algebra 2 quadradic word problems

Question

 company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out what price the widgets should be sold for, to the nearest cent, for the company to make the maximum profit.

y, equals, minus, 17, x, squared, plus, 631, x, minus, 3081

y=−17x2+631x−3081

Answer 1

If you graph this quadratic function the result is a downward opening parabola. That means the maximum y-value occurs at the vertex of the parabola. The vertex of the parabola lies on its axis of symmetry. The equation for the axis of symmetry is given by x = -b/2a where in this case a = -17, b= 631 (c = -3081).

So the x-value that generates the maximum is -631/2(-17) = 631/34 Assuming x is in $, to the nearest cent: $18.56.

The axis of symmetry is the vertical line that crosses the x-axis halfway between the roots. The roots can be obtained using the quadratic formula. If you add those roots together the result is -b/a. Take half of that to get the average or midpoint of the segment joining the roots. (1/2) (-b/a) = -b/2a which explains the formula for the axis of symmetry.

The following graph provides some details including the max profit.

desmos.com/calculator/quhfacmm6k