To solve the problem, let's go through it step-by-step:
1. Find the initial value of the house:
The given exponential function is:
[tex]\[
v(t) = 532,000 \cdot (0.77)^t
\][/tex]
The initial value of the house is determined by evaluating the function at [tex]\( t = 0 \)[/tex]:
[tex]\[
v(0) = 532,000 \cdot (0.77)^0 = 532,000 \cdot 1 = 532,000
\][/tex]
Therefore, the initial value of the house is:
[tex]\[
\boxed{532,000}
\][/tex]
2. Determine if the function represents growth or decay:
The factor being raised to the power of [tex]\( t \)[/tex] in the function is 0.77. Since this factor is less than 1, the function represents decay. Thus, the value of the house decreases over time.
Therefore, the function represents:
[tex]\[
\boxed{\text{decay}}
\][/tex]
3. Calculate the percent change in the value of the house each year:
The decay rate is given by 0.77. To find the percent change per year, we first calculate how much the value of the house decreases each year:
[tex]\[
\text{Percent change} = (1 - \text{decay rate}) \times 100\%
\][/tex]
In this case, the decay rate is 0.77:
[tex]\[
\text{Percent change} = (1 - 0.77) \times 100\% = 0.23 \times 100\% = 23\%
\][/tex]
Therefore, the value of the house changes by:
[tex]\[
\boxed{23\%}
\][/tex]
each year.
