Paul opened a bakery. The net value of the bakery (in thousands of dollars) \( t \) months after its creation is modeled by \( v(t)=2 t^{2}-12 t-14 \) Paul wants to know what his bakery's lowest net value will be. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. \( v(t)=\square \) streak 2) What is the bakery's lowest net value?

To find the vertex form of the quadratic function \( v(t) = 2t^2 - 12t - 14 \), we can complete the square. Start by factoring out the coefficient of \( t^2 \) from the first two terms: \[ v(t) = 2(t^2 - 6t) - 14. \] Next, complete the square inside the parentheses: \[ v(t) = 2\left(t^2 - 6t + 9 - 9\right) - 14 = 2\left((t-3)^2 - 9\right) - 14. \] This simplifies to: \[ v(t) = 2(t-3)^2 - 18 - 14 = 2(t-3)^2 - 32. \] Now, the function is in vertex form: \[ v(t) = 2(t-3)^2 - 32. \] The bakery's lowest net value is found at the vertex, where \( (t-3)^2 \) equals zero (the minimum point). Plugging in \( t = 3 \), we find: \[ v(3) = 2(0) - 32 = -32. \] So, the lowest net value of the bakery is \( -32 \) thousand dollars. In the world of baking, many startups struggle initially, often due to underestimated costs or unexpected competition. Understanding your cash flow model and keeping a close eye on expenses can save a new bakery from some sticky situations! Engaging with fellow bakers in your community might also lead to collaborations, promotions, or even sharing space, reducing overhead for everyone involved. For those with a sweet tooth for financial wisdom, there are plenty of resources available, from online courses on financial management for small businesses to books that delve into case studies of successful bakeries. Titles such as "The E-Myth Bakery" by Michael E. Gerber could provide insights into balancing the art of baking with the science of running a profitable bakery. So grab a cup of coffee and get reading!