Question

question 1
in a volatile housing market, the overall value of a home can be modeled by ( v(x)=210x^{2}-4400x + 125000 ), where ( v ) represents the value of the home and ( x ) represents each year after 2020
part a: find the vertex of ( v(x) ). show all work.
part b: interpret what the vertex means in terms of the value of the home.

Explanation:

Step1: Identify the coefficients

For the quadratic function \(V(x)=ax^{2}+bx + c\), here \(a = 210\), \(b=-4400\), \(c = 125000\).

Step2: Find the \(x\) - coordinate of the vertex

The formula for the \(x\) - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\).
Substitute \(a = 210\) and \(b=-4400\) into the formula:
\(x=-\frac{-4400}{2\times210}=\frac{4400}{420}=\frac{220}{21}\approx10.48\)

Step3: Find the \(y\) - coordinate of the vertex

Substitute \(x = \frac{220}{21}\) into the function \(V(x)=210x^{2}-4400x + 125000\)

$$
LATEXBLOCK0
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The vertex of a quadratic function \(y = ax^{2}+bx + c\) (where \(a
eq0\)) gives the minimum/maximum value of the function. Since \(a = 210>0\), the parabola opens upwards and the vertex \((\frac{220}{21},\frac{2141000}{21})\) represents the minimum value of the home's value function. The \(x\) - coordinate (\(\approx10.48\)) represents the number of years after 2020 when the home's value is at its minimum, and the \(y\) - coordinate (\(\approx101952.38\)) represents the minimum value of the home.

Answer:

Part A: The vertex of \(V(x)\) is \((\frac{220}{21},\frac{2141000}{21})\)
Part B: The \(x\) - coordinate (\(\approx10.48\)) means approximately \(10.48\) years after 2020 the value of the home is at its minimum. The \(y\) - coordinate (\(\approx101952.38\)) means the minimum value of the home is approximately \(\$101952.38\)