Let's solve the problem step-by-step.
1. Initial Value of the House:
The initial value of an exponential function [tex]\( v(t) = v_0 \cdot b^t \)[/tex] is given by [tex]\( v_0 \)[/tex], which is the coefficient in front of the exponential term.
For the given function [tex]\( v(t) = 532,000 (0.77)^t \)[/tex], the initial value [tex]\( v_0 \)[/tex] is 532,000.
Therefore, the initial value of the house is
[tex]$
532,000
$[/tex]
2. Determining if the Function Represents Growth or Decay:
In an exponential function of the form [tex]\( v(t) = v_0 \cdot b^t \)[/tex], if the base [tex]\( b \)[/tex] is less than 1, the function represents decay. If [tex]\( b \)[/tex] is greater than 1, the function represents growth.
Here, the base [tex]\( b \)[/tex] is 0.77, which is less than 1.
Therefore, the function represents
[tex]$
\text{decay}
$[/tex]
3. Percent Change per Year:
The percent change per year in an exponential decay or growth function can be calculated as [tex]\((1 - b) \times 100\%\)[/tex], where [tex]\( b \)[/tex] is the base of the exponential.
For [tex]\( b = 0.77 \)[/tex], the percent change per year is:
[tex]$
(1 - 0.77) \times 100\% = 0.23 \times 100\% = 23\%
$[/tex]
Therefore, the value of the house changes by
[tex]$
23 \%
$[/tex] per year.
