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In a volatile housing market, the overall value of a home can be modeled by [tex]V(x)=210x^2-4400x+125000[/tex], where [tex]V[/tex] represents the value of the home and [tex]x[/tex] represents each year after 2020.

Part A: Find the vertex of [tex]V(x)[/tex]. Show all work.

Part B: Interpret what the vertex means in terms of the value of the home.

Sagot :

GinnyAnsweridn
Sure, let's tackle this problem step by step.

### Part A: Finding the Vertex of [tex]\(V(x)\)[/tex]

The given equation is:
[tex]\[ V(x) = 210x^2 - 4400x + 125000 \][/tex]

To find the vertex of this quadratic function, we need to use the vertex formula for a parabola, which is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are coefficients from the quadratic equation [tex]\(ax^2 + bx + c\)[/tex].

In this case, we have:
[tex]\[ a = 210 \][/tex]
[tex]\[ b = -4400 \][/tex]
[tex]\[ c = 125000 \][/tex]

Plugging in the values into the vertex formula:
[tex]\[ x = -\frac{-4400}{2 \cdot 210} \][/tex]
[tex]\[ x = \frac{4400}{420} \][/tex]
[tex]\[ x = 10.476190476190476 \][/tex]

This is the x-coordinate of the vertex.

To find the y-coordinate of the vertex, we substitute [tex]\( x = 10.476190476190476 \)[/tex] back into the original equation:
[tex]\[ V(10.476190476190476) = 210(10.476190476190476)^2 - 4400(10.476190476190476) + 125000 \][/tex]

After carefully calculating, we obtain:
[tex]\[ V(10.476190476190476) = 101952.38095238095 \][/tex]

So the vertex of the function [tex]\( V(x) \)[/tex] is:
[tex]\[ (10.476190476190476, 101952.38095238095) \][/tex]

### Part B: Interpreting the Vertex

The vertex of the function [tex]\(V(x)\)[/tex] in this context represents the minimum value of the home over time.

Since [tex]\(x\)[/tex] represents the number of years after 2020, the x-coordinate of the vertex [tex]\(x = 10.476190476190476\)[/tex] indicates the number of years after 2020 when this minimum value occurs.

In numerical terms:
[tex]\[ 2020 + 10.476190476190476 \approx 2030.476 \][/tex]

This suggests that around mid-2030, the value of the home reaches its minimum based on this model.

The y-coordinate of the vertex represents the minimum value of the home. Given that the y-coordinate is [tex]\( 101952.38095238095 \)[/tex], it means that the minimum value of the home around mid-2030 is approximately \[tex]$101,952.38. In summary: - The minimum value of the home occurs approximately 10.48 years after 2020, which is around mid-2030. - At this point, the value of the home is about \$[/tex]101,952.38.
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