (Calc) and hit Intersect Application Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s? Time (min) 1 2 3 4 5 6 7 Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6 Step 2 Graph the data and the function model to verify that it fits the data. 10 20 Graph the line y = 8 on Y2. Zoom Out, then use 2nd Trace (Calc) and hit Intersect The time it will reach 8.0 m/sec is when x = 16.6 min.

Lesson 15.2 Practice B