The profit p, in dollars, of a small business can be modeled by the function p(x) =...
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Problem
The profit p, in dollars, of a small business can be modeled by the function p(x) = 5x^2 - 40x + 60, where x is the number of units sold. Part A: What is the vertex of the function? Part B: What does the vertex indicate about the profit? Part C: What do the zeros indicate about the function?
Step-by-step solution
For the function $p(x) = 5x^2 - 40x + 60$, the vertex x-coordinate is $-(-40)/(2*5) = 4$. The vertex y-coordinate is $p(4) = 5(4)^2 - 40(4) + 60 = 80 - 160 + 60 = -20$. The vertex is $(4, -20)$. The zeros are found by solving $5x^2 - 40x + 60 = 0$, which simplifies to $x^2 - 8x + 12 = 0$, factoring to $(x-2)(x-6) = 0$, so the zeros are $x=2$ and $x=6$.
Answer
vertex (4, -20), vertex indicates profit made when 4 units are sold, zeros indicate two points where the company breaks even