Question

writing the equation of a trend line
starting salaries
a researcher surveyed 8 people to see if there is a relationship between years of education and starting salaries. the data points are shown on the graph.
which best represents the equation of the trend line shown on the graph? (note that the graph has a break on the x - axis.)
$y = 1.25x - 10$
$y = 0.25x + 15$
$y = 1.25x + 7.5$
$y = 0.25x + 17.5$

Explanation:

Step1: Recall the slope - intercept form

The equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Estimate the \(y\) - intercept

Looking at the graph, when \(x = 8\), \(y\approx16\).

Step3: Calculate the slope

Take two points on the trend line. Let's assume \((x_1,y_1)=(8,16)\) and \((x_2,y_2)=(16,26)\)
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{26 - 16}{16 - 8}=\frac{10}{8}=1.25\)

Step4: Find the \(y\) - intercept \(b\)

Using \(y = mx + b\) and \(m = 1.25\), \(x = 8\), \(y = 16\)
\(16=1.25\times8 + b\)
\(16 = 10 + b\)
\(b=6\). The closest \(b\) value among the options to our estimate (considering graph reading approximations) is \(b = 7.5\) in the equation \(y = 1.25x+7.5\)

Answer:

\(y = 1.25x + 7.5\)