4.8 Fitting exponential models to data  (Page 3/12)

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 Month 1 2 3 4 5 6 7 8 Debt ($) 620 761.88 899.8 1039.93 1270.63 1589.04 1851.31 2154.92 1. Use exponential regression to fit a model to these data. 2. If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating? 1. The exponential regression model that fits these data is $\text{\hspace{0.17em}}y=522.88585984{\left(1.19645256\right)}^{x}.$ 2. If spending continues at this rate, the graduate’s credit card debt will be$4,499.38 after one year.

Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?

No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).

Building a logarithmic model from data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.

Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis , we use the form of the logarithmic function most commonly used on graphing utilities, $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right).\text{\hspace{0.17em}}$ For this function

• All input values, $\text{\hspace{0.17em}}x,$ must be greater than zero.
• The point $\text{\hspace{0.17em}}\left(1,a\right)\text{\hspace{0.17em}}$ is on the graph of the model.
• If $\text{\hspace{0.17em}}b>0,$ the model is increasing. Growth increases rapidly at first and then steadily slows over time.
• If $\text{\hspace{0.17em}}b<0,$ the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.

Logarithmic regression

Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,

$y=a+b\mathrm{ln}\left(x\right)$

Note that

• all input values, $\text{\hspace{0.17em}}x,$ must be non-negative.
• when $\text{\hspace{0.17em}}b>0,$ the model is increasing.
• when $\text{\hspace{0.17em}}b<0,$ the model is decreasing.

Given a set of data, perform logarithmic regression using a graphing utility.

1. Use the STAT then EDIT menu to enter given data.
1. Clear any existing data from the lists.
2. List the input values in the L1 column.
3. List the output values in the L2 column.
2. Graph and observe a scatter plot of the data using the STATPLOT feature.
1. Use ZOOM [9] to adjust axes to fit the data.
2. Verify the data follow a logarithmic pattern.
3. Find the equation that models the data.
1. Select “LnReg” from the STAT then CALC menu.
2. Use the values returned for a and b to record the model, $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right).$
4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

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