Question

in a volatile housing market, the overall value of a home can be modeled by v(x)=210x² - 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: Recall vertex - form of a quadratic function

The quadratic function is given in standard form \(V(x)=210x^{2}-4400x + 125000\). The vertex - form of a quadratic function is \(y=a(x - h)^{2}+k\), where the vertex is \((h,k)\). For a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex \(h=-\frac{b}{2a}\). Here, \(a = 210\) and \(b=-4400\).
\[h=-\frac{-4400}{2\times210}=\frac{4400}{420}=\frac{220}{21}\approx10.48\]

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

Substitute \(x = \frac{220}{21}\) into the function \(V(x)=210x^{2}-4400x + 125000\).
\[V(\frac{220}{21})=210\times(\frac{220}{21})^{2}-4400\times\frac{220}{21}+125000\]
\[=210\times\frac{48400}{441}-\frac{968000}{21}+125000\]
\[=\frac{484000}{21}-\frac{968000}{21}+125000\]
\[=\frac{484000 - 968000}{21}+125000\]
\[=-\frac{484000}{21}+125000\]
\[=\frac{-484000+2625000}{21}=\frac{2141000}{21}\approx101952.38\]
So the vertex is \((\frac{220}{21},\frac{2141000}{21})\) or approximately \((10.48,101952.38)\)

Step3: Interpret the vertex

The \(x\) - value of the vertex \(\frac{220}{21}\approx10.48\) represents the number of years after 2020. The \(y\) - value of the vertex \(\frac{2141000}{21}\approx101952.38\) represents the minimum value of the home. So approximately \(10.48\) years after 2020 (around 2030 - 2031), the value of the home will reach its minimum value of approximately \(\$101952.38\)

Answer:

Part A: The vertex is \((\frac{220}{21},\frac{2141000}{21})\approx(10.48,101952.38)\)
Part B: Approximately \(10.48\) years after 2020 (around 2030 - 2031), the value of the home will reach its minimum value of approximately \(\$101952.38\)