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?

We are given the function v(t) = 4t² – 24t – 28. Step 1. Rewrite in vertex form. To rewrite in vertex form, complete the square: 1. Factor out 4 from the quadratic and linear terms:   v(t) = 4(t² – 6t) – 28 2. Complete the square inside the parentheses. Take half of –6 (which is –3) and square it (9):   t² – 6t = (t² – 6t + 9) – 9 = (t – 3)² – 9 3. Substitute back:   v(t) = 4[(t – 3)² – 9] – 28      = 4(t – 3)² – 36 – 28      = 4(t – 3)² – 64 So the vertex form is:   v(t) = 4(t – 3)² – 64 Step 2. Find when the company reaches its lowest net value. In vertex form v(t) = 4(t – 3)² – 64, the vertex is at t = 3. Since the coefficient 4 is positive, the vertex represents the minimum value. Therefore, the company reaches its lowest net value 3 months after its creation.