The table lists the average annual cost (in dollars) of...

The table lists the average annual cost (in dollars) of tuition and fees at private 4 -year colleges for selected years. (a) Determine a function defined by $f(x)=m x+b$ that models the data, where $x=0$ represents $1994, x=1$ represents $1995,$ and so on. Use the points $(0,11,719)$ and $(12,22,218) .$ Graph $f$ and a scatter diagram of the data on the same coordinate axes. (You may wish to use a graphing calculator.) What does the slope of the graph of $f$ indicate? (b) Use this function to approximate tuition and fees in $2005 .$ Compare your approximation to the actual value of $\$ 20,980$ (c) Use the linear regression feature of a graphing calculator to find the equation of the line of best fit. (TABLE CANNOT COPY)

The table lists the average annual cost (in dollars) of tuition and fees at private 4 -year colleges for selected years.
(a) Determine a function defined by $f(x)=m x+b$ that models the data, where $x=0$ represents $1994, x=1$ represents $1995,$ and so on. Use the points $(0,11,719)$ and $(12,22,218) .$ Graph $f$ and a scatter diagram of the data on the same coordinate axes. (You may wish to use a graphing calculator.) What does the slope of the graph of $f$ indicate?
(b) Use this function to approximate tuition and fees in $2005 .$ Compare your approximation to the actual value of $\$ 20,980$
(c) Use the linear regression feature of a graphing calculator to find the equation of the line of best fit.
(TABLE CANNOT COPY)

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Key Concepts

Linear Function

A linear function is an algebraic expression representing a constant rate of change, typically written in the form f(x) = mx + b. It is used to model relationships where one variable changes at a steady rate with respect to another, making it ideal for approximating trends in data.

Slope (Rate of Change)

The slope of a linear function signifies the rate at which the dependent variable changes for each unit increase in the independent variable. In the context of modeling cost data, it indicates how much the tuition and fees increase per year.

Y-Intercept

The y-intercept in a linear equation is the value of the function when the independent variable is zero. This represents the initial condition or starting value, such as the tuition cost in the base year for the dataset.

Extrapolation

Extrapolation involves using a linear model to predict values outside the range of the observed data. It allows one to estimate future values based on the trend established by the available data.

Linear Regression

Linear regression is a statistical technique used to determine the line of best fit for a set of data points by minimizing the sum of the squared differences between observed and predicted values. It helps in understanding and quantifying the relationship between variables in real-world data.

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