Belinda is thinking about buying a house for $249,000. The table below shows the projected value of two different houses for three years: Number of years 1 2 3
House 1 (value in dollars) 253,980 259,059.60 264,240.79
House 2 (value in dollars) 256,000 263,000 270,000

Part A: What type of function, linear or exponential, can be used to describe the value of each of the houses after a fixed number of years? Explain your answer. (2 points)

Part B: Write one function for each house to describe the value of the house f(x), in dollars, after x years. (4 points)

Part C: Belinda wants to purchase a house that would have the greatest value in 45 years. Will there be any significant difference in the value of either house after 45 years? Explain your answer, and show the value of each house after 45 years. (4 points)

A) Part A:

For House 1: exponential functionFor House 2: linear function

B) Part B:

For house 1:       For house 2:      

C) Part C:

For house 1: $619,166.21For house 2: $571,000

Explanation:

Number of years   House 1 (value in dollars)   House 2 (value in dollars)

   1                                       253,980                           256,000

   2                                      259,059.60                      263,000

   3                                      264,240.79                       270,000

Part A: Type of function.

To state what type of function, linear or exponential, can be used to describe the value of the houses after a fixed number of years, you can look whether the pattern corresponds to an arithmetic or an geometric sequence.

Aritmetic sequences are linear functions and what characterizes them is that there is a common constant difference between any two consecutive terms of the sequence.

For the first house, when you subtract the first term from the second term, you find a difference of 5,079.6. When you subtract the second term from the third, you find a difference of 5,181.19, hence this does not represent a linear function.

Geometric sequences are exponential functions and what charaterizes them is that there is a common constant ratio between any two consecutive terms of the sequence.

For the first house that is:

ratio = 259,059.6 / 253,980 = 1.02ratio = 264,240.79 / 259,059.6 = 1.01999 ≈ 1.02

Hence, you conclude that an exponential function can be used to describe the value of the first house.

For the second house you will find that the difference of value after each year is constant:

difference = 263,000 - 256,000 = 7,000 (in dollars)difference = 270,000 - 263,000 = 7,000 (in dollars)

Hence, you conclude that a linear function can be used to describe the value of the second house.

Part B: Functions for each house to describe the value of the house f(x), in dollars, after x years.

For an exponential function (first house), the function f(x) will be the first term (the initial value of the house) multiplied by the ratio raised to the power equal to the number of years:

For a linear function (second house), the function f(x) wil lbe the initial term (the initial value of the house) plus the product of the common difference with the number of years:

Part C: Greatest value in 45 years

For the first house, the value in dollars will be:

         

For the second house, the value in dollars will be:

             

Therefore, there will be a significant difference in the value of both houses after 45 years.

The first house will have a value 619.166.21 - 571,000 = 48,166.21 dollars greater than the second house.