Average annual tuition and fees for​ in-state students at public​ four-year colleges in a given region are shown in the table for selected years and graphed as ordered pairs of​ points, where x=0 represents​ 2005, x=1 represents​ 2006, and so​ on, and​ f(x) represents the cost in dollars. Answer parts ​(a) and​ (b) below.

Question

Question content area bottom left
Part 1
​(a) Use the points from the years 2005 and 2010 to find a linear function f that models the data.
A linear function f that models the data is ​f(x)=458.8 x plus 5406. 458.8x + 5406.. ​(Simplify your​ answer.)
Part 2
​(b) Use the function from part​ (a) to estimate the cost of tuition and fees at public​ four-year colleges in 2012.
Tuition and fees will cost about ​$enter your response here.
​(Round to the nearest dollar as​ needed.)
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Question content area right
Part 1
Tuition and Fees
Year
Cost​ (in dollars)
2005
5406
2006
5805
2007
6211
2008
6650
2009
7167
2010
7700
0
2
4
6
8
10
1000
2000
3000
4000
5000
6000
7000
8000
9000
10,000
0
Year
Cost (in dollars)
y
x

Answer 1

To find the linear function that models the data, we will use the points from the years 2005 (x=0) and 2010 (x=5).

Using the formula for a linear function, which is y = mx + b, where m is the slope and b is the y-intercept, we can substitute the points (0, 5406) and (5, 7700) into the equation and solve for m and b.

Using the point (0, 5406):
5406 = m(0) + b
5406 = b

Using the point (5, 7700):
7700 = m(5) + 5406
7700 = 5m + 5406
2294 = 5m
m = 458.8

So the linear function that models the data is f(x) = 458.8x + 5406.

To estimate the cost of tuition and fees at public four-year colleges in 2012 (x=7), we can substitute x=7 into the linear function:

f(7) = 458.8(7) + 5406
f(7) = 3211.6 + 5406
f(7) = 8617.6

Therefore, the cost of tuition and fees at public four-year colleges in 2012 will be about $8,618 (rounded to the nearest dollar).