Problem 67 - A Graphical Approach to Precalculus with Limits

Question

The table lists the worldwide advertising revenue of Google (in billions of dollars). $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2012 & 2014 & 2016 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { ( 5 billions) } \end{array} & 28 & 44 & 60 & 79 \\ \hline \end{array}$$ (a) Find the point-slope form of the line that passes through the points \((2010,28)\) and \((2016,79) .\) Let \(\left(x_{1}, y_{1}\right)\) be \((2010,28)\) (b) Interpret the slope of the line. (c) Use this equation to estimate the revenue in 2012 and \(2014 .\) Compare these estimates with the actual values shown in the table.

Step-by-Step Solution

Answer

(a) \(y - 28 = 8.5(x - 2010)\). (b) Slope is 8.5 billion/year. (c) 2012 estimate: 45 billion (actual 44), 2014 estimate: 62 billion (actual 60).

Step 1: Find the Slope

We start by calculating the slope of the line that passes through the points \((2010, 28)\) and \((2016, 79)\). The formula for the slope \(m\) is \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. With \((x_1, y_1) = (2010, 28)\) and \((x_2, y_2) = (2016, 79)\), the calculation is \[m = \frac{79 - 28}{2016 - 2010} = \frac{51}{6} = 8.5\].

Step 2: Write Point-Slope Form

Using the point-slope form equation \(y - y_1 = m(x - x_1)\), and our point \((2010, 28)\) with slope \(m = 8.5\), we write \[y - 28 = 8.5(x - 2010)\].

Step 3: Interpret the Slope

The slope of the line, \(8.5\), represents the rate of change of Google's advertising revenue per year, indicating that the revenue increases by 8.5 billion dollars each year.

Step 4: Estimate Revenue in 2012

To estimate the revenue for 2012, substitute \(x = 2012\) into the equation: \(y - 28 = 8.5(2012 - 2010)\). Calculate \[y - 28 = 8.5 \times 2 = 17\], so \[y = 28 + 17 = 45\].

Step 5: Compare 2012 Estimate with Actual

The estimated revenue for 2012 is 45 billion dollars. The actual revenue from the table is 44 billion dollars, indicating our estimate is 1 billion too high.

Step 6: Estimate Revenue in 2014

Substitute \(x = 2014\) into the equation: \(y - 28 = 8.5(2014 - 2010)\). Calculate \[y - 28 = 8.5 \times 4 = 34\], so \[y = 28 + 34 = 62\].

Step 7: Compare 2014 Estimate with Actual

The estimated revenue for 2014 is 62 billion dollars. The actual revenue from the table is 60 billion dollars, indicating our estimate is 2 billion too high.

Key Concepts

Point-Slope Form

Understanding the point-slope form is crucial in linear regression because it helps describe a line's equation when given a point and a slope. The point-slope form is expressed as \(y - y_1 = m(x - x_1)\). This equation is handy when you are given a point, \((x_1, y_1)\), and a slope, \(m\), and wish to describe all other points \((x, y)\) on the line.

This method is especially useful in problems related to estimating and predicting trends over time because it allows you to plug in values for \(x\) (like a specific year) and determine the corresponding \(y\) value (like revenue) easily. In our exercise, the point-slope form \(y - 28 = 8.5(x - 2010)\) showcases how you can express Google's revenue growth starting from 2010. Here, the starting point or initial value is 2010 with a revenue of 28 billion dollars. The slope 8.5 indicates how the revenue changes per year.

Slope Interpretation

Interpreting the slope in any linear regression context is about understanding the rate of change. In general terms, the slope is the "rise over run" or the change in \(y\) for a unit change in \(x\).

In this exercise, the slope is 8.5, representing that Google's worldwide advertising revenue increases at a rate of 8.5 billion dollars per year. This is calculated between two points, \((2010, 28)\) and \((2016, 79)\), using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Understanding this concept can be useful for business forecasts because it models how fast or slow a company's revenue or other metrics might change over time, helping decision-makers to plan effectively.

Revenue Estimation

Revenue estimation leverages the equation derived from the linear regression model to predict revenues for years that are not explicitly given in the data. This is done by simply substituting the year into the point-slope equation.

For instance, to estimate the revenue in 2012, you substitute \(x = 2012\) in the point-slope equation \(y - 28 = 8.5(x - 2010)\). Solving this gives \(y = 45\), which is close to the actual value of 44 billion dollars, showing just a slight overestimation.

Similarly, substituting \(x = 2014\) gives \(y = 62\), which again slightly overestimates the actual figure of 60 billion dollars by 2 billion. Despite these minor discrepancies, revenue estimation through linear regression is a powerful tool for forecasting and budget planning. It can help stakeholders anticipate future results and make informed decisions based on data-driven insights.