### Part A: Finding the Vertex
To find the vertex of the quadratic function [tex]\( V(x) = 415x^2 - 4600x + 200000 \)[/tex], we need to use the vertex formula for a parabola, which is [tex]\( x = -\frac{b}{2a} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the coefficients from the standard quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
1. Identify the coefficients from the given quadratic function:
- [tex]\( a = 415 \)[/tex]
- [tex]\( b = -4600 \)[/tex]
- [tex]\( c = 200000 \)[/tex]
2. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula:
[tex]\[
x = -\frac{-4600}{2 \cdot 415} = \frac{4600}{830}
\][/tex]
3. Simplify the fraction:
[tex]\[
x = \frac{4600}{830} \approx 5.542168674698795
\][/tex]
So, the x-coordinate of the vertex is approximately [tex]\( 5.542168674698795 \)[/tex].
Next, we'll find the y-coordinate (V(x)) of the vertex by substituting the x-coordinate back into the quadratic function.
4. Substitute [tex]\( x \approx 5.542168674698795 \)[/tex] back into the quadratic function:
[tex]\[
V(5.542168674698795) = 415(5.542168674698795)^2 - 4600(5.542168674698795) + 200000
\][/tex]
5. Calculate the value:
[tex]\[
V(5.542168674698795) \approx 187253.01204819276
\][/tex]
So, the coordinates of the vertex are approximately [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex].
### Part B: Interpreting the Vertex
The vertex [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex] represents a key point in the context of the home's value over time.
- x-coordinate [tex]\( 5.542168674698795 \)[/tex]: This represents the number of years after 2020 when the home reaches its minimum value. Approximately 5.54 years after 2020 is around mid-2025.
- y-coordinate [tex]\( 187253.01204819276 \)[/tex]: This represents the minimum value of the home. The value of the home will be at its lowest point, which is approximately [tex]$187,253.01, during mid-2025.
Thus, in terms of the value of the home, the vertex indicates that the home's value will decrease and reach its minimum value of about $[/tex]187,253.01 around the middle of the year 2025. After this point, if the model holds, the value of the home would begin to increase again.
