Respuesta :
now, we know that he's charging $12 per session to each of his students, and he has 14 students currently, so his revenue is just 14 * 12 or 168 bucks.
now, let's take a peek as the session price goes up in jumps of 2, from 12, to 14, 16, 18 and so on, as each jumps happen, the students drop by 1, from 14, to 13, to 12 and so on.
[tex]\bf \begin{array}{ccccllll} \stackrel{increase}{x}&\stackrel{new}{price}&students&\stackrel{revenue}{y}\\ ------&------&------&------\\ 0&12&14&168\\ 1&14&13&182\\ 2&16&12&192\\ 3&18&11&198\\ \boxed{4}&20&10&\boxed{200}\\ 5&22&9&198\\ 6&24&8&192\\ 7&26&7&182\\ 8&28&6&168\\ ..&..&..&.. \end{array}[/tex]
notice, the revenue starts off at 168, goes up up, reaches 200 bucks and then starts to drop back down.
thus, that means the U-turn or vertex of that revenue function is at 4,200, namely h = 4, and k = 200
[tex]\bf \qquad \textit{parabola vertex form}\\\\ \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ \stackrel{c(x)}{y}=a(x-\stackrel{h}{4})\stackrel{k}{+200} \\\\\\ \textit{now, we also know that } \begin{cases} x=2\\ y=192 \end{cases}\implies 192=a(2-4)^2+200 \\\\\\ -8=a(-2)^2\implies -8=4a\implies \cfrac{-8}{4}=a\implies -2=a \\\\\\ [/tex]
[tex]\bf therefore\qquad \stackrel{ c(x)}{y}=-2(x-4)^2+200\impliedby \textit{let's expand that} \\\\\\ y=-2(x^2-8x+16)+200\implies y=-2x^2+16x-32+200 \\\\\\ \stackrel{y}{c(x)}=-2x^2+16x+168[/tex]
now, let's take a peek as the session price goes up in jumps of 2, from 12, to 14, 16, 18 and so on, as each jumps happen, the students drop by 1, from 14, to 13, to 12 and so on.
[tex]\bf \begin{array}{ccccllll} \stackrel{increase}{x}&\stackrel{new}{price}&students&\stackrel{revenue}{y}\\ ------&------&------&------\\ 0&12&14&168\\ 1&14&13&182\\ 2&16&12&192\\ 3&18&11&198\\ \boxed{4}&20&10&\boxed{200}\\ 5&22&9&198\\ 6&24&8&192\\ 7&26&7&182\\ 8&28&6&168\\ ..&..&..&.. \end{array}[/tex]
notice, the revenue starts off at 168, goes up up, reaches 200 bucks and then starts to drop back down.
thus, that means the U-turn or vertex of that revenue function is at 4,200, namely h = 4, and k = 200
[tex]\bf \qquad \textit{parabola vertex form}\\\\ \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ \stackrel{c(x)}{y}=a(x-\stackrel{h}{4})\stackrel{k}{+200} \\\\\\ \textit{now, we also know that } \begin{cases} x=2\\ y=192 \end{cases}\implies 192=a(2-4)^2+200 \\\\\\ -8=a(-2)^2\implies -8=4a\implies \cfrac{-8}{4}=a\implies -2=a \\\\\\ [/tex]
[tex]\bf therefore\qquad \stackrel{ c(x)}{y}=-2(x-4)^2+200\impliedby \textit{let's expand that} \\\\\\ y=-2(x^2-8x+16)+200\implies y=-2x^2+16x-32+200 \\\\\\ \stackrel{y}{c(x)}=-2x^2+16x+168[/tex]
