Given:
The net value of the bakery (in thousands of dollars) t months after its creation is modeled by
[tex]v(t)=2t^2-12t-14[/tex]
Paul wants to know what his bakery's lowest net value will be.
To find:
The function in a different form (factored or vertex) where the answer appears as a number in the equation.
Solution:
Factor form is used to find the x-intercepts and vertex form is used to find the extreme values (maximum or minimum). So, here we need to find the vertex form.
We have,
[tex]v(t)=2t^2-12t-14[/tex]
[tex]v(t)=2(t^2-6t)-14[/tex]
Adding and subtract square of half of 6 in the parenthesis, we get
[tex]v(t)=2(t^2-6t+3^2-3^2)-14[/tex]
[tex]v(t)=2(t^2-6t+3^2)+2(-9)-14[/tex]
[tex]v(t)=2(t-3)^2-18-14[/tex] [tex][\because (a-b)^2=a^2-2ab+b^2][/tex]
[tex]v(t)=2(t-3)^2-32[/tex]
Vertex form:
[tex]f(x)=a(x-h)^2+k[/tex]
where, (h,k) is vertex.
On comparing this equation with vertex form, we get the of the function is (3,-32).
Therefore, the vertex form is [tex]v(t)=2(t-3)^2-32[/tex] and the function has minimum value at (3,-32). It means, minimum net value of the bakery is -32 after 3 months.
Answer:
v(t) = 2(t - 3)^2 - 32. and -32
Please mark the other one Brainiest.
Hope that this helps!
