Question

Kwan's parents bought a home for $50,000 in 1997 just as real estate values in the area started to rise quickly. Each year, their house was worth more until they sold the home for $309,587. Model the growth of the home's value from 1997 to 2007 with both linear and an exponential equation. Graph the two models below.

Answer 1

A. First let us start with the linear model.

The equation is in the form of y = m x + b

Calculating for the slope m:

m = (309,587 – 50,000) / (2007 – 1997)

m = 25,958.7

 

Subsituting:

y = 25,958.7 x + b

Taking x = 1997, y = 50,000. Solve for b:

50,000 = 25,958.7 (1997) + b

b = -51,789,523.9

 

The complete equation is therefore:

y = 25,958.7 x - 51,789,523.9

 

 

B. The exponential model has the following form:

y = a b^x

where a and b are constants

 

Taking x1= 1997, y1 = 50,000; x2 = 2007, y2 = 309,587

50,000 = a b^1997

309,587 = a b^2007

 

Combining in terms of a:

50,000 / b^1997 = 309,587 / b^2007

b^2007 / b^1997 = 309,587 / 50,000

b^10 = 6.19174

b = 1.2

 

Substituting:

y = a 1.2^x

 

Solving for a:

50,000 = a 1.2^1997

a = 3.75 x 10^-154

 

The complete equation is:

y = 3.75 x 10^-154 * 1.2^x