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5b
score: 25.63/30 answered: 12/13
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tom started an entertainment company. the net value, v, of the company (in thousands of dollars) t
months after its creation is modeled by: ( v ( t ) = 4 t ^ { 2 } - 24 t - 28 )
identify the variables.
( x : ) months and ( y : ) value (thousands of dollars)
what is the ( y ) - intercept in context? ( ( 0, - 28 ) )
which means toms company has cost ( $ 28,000 ) to start up.
what was the lowest net value for toms company? ( - 64 )
when did this happen? 3 months after creation

Explanation:

Step1: Encontrar la forma estándar del vértice

La función \(v(t) = 4t^{2}-24t - 28\) se puede escribir en forma \(v(t)=a(t - h)^{2}+k\), donde \((h,k)\) es el vértice.
Primero, factorizamos \(a = 4\):
\(v(t)=4(t^{2}-6t)-28\)
Completamos el trinomio cuadrado perfecto dentro del paréntesis.
Para \(t^{2}-6t\), el término que completará el cuadrado es \((\frac{-6}{2})^{2}=9\)
\(v(t)=4(t^{2}-6t + 9-9)-28\)
\(v(t)=4((t - 3)^{2}-9)-28\)
\(v(t)=4(t - 3)^{2}-36-28\)
\(v(t)=4(t - 3)^{2}-64\)

Step2: Identificar el vértice

Como \(a = 4>0\), la parábola abre hacia arriba y el vértice \((h,k)=(3,-64)\) es el mínimo.
El valor \(y\) del vértice es \(k=-64\), pero la función \(v(t)\) está en miles de dólares.
El valor mínimo real es \(-64\times1000=-64000\) dólares.

Answer:

\(-64000\) dólares.