The function that describes the house value is an exponential function in the form y=ab^x.
B
b
Set y equal to 800 000 and solve for x.
C
c
Create a new function that describes the houses in Jacksonville. Notice that a multiplier that describes a depreciation is below 1.
A
a
About $564 240
B
b
In the year 2025
C
c
$36 586
Practice makes perfect
a
Since the house appreciates in value with a percentage, the price of the house can be modeled by an exponential function.
y=ab^xIn this equation a is the initial value and b is the multiplier. The initial value of the house in 2005 was a= 400 000. Also, the rate at which the house increases in value is 3.5 % per year. This equals a multiplier of b= 1.035. Now, we can write the function that describes the house's value.
y= 400 000( 1.035)^x
Since 2015 is 10 years after 2005, we can determine the house's value at this time by substituting t=10 into the equation and simplifying.
After about 20.15 years, the house is worth $800 000. This puts the year at 2025.
c
Before we can answer the question, we have to write a new equation describing the houses in Jacksonville. The houses depreciates each year by 2 %, which means it loses value. We can write this as the multiplier b=0.98. Also, the house we are examining is currently worth 200 000. Now we can write the function.
y=200 000(0.98)^xTo determine how much it is worth in 10 years, we have to set x equal to 10 and simplify.
In 10 years, a house in Jacksonville that is originally worth $200 000 will be worth $163 414. By subtracting the lower value from the greater value, we can determine how much value was lost.
200 000-163 414=$36 586
The house has lost $36 586.
