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.
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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)=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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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
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).
To find a linear function that models the data for the years 2005-2010, we can use the formula for the equation of a line: y = mx + b, where m is the slope and b is the y-intercept.
First, we need to find the slope (m) using the points (2005, 5406) and (2010, 7700). The change in y-values (5406 to 7700) is 7700 - 5406 = 2294, and the change in x-values (2005 to 2010) is 2010 - 2005 = 5. So the slope (m) is 2294 / 5 = 458.8.
Next, we need to find the y-intercept (b) by substituting the values of one of the points into the equation and solving for b. Let's use the point (2005, 5406):
y = mx + b
5406 = 458.8 * 0 + b
5406 = b
So the y-intercept (b) is 5406.
Therefore, the linear function (f) that models the data is f(x) = 458.8x + 5406.
Now, to estimate the cost of tuition and fees at public four-year colleges in 2012, we substitute x = 7 (since x = 0 represents 2005, and x = 7 represents 2012) 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.
To find a linear function that models the data using the points from 2005 and 2010, we need to determine the slope and y-intercept of the line.
Step 1: Find the slope
The slope of a line can be found using the formula:
slope = (change in y) / (change in x)
In this case, the change in y is the difference in cost between 2010 and 2005, and the change in x is 5 (since we are going from 2005 to 2010).
Change in y = 7700 - 5406 = 2294
Change in x = 5
Slope = 2294 / 5 = 458.8
Step 2: Find the y-intercept
The y-intercept is the value of y when x = 0. In this case, we already have the data point for 2005 (x = 0) with a cost of 5406.
Y-intercept = 5406
Step 3: Write the linear function
Using the slope and y-intercept, the linear function can be written as follows:
f(x) = 458.8x + 5406
This is the linear function that models the data.
For part (b), we need to estimate the cost of tuition and fees at public four-year colleges in 2012 using the function from part (a).
To do this, we plug in x = 7 into the linear function:
f(7) = 458.8 * 7 + 5406 = 7860.6 + 5406 = 13267.6
Therefore, the cost of tuition and fees at public four-year colleges in 2012 is approximately $13,268.
(Rounded to the nearest dollar)