Let's address the problem piece by piece.
### Part A: Determine the Type of Function (Linear or Exponential)
House 1:
1. The initial value is [tex]$286,000, and the values for the next three years are $[/tex]294,580, [tex]$303,417.40, and $[/tex]312,519.92.
2. Calculate the year-over-year growth:
- Year 1 to Year 2: [tex]$\frac{303,417.40}{294,580} \approx 1.030$[/tex]
- Year 2 to Year 3: [tex]$\frac{312,519.92}{303,417.40} \approx 1.030$[/tex]
- Since the growth rate is consistent, the values suggest an exponential function.
House 2:
1. The initial value is [tex]$286,000, and the values for the next three years are $[/tex]295,000, [tex]$304,000, and $[/tex]313,000.
2. Calculate the year-over-year growth:
- Year 1 to Year 2: [tex]$\frac{304,000}{295,000} \approx 1.0305$[/tex]
- Year 2 to Year 3: [tex]$\frac{313,000}{304,000} \approx 1.0296$[/tex]
- Although there's a slight variation, it maintains an approximate linear growth rate. Examining differences:
- Year 1 to Year 2: [tex]$304,000 - 295,000 = 9,000$[/tex]
- Year 2 to Year 3: [tex]$313,000 - 304,000 = 9,000$[/tex]
- Since the differences are consistent, the trend suggests a linear function.
### Part B: Write Functions Describing Each House's Value
House 1:
- The growth rate, calculated approximately as 1.03, showcases exponential growth:
[tex]\[
f_1(x) = 286,000 \times (1.03)^x
\][/tex]
House 2:
- The consistent increase of \[tex]$9,000 per year describes linear growth:
\[
f_2(x) = 286,000 + 9,000x
\]
### Part C: Value of Each House After 25 Years
House 1:
\[
f_1(25) = 286,000 \times (1.03)^{25}
\]
Calculate:
\[
(1.03)^{25} \approx 2.093 \, \text{(using a calculator)}
\]
Hence,
\[
f_1(25) = 286,000 \times 2.093 \approx 598,598
\]
House 2:
\[
f_2(25) = 286,000 + 9,000 \times 25
\]
Calculate:
\[
9,000 \times 25 = 225,000
\]
Hence,
\[
f_2(25) = 286,000 + 225,000 = 511,000
\]
### Conclusion
Belinda should purchase House 1, as its projected value after 25 years ($[/tex]598,598[tex]$) is higher than that of House 2 ($[/tex]511,000$). The exponential growth of House 1 leads to a significantly higher value in the long term.
