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The owner of a video game store creates the expression −2x2 + 24x + 174 to represent the store's weekly profit in dollars, where x represents the price of a new video game. Choose the equivalent expression that reveals the video game price that produces the highest weekly profit, and use it to determine the price.
−2(x2 − 12x) + 174; x = $12
−2(x2 − 12x − 87); x = $87
−2(x − 6)2 + 246; x = $246
−2(x − 6)2 + 246; x = $6

Recall that the maximum of a parabola resides at its vertex. So let's find the x and y position of that vertex, by using first the fact that the x value of the vertex of a parabola of general form:

[tex]y=ax^2+bx+c[/tex]

is given by:

[tex]x_{vertex}=\frac{-b}{2\,a}[/tex]

In our case, the quadratic expression that generates the parabola is:

[tex]y=-2x^2+24x+174[/tex]

then the x-position of its vertex is:

[tex]x_{vertex}=\frac{-b}{2\,a}\\x_{vertex}=\frac{-24}{2\,(-2)}\\x_{vertex}=\frac{-24}{-4)}\\x_{vertex}=6[/tex]

This is the price of the video game that produces the maximum profit (x = $6). Now let's find the y-position of the vertex using the actual equation for this value of x:

[tex]y=-2x^2+24x+174\\y_{vertex}=-2\,(6)^2+24\,(6)+174\\y_{vertex}=-72+144+174\\y_{vertex}=246[/tex]

This value is the highest weekly profit (y = $246).

Now, recall that we can write the equation of the parabola in what is called "vertex form" using the actual values of the vertex position [tex](x_{vertex},y_{vertex})[/tex]:

[tex]y-y_{vertex}=a\,(x-x_{vertex})^2\\y-246=-2\,(x-6)^2\\y=-2\,(x-6)^2+246[/tex]

Therefore the answer is:

[tex]y=-2\,(x-6)^2+246[/tex]

with the video game cost of x = $6