Question

1. a certain company wants to maximize profits. they need to price (x) the product to maximize profits (y). which of the following quadratic equations would be best for the job.

a) y = - 4x² + 40x - 104
b) y = - 4(x - 4)(x - 8)
c) y = - 4(x - 4)² + 24

Explanation:

Step1: Recall vertex - form of quadratic function

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. If $a<0$, the parabola opens downwards and the vertex represents the maximum point of the function.

Step2: Analyze each option

• Option a: $y=-4x^{2}+40x - 104$ is in standard form $y = ax^{2}+bx + c$. We would need to complete the square to find the vertex.
• Option b: $y=-4(x - 4)(x - 8)=-4(x^{2}-12x + 32)=-4x^{2}+48x-128$, also in standard form and requires completing the square.
• Option c: $y=-4(x - 4)^{2}+24$ is in vertex - form $y = a(x - h)^{2}+k$ with $a=-4<0$, $h = 4$ and $k = 24$. The vertex of the parabola is $(4,24)$, which means the maximum value of the function is $y = 24$ when $x = 4$.

Answer:

c) $y=-4(x - 4)^{2}+24$