The number of portable widgets produced weekly by Widget, Inc., is related to the weekly profit in the way: P(x) = -2x^2 + 88x - 384
where P(x) is the weekly profit in hundreds of dollars and x is the number of widgets must be produced weekly. How many widgets must be produced weekly for the maximum weekly profit? What is the maximum weekly profit? Find the analytically.

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Step-by-step explanation:

The turning point (maximum or minimum) of quadratic function ...

  f(x) = ax² +bx +c

is at x = -b/(2a).

Here, this means the maximum profit will be had when the number of widgets produced weekly is ...

  x = -88/(2(-2)) = 22

The profit at that level of production is ...

  P(22) = (-2·22 +88)(22) -384 = 44(22) -384 = 968 -384 = 584 . . . hundreds

22 widgets must be produced weekly for the maximum weekly profit of $58,400.

horizontal translation of 2 units in the positive x direction

followed by

vertical translation of 3 units in the positive y direction

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