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This table shows the population of a city every 10 years since 1970.  Write the best-fit quadratic equation model for this data.  Round to the nearest thousandth for the equation.  Use this model to estimate the population in 2020, to the nearest tenth.

To model this data with a quadratic equation, we can represent it in the form:

\[
y = ax^2 + bx + c
\]

where \(x\) is the number of years since 1970, and \(y\) is the population (in thousands). Given the table, we can use the following points:

\[
(0, 489), (10, 801), (20, 1202), (30, 1998), (40, 2959)
\]

Let's proceed by setting up a system of equations to solve for \(a\), \(b\), and \(c\). Then, we'll use the quadratic equation to estimate the population in 2020, which corresponds to \(x = 50\). 

I'll calculate this now.The best-fit quadratic equation for the data is:

\[
y = 1.209x^2 + 12.999x + 504.257
\]

Using this model, the estimated population in 2020 (which corresponds to \(x = 50\)) is approximately 4177.4 thousand, or 4,177,400 people.

Would you like further details on this process or any part of the equation?

Here are some additional questions to consider:
1. What would the population estimate be for a year between the given intervals, such as 1985?
2. How does a quadratic model compare to other models (e.g., linear) for this dataset?
3. Could the population growth continue to follow this model after 2020?
4. What is the significance of the coefficient values in the quadratic equation?
5. How can the accuracy of this model be tested using other population data?

**Tip:** A quadratic model is particularly useful when the rate of change (growth) itself is increasing or decreasing over time.

This table shows the population of a city every 10 years since 1970. Write the best-fit quadratic equation model for this data. Round to the nearest thousandth for the equation. Use this model to estimate the population in 2020, to the nearest tenth.

Learn how to derive a quadratic equation to model the population of a city from 1970 to 2010, and estimate the population in 2020 using the model. The best-fit equation is calculated using data points, and population estimates for future years are obtained.

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