Question

the table lists average annual cost (in dollars) of tuition and fees at private four-year colleges for selected years.
(a) determine a linear function f(x) = ax + b that models the data, where x = 0 represents 2013, x = 1 represents 2014, and so on. use the points (0, 24552) and (4, 29468) to graph f and a scatter diagram of the data on the same coordinate axes. what does the slope of the graph indicate?
(b) use the function from part (a) to approximate average tuition and fees, to the nearest dollar, in 2016. compare the approximation to the actual figure given in the table, $27,958.
(c) use the linear regression feature of a graphing calculator to find the equation of the line of best fit.

(a) what is the linear model for the data?
f(x) = \square
(type your answer in slope-intercept form. use integers or decimals for any numbers in the expression. round to the nearest hundredth as needed.)

year | cost (in dollars)
2013 | 24,552
2014 | 25,717
2015 | 26,754
2016 | 27,958
2017 | 29,468

Explanation:

Step1: Find the slope \(a\)

The slope formula is \(a=\frac{y_2 - y_1}{x_2 - x_1}\). Using points \((0, 24552)\) and \((4, 29468)\), we have \(x_1 = 0\), \(y_1 = 24552\), \(x_2 = 4\), \(y_2 = 29468\).
\[a=\frac{29468 - 24552}{4 - 0}=\frac{4916}{4}=1229\]

Step2: Find the y - intercept \(b\)

Since the line passes through \((0, 24552)\), when \(x = 0\), \(f(x)=b\). So \(b = 24552\).

Step3: Write the linear function

The linear function is \(f(x)=ax + b\), substituting \(a = 1229\) and \(b = 24552\), we get \(f(x)=1229x+24552\).

Answer:

\(f(x)=1229x + 24552\)