Module 3  Functions and Transformations 
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest 
Lesson 3.4: Cooling Liquids 
Lesson 3.1 and Lesson 3.2 discussed transformations and Lesson 3.3 explored finding the solutions of transformed exponential functions. This lesson will investigate finding the equation that models a set of data. The Cooling Problem A cup of hot water was left on a kitchen table to cool. The initial temperature of the water was 77.8°C and the room temperature in the kitchen was 23°C. The following table shows data for the water as it cooled. Enter the data into lists.
The lists are accessed by pressing and selecting 1:Edit. Display the scatter plot of the data.
The scatter plot looks like an exponential decay graph with one important difference: the horizontal asymptote for the scatter plot appears to be about y = 23 while the horizontal asymptote for an exponential decay curve is y = 0. If the scatter plot is shifted down 23, the resulting scatter plot appears to approximate an exponential decay curve. You may calculate an exponential regression equation for the transformed scatter plot and then shift the equation up 23 units to fit the original scatter plot. Shifting the Scatter Plot Down Shift the scatter plot down 23 units by subtracting 23 from each L_{2}value.
Notice as you type that the expression appears in the edit line at the bottom of the screen.
Every element of L_{2} is reduced by 23 and the corresponding value stored in L_{3}. Display the transformed scatter plot with the original scatter plot.
Calculate the exponential regression equation for the transformed scatter plot.
The equation is calculated, displayed on the Home screen and stored in Y_{1}. The equation that fits the transformed data is approximately y = 49.3(0.948)^{x}. Shift this equation up 23 to fit the original data by adding 23: y = 49.3(0.948)^{x} + 23.
Press to display both equations along with the scatter plots. 3.4.1 Estimate the time needed for the water to cool to 30°C using the table of values of the transformed regression equation. Click here for the answer. 
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