Tom started an entertainment company. The net value of the company (in thousands of dollars) \( t \) months after its creation is modeled by \( v(t)=4 t^{2}-24 t-28 \) 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?

To rewrite the function in vertex form, we can complete the square. Start with \( v(t) = 4t^2 - 24t - 28 \). Factoring out a 4 from the first two terms gives us \( v(t) = 4(t^2 - 6t) - 28 \). Completing the square inside the parentheses yields \( v(t) = 4((t - 3)^2 - 9) - 28 \), which simplifies to \( v(t) = 4(t - 3)^2 - 36 \). The company reaches its lowest net value at \( t = 3 \) months after its creation. This is found by analyzing the vertex of the parabola, where the minimum point occurs. Since we rewrote the function to identify the vertex, we see that the lowest point occurs precisely when \( t = 3 \).