A. (7.077,128,723.08)
B. The vertex is the minimum value of the home of $128,723.08 in 2027.
Part A:
We can use the vertex formula to solve for the vertex. D = b² - 4ac.
[tex]\boxed{\text{Vertex Formula}: \text{For }y=ax^2+bx+c,\;\;\;\;\; \text{Vertex at (h, k) is } (\frac{-b}{2a},\frac{-D}{4a})}}}}[/tex]
Formula:
[tex](\dfrac{-b}{2a},\dfrac{-D}{4a})[/tex]
Substitute for D:
[tex](\dfrac{-b}{2a},\dfrac{-(b^2-4ac)}{4a})[/tex]
Subsiute given values, where a = 325, b = -4,600, and c = 145,000:
[tex](\dfrac{-(-4,600)}{2(325)},\dfrac{-((-4,600)^2-4(325)(145,000))}{4(325)})[/tex]
Distribute negatives:
[tex](\dfrac{4,600}{2(325)},\dfrac{(-(-4,600)^2+4(325)(145,000))}{4(325)})[/tex]
Square:
[tex](\dfrac{4,600}{2(325)},\dfrac{-2,116,0000+4(325)(145,000)}{4(325)})[/tex]
Multiply:
[tex](\dfrac{4,600}{650},\dfrac{-21,160,000+188,500,000}{1,300})[/tex]
Addition:
[tex](\dfrac{4,600}{650},\dfrac{167340000}{1,300})[/tex]
Division:
[tex](7.0769230769,128,723.076923)[/tex]
Round years to the nearest thousandths and y to the nearest cent:
[tex](7.077,128,723.08)[/tex]
We can also solve by graphing. See the attached image.
Part B:
This vertex is a minimum, so the vertex is the minimum value of the home of $128,723.08 in 2027.
