Alright, let's work through the problem step by step.
### Part A: Find the vertex of [tex]\( V(x) \)[/tex]
The equation given is:
[tex]\[ V(x) = 325x^2 - 4600x + 145000 \][/tex]
This is a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 325 \)[/tex]
- [tex]\( b = -4600 \)[/tex]
- [tex]\( c = 145000 \)[/tex]
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x_{\text{vertex}} = -\frac{-4600}{2 \cdot 325} = \frac{4600}{650} \approx 7.08 \][/tex]
So, the x-coordinate of the vertex is approximately [tex]\( 7.08 \)[/tex].
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]\[
y_{\text{vertex}} = V(7.08) = 325(7.08)^2 - 4600(7.08) + 145000
\][/tex]
Calculating this, we approximately get:
[tex]\[ y_{\text{vertex}} = 325 \cdot (7.08)^2 - 4600 \cdot 7.08 + 145000 \approx 128723.08 \][/tex]
So, the coordinates of the vertex are approximately [tex]\( (7.08, 128723.08) \)[/tex].
### Part B: Interpret what the vertex means in terms of the value of the home
The vertex [tex]\( (7.08, 128723.08) \)[/tex] of the quadratic function [tex]\( V(x) \)[/tex] is significant in the context of the value of the home. Here’s how to interpret it:
- The x-coordinate of the vertex, [tex]\( x = 7.08 \)[/tex], represents the number of years after 2020 when the minimum value of the home occurs. Since [tex]\( x \)[/tex] is 7.08, this indicates that the year when the minimum value occurs is 2020 + 7.08, which is roughly 2027.
- The y-coordinate of the vertex, [tex]\( y = 128723.08 \)[/tex], represents the minimum value of the home in dollars.
Therefore, in terms of the value of the home:
- In the year approximately 2027 (7.08 years after 2020), the value of the home is at its minimum.
- The minimum value of the home is approximately $128,723.08.
