# College Costs Project

### By: Michelle Zhang and Aparna Bejoy

## Yale

## 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

## Graphs

## 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.

## Calculations for % Error for 1990-1991 and 2014-2015

## Function Analysis

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

**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

## Adding 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.

## UTSA

## 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

## Graph

## 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%

## Calculations for Percent Error

## 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.

**Was one model better consistently? Was this what you expected? **

## UTSA 1969

## Adding 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

*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/>.