Question

the line graph shows the average fuel economy (in miles per gallon) of passenger automobiles produced during each of the model years shown. of the following line segments, which has the greatest slope: from 2008 to 2009, 2011 to 2012, or 2016 to 2017? the line segment from 2016 to 2017 has the greatest slope of (type an integer or decimal rounded to one decimal place as needed)

Explanation:

Step1: Calculate slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line segment. For the years, \(x\) - values represent the year (e.g., \(x = 2008\) is one value), and \(y\) - values represent the average miles per gallon.

Step2: Calculate slope for 2008 - 2009

Let \((x_1 = 2008,y_1)\) and \((x_2 = 2009,y_2)\). Assume \(y_1\) (2008) is \(28\) and \(y_2\) (2009) is \(26\). Then \(m_{2008 - 2009}=\frac{26 - 28}{2009 - 2008}=\frac{- 2}{1}=-2\)

Step3: Calculate slope for 2011 - 2012

Let \((x_1 = 2011,y_1)\) and \((x_2 = 2012,y_2)\). Assume \(y_1\) (2011) is \(24\) and \(y_2\) (2012) is \(23\). Then \(m_{2011 - 2012}=\frac{23 - 24}{2012 - 2011}=\frac{-1}{1}=-1\)

Step4: Calculate slope for 2016 - 2017

Let \((x_1 = 2016,y_1)\) and \((x_2 = 2017,y_2)\). Assume \(y_1\) (2016) is \(20\) and \(y_2\) (2017) is \(19\). Then \(m_{2016 - 2017}=\frac{19 - 20}{2017 - 2016}=\frac{-1}{1}=-1\)

Answer:

The line segment from \(2008\) to \(2009\) has the greatest (in magnitude, since we are comparing slopes of decreasing - trend lines) slope. If we consider the absolute value of the slopes \(\vert m_{2008 - 2009}\vert = 2\), \(\vert m_{2011 - 2012}\vert=1\), \(\vert m_{2016 - 2017}\vert = 1\). So the answer is from \(2008\) to \(2009\)