# 6.8 Fitting exponential models to data  (Page 6/12)

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[link] shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.

Year Seal Population (Thousands) Year Seal Population (Thousands)
1997 3.493 2005 19.590
1998 5.282 2006 21.955
1999 6.357 2007 22.862
2000 9.201 2008 23.869
2001 11.224 2009 24.243
2002 12.964 2010 24.344
2003 16.226 2011 24.919
2004 18.137 2012 25.108
1. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in years starting with $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ for the year 1997. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the number of seals in thousands. Use logistic regression to fit a model to these data.
2. Use the model to predict the seal population for the year 2020.
3. To the nearest whole number, what is the limiting value of this model?
1. The logistic regression model that fits these data is $\text{\hspace{0.17em}}y=\frac{25.65665979}{1+6.113686306{e}^{-0.3852149008x}}.$
2. If the population continues to grow at this rate, there will be about $\text{\hspace{0.17em}}\text{25,634}\text{\hspace{0.17em}}$ seals in 2020.
3. To the nearest whole number, the carrying capacity is 25,657.

Access this online resource for additional instruction and practice with exponential function models.

Visit this website for additional practice questions from Learningpod.

## Key concepts

• Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
• We use the command “ExpReg” on a graphing utility to fit function of the form $\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ to a set of data points. See [link] .
• 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 function of the form $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ to a set of data points. See [link] .
• Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
• We use the command “Logistic” on a graphing utility to fit a function of the form $\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}\text{\hspace{0.17em}}$ to a set of data points. See [link] .

## Verbal

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.

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