The graph shows the sales (in billions of dollars) for...

The graph shows the sales (in billions of dollars) for Goodyear Tire for the years 1995 through 2004, where $t=5$ represents 1995. (a) Use the slopes to determine the years in which the sales for Goodyear Tire showed the greatest increase and the smallest increase. (b) Find the equation of the line between the years 1995 and 2004. (c) Interpret the meaning of the slope of the equation from part (b) in the context of the problem. (d) Use the equation from part (b) to estimate the sales for Goodyear Tire in the year 2010. Do you think this is an accurate estimation? Explain.

The graph shows the sales (in billions of dollars) for Goodyear Tire for the years 1995 through 2004, where $t=5$ represents 1995.
(a) Use the slopes to determine the years in which the sales
for Goodyear Tire showed the greatest increase and the smallest increase.
(b) Find the equation of the line between the years 1995 and 2004.
(c) Interpret the meaning of the slope of the equation from part (b) in the context of the problem.
(d) Use the equation from part (b) to estimate the sales for Goodyear Tire in the year 2010. Do you think this is an accurate estimation? Explain.

Key Concepts

Rate of Change

This concept involves determining how a quantity changes with respect to another variable. In the given context, it refers to the change in sales (in billions of dollars) over time (years), calculated as the difference in sales divided by the difference in years. The resulting slopes help identify periods with different rates of growth.

Linear Function

A linear function is a mathematical model represented by the equation y = mx + b, where m is the slope and b is the y-intercept. It is used to describe trends where changes occur at a constant rate. In this case, it models the trend in sales over a period of time, allowing for predictions and a better understanding of the overall behavior of the data.

Interpretation of Slope

The slope in a linear equation represents the rate of change of the dependent variable with respect to the independent variable. Here, it indicates how many billions of dollars in sales increase or decrease per year. This interpretation helps in understanding the magnitude and direction of the trend in sales.

Extrapolation

Extrapolation involves using a linear model to estimate values outside the range of the observed data. Although it can provide useful predictions, it assumes that the established trend continues unchanged beyond the available data, which may not always be the case in real-life situations.

Graphical Analysis

Graphical analysis involves examining the plotted data to extract meaningful information such as identifying periods of significant growth or decline. By comparing slopes over different intervals in the graph, one can determine which periods experienced the greatest or smallest increases, offering insights into the business performance over time.

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