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

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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)=enter your response here. ​(Simplify your​ answer.)
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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 f that models the data, we can use the points from the years 2005 and 2010: (0, 5406) and (5, 7700).

We can use the slope-intercept form of a linear function, y = mx + b, where m is the slope and b is the y-intercept.

First, let's find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1)

Using the coordinates (0, 5406) and (5, 7700):
m = (7700 - 5406) / (5 - 0)
m = 2294 / 5
m = 458.8

Now that we have the slope, we can substitute one of the points into the slope-intercept form to find the y-intercept (b).

Using the point (0, 5406):
5406 = 458.8(0) + b
b = 5406

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

Note: Since the cost of tuition and fees cannot be negative, we can assume that the function is only valid for values of x that are non-negative.