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In a volatile housing market, the overall value of a home can be modeled by [tex]\(V(x) = 325x^2 - 4600x + 145000\)[/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 :

GinnyAnswer
Certainly, let's go through the solution step-by-step.

### Part A: Finding the Vertex of [tex]\( V(x) \)[/tex]
The quadratic function representing the value of the home is given by:
[tex]\[ V(x) = 325x^2 - 4600x + 145000 \][/tex]

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

Here, [tex]\( a = 325 \)[/tex], [tex]\( b = -4600 \)[/tex], and [tex]\( c = 145000 \)[/tex].

Step 1: Calculate the x-coordinate of the vertex
[tex]\[ x = -\frac{b}{2a} = -\frac{-4600}{2 \cdot 325} = \frac{4600}{650} \approx 7.0769 \][/tex]

So, the x-coordinate of the vertex is approximately 7.0769.

Step 2: Calculate the y-coordinate of the vertex
The y-coordinate is found by substituting [tex]\( x \)[/tex] back into the quadratic function:
[tex]\[ V(7.0769) = 325(7.0769)^2 - 4600(7.0769) + 145000 \][/tex]
[tex]\[ V(7.0769) \approx 128723.08 \][/tex]

So, the vertex of the quadratic function [tex]\( V(x) \)[/tex] is approximately [tex]\( (7.0769, 128723.08) \)[/tex].

### Part B: Interpretation of the Vertex
The x-coordinate of the vertex represents the number of years after 2020 when the home value reaches its maximum or minimum. Since the coefficient of [tex]\( x^2 \)[/tex] (i.e., [tex]\( 325 \)[/tex]) is positive, this indicates the parabola opens upwards, meaning the vertex represents the minimum value of the home.

Interpretation:
The x-coordinate of the vertex is approximately 7.0769, which corresponds to the year:
[tex]\[ 2020 + 7.0769 \approx 2027.0769 \][/tex]

So, around the year 2027, the value of the home reaches its minimum.

The y-coordinate of the vertex indicates the minimum value of the home at that time, which is approximately [tex]\( \$128723.08 \)[/tex].

Summary:
- Vertex: [tex]\( (7.0769, 128723.08) \)[/tex]
- Year of Minimum Value: Approximately 2027
- Minimum Home Value: Approximately [tex]\( \$128723.08 \)[/tex]

Therefore, in around 2027, the home's value reaches its minimum, which is approximately [tex]\( \$128723.08 \)[/tex].
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