Tom started an entertainment company. The net value of the company (in thousands of dollars)

Question

\[t\] months after its creation is modeled by
\[v(t)=4t^2-24t-28\]
Tom wants to know when his company will be at its lowest net value.
1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.
\[v(t)=\]
2) How many months after its creation does the company reach its lowest net value?
months

Answer 1

To find when Tom's entertainment company reaches its lowest net value using the function \( v(t) = 4t^2 - 24t - 28 \), we will rewrite the equation in vertex form and find the vertex.

1. Rewriting the function in vertex form:

The vertex form of a quadratic function \( v(t) = a(t - h)^2 + k \) allows us to easily find the vertex \((h,k)\). We can start by completing the square for the quadratic \(v(t)\):

1. Factor out the coefficient of \(t^2\) (which is 4) from the first two terms:

\[ v(t) = 4(t^2 - 6t) - 28 \]

2. Next, to complete the square inside the parentheses, we take half of the coefficient of \(t\) (which is -6), square it, and add and subtract that square:

\[ \left(-\frac{6}{2}\right)^2 = 9 \]

3. We add and subtract 9 inside the parentheses:

\[ v(t) = 4(t^2 - 6t + 9 - 9) - 28 \]

This simplifies to:

\[ v(t) = 4((t - 3)^2 - 9) - 28 \]

4. Expanding this gives:

\[ v(t) = 4(t - 3)^2 - 36 - 28 \]

\[ v(t) = 4(t - 3)^2 - 64 \]

Now, we have the function in vertex form:

\[ \boxed{v(t) = 4(t - 3)^2 - 64} \]

2. Finding the time when the company reaches its lowest net value:

The minimum value of a parabola \(a(t - h)^2 + k\) occurs at \(t = h\). From our vertex form \(v(t) = 4(t - 3)^2 - 64\), we see that:

\[ h = 3 \]

Thus, the company reaches its lowest net value:

\[ \boxed{3} \text{ months} \] after its creation.