Question

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The function f(x) = −x^2 + 28x − 192 models the hourly profit, in dollars, a shop makes for selling sodas, where x is the number of sodas sold. Determine the vertex, and explain what it means in the context of the problem.

(12, 16); The vertex represents the maximum profit.
(12, 16); The vertex represents the minimum profit.
(14, 4); The vertex represents the maximum profit.
(14, 4); The vertex represents the minimum profit.

Answer 1

Answer:

Part A:

The vertex :  x = - b / 2 a = - 28 : ( - 2 ) = 14

y = f ( 14 ) = - 14² + 28 · 14 - 192 = 196 + 392 - 192 = 4

The vertex is ( 14, 4 ). It means that the maximum hourly profit is $4, and that the shop should sell 14 sodas to reach the maximum.

Part B :

We will solve the quadratic equation:

x 1/2 = ( - 28 +/- sqrt ( 784 - 768 ) ) / ( - 2 )

x 1 = 12,  x 2 = 16.

It means that if the shop must sell more than 12 and less then 16 sodas if they want to make a profit ( 13, 14, or 15 ). If they sell 12, or 16 they will brake even. 

I think

Answer 2

The vertex of the given function is option(C) (14,4) The vertex represents the maximum profit

What is a Quadratic function?

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two

What is Vertex of Quadratic function?

The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry

Given,

$f(x) =-x^{2} +28x-192$

Vertex

x= $-\frac{b}{2a}$

x= $-\frac{28}{2(-1)} =14$

y= $-14^{2}+28(14)-192 =4$

Vertex (14,4)

It means that the maximum hourly profit is $4, and that the shop should sell 14 sodas to reach the maximum.

Hence, the vertex of the given function is option(C) (14,4) The vertex represents the maximum profit

Learn more about Quadratic function and Vertex of quadratic function here

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