Sure, let's solve this step-by-step.
### Part A: Find the vertex of [tex]\( V(x) \)[/tex]
To find the vertex of a quadratic function [tex]\( V(x) = ax^2 + bx + c \)[/tex], we use the formulas for the x-coordinate and y-coordinate of the vertex:
1. Find the x-coordinate:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
Given the quadratic function [tex]\( V(x) = 325x^2 - 4600x + 145000 \)[/tex]:
- [tex]\( a = 325 \)[/tex]
- [tex]\( b = -4600 \)[/tex]
- [tex]\( c = 145000 \)[/tex]
Plug these values into the formula for the x-coordinate:
[tex]\[ x_{\text{vertex}} = -\frac{-4600}{2 \times 325} \][/tex]
[tex]\[ x_{\text{vertex}} = \frac{4600}{650} \][/tex]
[tex]\[ x_{\text{vertex}} \approx 7.076923076923077 \][/tex]
2. Find the y-coordinate:
To find the y-coordinate of the vertex, we substitute [tex]\( x_{\text{vertex}} \)[/tex] back into the original equation [tex]\( V(x) \)[/tex]:
[tex]\[ V(x_{\text{vertex}}) = 325(x_{\text{vertex}})^2 - 4600(x_{\text{vertex}}) + 145000 \][/tex]
Using [tex]\( x_{\text{vertex}} \approx 7.076923076923077 \)[/tex]:
[tex]\[ V(7.076923076923077) = 325(7.076923076923077)^2 - 4600(7.076923076923077) + 145000 \][/tex]
[tex]\[ V(7.076923076923077) \approx 325 \times 50.081784 \][/tex]
[tex]\[ V(7.076923076923077) \approx 16276.080 \][/tex]
[tex]\[ V(7.076923076923077) \approx 128723.07692307692 \][/tex]
So, the vertex of the function [tex]\( V(x) \)[/tex] is:
[tex]\[ \left( 7.076923076923077, 128723.07692307692 \right) \][/tex]
### Part B: Interpret what the vertex means in terms of the value of the home
The vertex of the quadratic function [tex]\( V(x) = 325x^2 - 4600x + 145000 \)[/tex] can provide significant insights into the value trend of the home over time.
- The x-coordinate of the vertex, approximately [tex]\( 7.08 \)[/tex], represents the number of years after 2020 when the value of the home reaches its minimum. This is around the year [tex]\( 2020 + 7.08 \approx 2027 \)[/tex].
- The y-coordinate of the vertex, approximately [tex]\( 128723.08 \)[/tex], represents the minimum value of the home at that time.
Interpretation:
The vertex indicates that around the year [tex]\( 2027 \)[/tex], the value of the home will hit a minimum of about [tex]\(\$128,723.08 \)[/tex]. This suggests that starting from 2020, the home value will decrease, reaching its lowest point in 2027, and then start to increase after that period.
So, in summary:
- The home’s value hits its minimum approximately 7.08 years after 2020, which is around the year 2027.
- At that time, the minimum value of the home is approximately [tex]\(\$128,723.08\)[/tex].
This completes the detailed solution and interpretation for the given problem.
