To solve the problem, let's break it down step-by-step:
1. Initial Value of the House:
- The function [tex]\( v(t) = 532,000 (1.02)^t \)[/tex] is an exponential function that models the value of the house over time.
- The initial value of the house is found by evaluating the function when [tex]\( t = 0 \)[/tex].
- Plugging [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
v(0) = 532,000 \times (1.02)^0
\][/tex]
- Since [tex]\( (1.02)^0 = 1 \)[/tex]:
[tex]\[
v(0) = 532,000 \times 1 = 532,000
\][/tex]
- Therefore, the initial value of the house is [tex]\( \$532,000 \)[/tex].
2. Growth or Decay:
- In an exponential function of the form [tex]\( v(t) = a(b)^t \)[/tex]:
- If [tex]\( b > 1 \)[/tex], the function represents growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents decay.
- In the given function, [tex]\( b = 1.02 \)[/tex], which is greater than 1.
- Therefore, the function represents growth.
3. Percent Change Each Year:
- The base of the exponent, [tex]\( 1.02 \)[/tex], indicates the factor by which the house value changes each year.
- To find the percent change, we convert the factor to a percentage:
- Since [tex]\( 1.02 \)[/tex] represents a 2% increase (because [tex]\( 1.02 = 1 + 0.02 \)[/tex]).
- Therefore, the value of the house changes by 2% each year.
Let's summarize the findings:
1. Initial value of the house: [tex]\( \$532,000 \)[/tex]
2. The function represents: growth
3. Percent change each year: [tex]\( 2\% \)[/tex]
