The following table gives the sales, in billions of current...

The following table gives the sales, in billions of current dollars, for restaurants in the United States for selected years from 1970 to $2005 .$ (Source: National Restaurant Association Fact Sheet, 2005 ) $$\begin{aligned}&\text { Year } 1970 \quad 1985 \quad 1995 \quad 2005\\&\text { Sales } 42.8 \quad 173.7 \quad 295.7 \quad 475.8\end{aligned}$$ (a) Make a scatter plot of the data and find the exponential function of the form $f(x)=C a^{x}$ that best fits the data. Let $x$ be the number of years since 1970 (b) Why must $a$ be greater than 1 in your model? (c) Using your model, what are the projected sales for restaurants in the year $2008 ?$ (d) Do you think this model will be accurate over the long term? Explain.

The following table gives the sales, in billions of current dollars, for restaurants in the United States for selected years from 1970 to $2005 .$ (Source: National Restaurant Association Fact Sheet, 2005 )
$$\begin{aligned}&\text { Year } 1970 \quad 1985 \quad 1995 \quad 2005\\&\text { Sales } 42.8 \quad 173.7 \quad 295.7 \quad 475.8\end{aligned}$$
(a) Make a scatter plot of the data and find the exponential function of the form $f(x)=C a^{x}$ that best fits the data. Let $x$ be the number of years since 1970
(b) Why must $a$ be greater than 1 in your model?
(c) Using your model, what are the projected sales for restaurants in the year $2008 ?$
(d) Do you think this model will be accurate over the long term? Explain.

Key Concepts

Scatter Plot

A scatter plot is a graphical representation of data points where each point corresponds to a pair of values. In the context of modeling trends over time, it helps visualize the relationship between the independent variable (e.g., years since a baseline) and the dependent variable (e.g., sales) to assess the pattern and appropriateness of a chosen model.

Exponential Function Modeling

Exponential modeling involves finding a function of the form f(x) = C · a^x to approximate data that grows or decays by a constant percentage over equal increments of the independent variable. In this approach, C represents the initial value at the baseline, and a is the growth (or decay) factor that determines the rate of change.

Growth Factor a Greater Than 1

In an exponential model that describes growth, the parameter a must be greater than 1 because this indicates that the quantity increases with each time step. Conversely, if 0 < a < 1, the function would represent exponential decay, where the quantity decreases over time.

Model Extrapolation

Model extrapolation is the process of using an established function to predict or estimate values outside the range of the original data, such as forecasting future sales by substituting a value of x corresponding to a future year. It relies on the assumption that the established trend will continue beyond the observed data.

Model Limitations and Long-Term Accuracy

While an exponential model may fit historical data well, its accuracy over the long term can be limited. Real-world phenomena often experience constraints such as market saturation, resource limitations, or changes in behavior that cause deviations from the exponential trend, making long-term projections less reliable.

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