Problem 68 - A Graphical Approach to College Algebra

Question

The table lists U.S. print newspaper advertising revenue (in billions of dollars). $$\begin{array}{|l|c|c|c|c|}\hline \text { Year } & 2006 & 2008 & 2010 & 2012 \\\\\hline \begin{array}{l}\text { Revenue } \\\\\text { (\$billions) }\end{array} & 48 & 35 & 22 & 10 \\\\\hline\end{array}$$ (a) Find the point-slope form of the line that passes though \((2006,48)\) and \((2010,22) .\) Let \(\left(x_{1}, y_{1}\right)\) be \((2006,48)\) (b) Find the point-slope form of the line that passes though \((2008,35)\) and \((2012,10) .\) Let \(\left(x_{1}, y_{1}\right)\) be \((2008,35)\) (c) Interpret the slope of the line from part (b). (d) Use equations from parts (a) and (b) to predict the revenue for 2009

Step-by-Step Solution

Answer

In both cases, the predicted revenue for 2009 is approximately $28-29 billion.

Step 1: Calculate the slope for part (a)

Given two points: \((2006,48)\) and \((2010,22)\). The formula for the slope \(m\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{22 - 48}{2010 - 2006} = \frac{-26}{4} = -6.5 \]

Step 2: Write the point-slope form for part (a)

Using the point-slope formula \(y - y_1 = m(x - x_1)\), with \(m = -6.5\), \((x_1, y_1) = (2006, 48)\):\[ y - 48 = -6.5(x - 2006) \]

Step 3: Calculate the slope for part (b)

Given two points: \((2008,35)\) and \((2012,10)\). Using the slope formula:\[ m = \frac{10 - 35}{2012 - 2008} = \frac{-25}{4} = -6.25 \]

Step 4: Write the point-slope form for part (b)

Using the point-slope formula with \(m = -6.25\), \((x_1, y_1) = (2008, 35)\):\[ y - 35 = -6.25(x - 2008) \]

Step 5: Interpret the slope from part (b)

The slope \(-6.25\) indicates that for each year from 2008 to 2012, the revenue decreases by \$6.25 billion.

Step 6: Predict revenue for 2009 using part (a)

Substitute \(x = 2009\) into the point-slope form from part (a): \[ y - 48 = -6.5(2009 - 2006) \]\[ y - 48 = -6.5 \times 3 = -19.5 \]\[ y = 48 - 19.5 = 28.5 \] billion dollars

Step 7: Predict revenue for 2009 using part (b)

Substitute \(x = 2009\) into the point-slope form from part (b): \[ y - 35 = -6.25(2009 - 2008) \]\[ y - 35 = -6.25 \times 1 = -6.25 \]\[ y = 35 - 6.25 = 28.75 \] billion dollars

Key Concepts

Slope Calculation

Calculating the slope between two points is an essential part of determining how much a quantity changes over time. In any context, including economics like in our exercise, understanding this rate of change can inform predictions and decision-making. To calculate the slope, you need two points from the dataset. Let's use the points

• \((x_1, y_1) = (2006, 48)\)
• \((x_2, y_2) = (2010, 22)\)

With these, apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].Inserting our values gives: \[ m = \frac{22 - 48}{2010 - 2006} = \frac{-26}{4} = -6.5 \].
This slope of \(-6.5\) indicates that U.S. print newspaper advertising revenue decreased by \$6.5 billion per year between 2006 and 2010.

Linear Equation

A linear equation in the point-slope form connects two variables with a constant rate of change represented by the slope. This form is especially useful for creating linear models when you have a data point and a slope. The point-slope form of a linear equation is: \[ y - y_1 = m(x - x_1) \].
Here,

• \(y\) and \(x\) are variables in the equation,
• \(y_1\) and \(x_1\) are the coordinates of a point the line passes through, and
• \(m\) is the slope.

Using

• the point \((2006, 48)\) and the slope \(-6.5\),

the equation becomes \[ y - 48 = -6.5(x - 2006) \] for the year 2006 to 2010. This enables calculation for any value within this range, illustrating how revenue changes over time.

Interpretation of Slope

Interpreting the slope is about understanding what it tells us concerning the relationship between variables. In our specific case, the slope calculated from the points

• \((2008, 35)\)
• \((2012, 10)\)

was \(-6.25\). This negative value reveals a decreasing trend, meaning that every year between 2008 and 2012, the revenue from U.S. print newspaper advertisement declined by about \$6.25 billion.
This trend not only reflects diminishing advertising spending in the print sector but also indicates possible shifts towards alternative, maybe more viable, platforms like digital media. Recognizing this change highlights the importance of adapting to technological and societal trends to sustain market viability.