Charlie sells beaded bracelets as a side job. Charlie will have a maximum profit of $12 when he sells 5 bracelets. Construct Charlie’s profit function, p(x) , to model his sales, in dollars, as a function of the number of bracelets he sells. p(x)= [a](x−[h])2+[k] Find [a]
To find the value of [a], we can plug in the given values of 5 bracelets sold for a maximum profit of $12.
p(x) = a(x - h)^2 + k
12 = a(5 - h)^2 + k
Since we are given that Charlie has a maximum profit of $12 when he sells 5 bracelets, we have:
p(5) = 12
12 = a(5 - h)^2 + k
Since p(5) = 12:
12 = a(5 - h)^2 + k
Next, we can substitute in the number of bracelets sold (5) for x in the formula:
12 = a(5 - h)^2 + k
Now, we need to solve for the value of a.
Since we are only given one data point, we can conclude that the maximum profit is at the vertex of the parabola. That means the x-coordinate of the vertex is 5.
Therefore, h = 5.
Plugging in the value of h:
12 = a(5 - 5)^2 + k
12 = a(0)^2 + k
12 = 0 + k
k = 12
Now, we can plug h = 5 and k = 12 back into the equation:
12 = a(5 - 5)^2 + 12
12 = a(0)^2 + 12
12 = a(0) + 12
12 = 0 + 12
a = 0
Therefore, the value of [a] is 0.