Finding an Exponential Regression Use the data in the program file COOL to find an exponential model.

Finding an exponential model y = k + a*r x The data was collected from a hot water bath as it cooled to an ambient temperature of 27.7 degrees Celcius. One sample was taken every 60 seconds for an hour. CoCo Time sec Pressing STAT - EDIT, one can see the data is in seconds. For easier analysis, convert the seconds to minutes.

Finding an exponential model y = k + a*r x To convert time from seconds to minutes, divide by 60. Place the cursor on L1. Enter the formula L1/60 and press ENTER. To redraw the graph with the new coordinates, press ZOOMSTAT (Zoom - 9) The data will reflect the conversion and appear similar to the table at the right.

Finding an exponential model y = k + a*r x The graph will now be degrees Celcius vs Time in minutes. CoCo Time min

Finding an exponential model y = k + a*r x Solving Algebraically: Step 1 - Find k The exponential model y = k + a*r x will approach y = k as x approaches infinity if 0 < r < 1. Since the ambient room temperature was reported to be 27.7 o C, the water will approach room temperature and k = 27.7 o C

Finding an exponential model y = k + a*r x Solving Algebraically: Step 2 - Find a If x=0, the model y = k + a*r x is y = k + a since r 0 = 1. Tracing the graph where x=0, finds the y-value to be o Solving: = a a = o C

Finding an exponential model y = k + a*r x Solving Algebraically: Step 3 - Find r Choose another ordered pair (x, y) and solve for ‘r’. Trace to find another point on the graph to substitute.

Finding an exponential model y = k + a*r x Solving Algebraically: Step 3 - Find ‘r’ (con’t) Substituting (50, 43.92) and solving for ‘r’: o =27.7 o o * r o o o = r 50 ) 1 50 (43.92 o o ) o = r ( r =

Finding an exponential model y = k + a*r x Solving Algebraically: Step 4 - Graph Graph y = *.9740 x

Finding an exponential model y = k + a*r x Using a regression: Step 1 - Shift the data to approach zero The TI-83 exponential regression is in the form of y = a*b x which assumes the model will approach 0 as x approaches infinity. Since our data approaches 27.7, we must shift the data down so that it will approach 0. After the regression is complete, the model will be shifted back up the 27.7 degrees. Press STAT - EDIT and place the cursor on L3. Enter the formula L and press ENTER. L3 will be the temperature reduced to approach 0.

Finding an exponential model y = k + a*r x Using a regression: Step 1 - Shift the data to approach zero The data in L3 will appear as: Using a regression: Step 2 - Fit the regression. Press STAT - CALC and choose ExpReg.

Finding an exponential model y = k + a*r x Using a regression: Step 2 - Fit the regression. (con’t) The y-values are in L3. Set the regression for the x-values in L1 and the y- values in L3. The resulting equation was stored in Y2 The resulting regression must be shifted up by 27.7 degrees to fit the original data.

Finding an exponential model y = k + a*r x Using a regression: Step 3 - Graph Graphing the regression determined in Step 2, one finds the graph at the right. Using a regression: Step 4 - Write a mathematical model T o C = 27.7 o C o C *.9758 T min Using a regression: Step 4 - Explain the mathematical model The ambient temperature is 27.7 o. The initial temperature is o greater than the ambient temperature and retains 97.58% of its temperature each minute or it decreases its temperature 2.42% each minute.