Find the vertex, the y-intercept, and the x-intercepts (if any exist), and graph the function.
-The daily profit in dollars of a specialty cake shop is described by the function $\mathrm { P } ( \mathrm { x } ) = - 6 \mathrm { x } ^ { 2 } + 2028 \mathrm { x } - 27,216$, where $x$ is the number of cakes prepared in one day. The maximum profit for the company occurs at the vertex of the parabola. What is the maximum profit?
A) $\$ 144,150$
B) $\$ 169.00$
C) $\$ 198,582$
D) $\$ 28,561$

Question

Quizplus · Accepted Answer

The vertex of a parabola given by $P(x) = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$. For the given function $P(x) = -6x^2 + 2028x - 27216$, we have $a = -6$ and $b = 2028$, so the x-coordinate of the vertex is $x = -\frac{2028}{2(-6)} = 169$. Substituting $x = 169$ into the function gives the maximum profit, $P(169) = -6(169)^2 + 2028(169) - 27216 = \$144,150$, which is the maximum profit.

Find the Vertex, the Y-Intercept, and the X-Intercepts (If Any P(x)=−6x2+2028x−27,216\mathrm { P } ( \mathrm { x } ) = - 6 \mathrm { x } ^ { 2 } + 2028 \mathrm { x } - 27,216P(x)=−6x2+2028x−27,216

Find the Vertex, the Y-Intercept, and the X-Intercepts (If Any $\mathrm { P } ( \mathrm { x } ) = - 6 \mathrm { x } ^ { 2 } + 2028 \mathrm { x } - 27,216$