# College Costs Project

## Equations

Linear Equation (ax+b): Blue

f(x) = (646.227272x + 31197.2272)

R^2: 0.98044305

Exponential Equation (a (b^x)): Red

f(x) = (31289.0491x (1.01894136^x))

R^2: 0.98376952

Cubic Equation (ax^3 + bx^2 + cx + d): Green

f(x) = (5.10003884x^3 -67.723193x^2 + 850.175602x + 31145.2867)

R^2: 0.98528335

## Predictions

1990-1991

Linear: When the year is in 1990, the tuition using the linear function is \$24734.95455.

Exponential: When the year is in 1990, the tuition using the exponential function is \$25935.84791.

Cubic: When the year is in 1990, the tuition using the cubic function is \$10771.17249.

2014-2015:

Linear: When the year is in 2014, the tuition using the linear function is \$40244.40909

Exponential: When the year is in 2014, the tuition using the exponential function is \$40689.3775.

Cubic: When the year is in 2014, the tuition using the cubic function is \$43768.50583.

2018-2019:

Linear: When the year is in 2018, the tuition using the linear function is \$42829.31818.

Exponential: When the year is in 2018, the tuition using the exponential function is \$43860.92753.

Cubic: When the year is in 2018, the tuition using the cubic function is \$54249.55944.

2040-2041:

Linear: When the year is in 2040, the tuition using the linear function is \$57046.31818.

Exponential: When the year is in 2040, the tuition using the exponential function is \$66276.56987.

Cubic: When the year is in 2040, the tuition using the cubic function is \$283197.6876.

## Actual Tuition

1990-1991: Actual tuition for Yale University is \$14,000.

2014-2015: Actual tuition for Yale University is \$44,800.

## Function Analysis

Which function best models the cost of tuition for Yale? Why? Justify mathematically.

The cubic function was the best to use in both years of 1990-1991 and through 2014-2015, as well as best modeling the cost of tuition for Yale, because not only was the r^2 value the closest to 1, but there was also the smallest percent error between the theoretical calculated tuition using the functions and the actual college tuition rate for Yale when you actually researched the real value.

Was one model better consistently? Why?

Yes, unlike the linear model which was an inaccurate representation of the college costs (they assume all college tuition will go up on a constant slope, which this project disproves), the exponential equation will be better consistently. The cubic could potentially be a good model, but since the point of inflection decreases and has the tuition stay constant for a bit, it cannot be an accurate representation of the college tuition, because Yale's tuition only ever has gone up, never decreased. Therefore, the exponential equation is better consistently because it doesn't inflect, nor is it an unrealistic straight line. It has a shape to it that best models the pattern of the tuition costs.

## Yale University 1964

Linear:

f(x) = 599.636652x + 31438.6513

R^2: 0.99887781

Exponential:

f(x) = 25944.8835 (1.05644406x)

R^2: 0.98303017

Cubic:

f(x) = 0.15575647x^3 + 7.08997255x^2 + 560.781011x + 31333.4932

R^2: 0.99918969

This changed little regarding the R^2 values, and while the equations changed, the graph for them remained around the same; the equation was just changed to accomodate for the new coordinate of (-50, 1550) because if x = 0 in 2000, then x = -50 in 1964.

## Equations

Linear Equation (ax+b): Blue

f(x) = 438.373x + 3851.318

R^2: 0.9753002

Cubic Equation(ax^3 + bx^2 + cx + d):Green

f(x) = -1.355 x^3 + 20.212 x^2 + 361.968 x + 3898.462

R^2: 0.97582374

Exponential Equation ( P * (a)^x):Red

f(x) = (4038.089 * (1.078^x))

R^2: 0.96455301

## Predictions

1990-1991

Linear: When the year is in 1990, the tuition using the linear function is \$-532.41 (which is not a possible tuition).

Exponential: When the year is in 1990, the tuition using the exponential function is \$1908.23 .

Cubic: When the year is in 1990, the tuition using the cubic function is \$3654.70 .

2014-2015:

Linear: When the year is in 2014, the tuition using the linear function is \$9988.54

Exponential: When the year is in 2014, the tuition using the exponential function is \$11552.91

Cubic: When the year is in 2014, the tuition using the cubic function is \$9210.29.

2018-2019:

Linear: When the year is in 2018, the tuition using the linear function is \$11742.03

Exponential: When the year is in 2018, the tuition using the exponential function is \$15,565.28

Cubic: When the year is in 2018, the tuition using the cubic function is \$9061.99

2040-2041:

Linear: When the year is in 2040, the tuition using the linear function is \$21,386.23

Exponential: When the year is in 2040, the tuition using the exponential function is \$80975.96

Cubic: When the year is in 2040, the tuition using the cubic function is \$35984.29

## Actual Tuitions

1990-1991: The actual tuition for UTSA was \$3,400.

2014-2015: The actual tuition for UTSA is \$8,700.

## Percent Error in Tuitions

1990-1991

Linear: 738.634%

Cubic: -6.927%

Exponential: 78.136%

2014-2015

Linear: -12.922%

Cubic: -5.518%

Exponential: -29.877%

## Function Anlaysis

Which function best models the cost of tuition for each school? Why? Justify mathematically.

The cubic function best modeled the cost of tuition for each school because it had the highest r^2 value. After running a regression for each equation, a r^2 value is obtained which depicts the accuracy of the graph. The cubic equation had the highest r^2 value at 0.978.

The exponential model was better consistently and this was expected because on the exponential graph, the y-values, or the college tuition, continuously increased which contrasts with the cubic graph which eventually, at its point of inflection, begins to decrease. The exponential model is also better because it realistically portrays the increase of college tuition, unlike the linear model which depicts college tuition as increasing at a constant slope.

## UTSA 1969

Linear:

f(x) = 327.698x + 3746.45

R^2: 0.975

Exponential:

f(x) = 3856* (1.028^x)

R^2: 0.978

Cubic:

f(x) = -1.574x^3 + 18.067x^2 + 340.984x + 3764.211

R^2: 0.986

The change in r^2 values between the regressions of the old and the new equations is very insignificant. Though the slopes of the equations slightly changed, the graphs appear to look mostly similar. The only purpose of the new equations was to accommodate for the new coordinate of (-45,355) because if x = 0 in 2000, then x = -50 in 1969.

## Reflection

This project has significantly influenced our college search as it has portrayed the considerable difference between the cost of tuition between a private, Ivy League school and a public school within Texas. This project has also aided us in understanding the gradual but continuous increase of college tuitions across America. We enjoyed this project because it allowed us to utilize various different math skills from Pre-Calculus, and apply them to a real life situation that all teenagers can find relevant. We also liked that we could choose the colleges we wanted to research and therefore we gained information about universities that we have interest in. We both worked very hard on this project, as our completion of this project spanned over the course of three days, due to the analysis that this project required of us. In 40 weeks, we believe that we will remember the college tuitions of Yale and UTSA when we are considering what universities to apply to. In 40 months, we will remember how to utilize linear, exponential, and cubic equations in real life applications, when we are taking math courses in college. In 40 years, we will remember the increase in college tuitions when we are helping our own children decide what universities to apply to.

## Bibliography

Bogumuil, Helen. "College, Graduate Schools Will Raise Tuition in 1964 | News | The Harvard Crimson." The Harvard Crimson. Harvard University, 21 May 1963. Web. 18 Nov. 2014. <http://www.thecrimson.com/article/1963/5/21/college-graduate-schools-will-raise-tuition/>.