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

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 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)=enter your response here. (Simplify your answer.)
.
.
.
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.

Answer 2

To find a linear function f that models the data for the years 2005 and 2010, we need to find the equation of the line that passes through the points (x=0, y=5406) and (x=5, y=7700).

Using the formula for the equation of a line (y = mx + b), where m is the slope and b is the y-intercept, we can calculate the slope:

slope (m) = (y2 - y1) / (x2 - x1)
= (7700 - 5406) / (5 - 0)
= 2294 / 5
= 458.8

Now we can substitute one of the points (using the first point) and the value of the slope into the equation:

y = mx + b
5406 = (458.8 * 0) + b
b = 5406

Therefore, the linear function f that models the data is:

f(x) = 458.8x + 5406.

Answer 3

To find a linear function that models the data, we need to find the equation for a line that passes through the points (2005, 5406) and (2010, 7700).

First, let's find the slope of the line. The formula for slope is:

slope = (change in y) / (change in x)

Using the two given points, we can calculate the change in y and change in x:

change in y = 7700 - 5406 = 2294
change in x = 2010 - 2005 = 5

So, the slope is given by:

slope = 2294 / 5 = 458.8

Now, let's find the y-intercept. The y-intercept is the value of y when x = 0. To find this, we can use one of the points given in the table. Let's use the point (2005, 5406).

The equation for a line in slope-intercept form is:

y = mx + b

where m is the slope and b is the y-intercept.

Substituting the values:
5406 = (458.8)(2005) + b

Now, solve for b:
b = 5406 - (458.8)(2005)
b ≈ -647932

So, the y-intercept is approximately -647932.

Now that we have the slope and the y-intercept, we can write the equation for the linear function f:

f(x) = 458.8x - 647932

Therefore, the linear function f that models the data is:

f(x) ≈ 458.8x - 647932.