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]
Find [h]
Find [k]
To find the values of [a], [h], and [k] in the profit function p(x) = [a] (x - [h])² + [k], we can use the information given that Charlie will have a maximum profit of $12 when he sells 5 bracelets.
First, we know that the profit function is in the form of a quadratic equation, where the x-coordinate of the vertex represents the number of bracelets sold to maximize profit. Since the vertex is at x = 5, we can use this information to find [h].
[h] = 5
Next, we know that the maximum profit is $12, which means that when x = 5, p(x) = 12. We can use this information to find [a] and [k].
p(5) = [a] (5 - 5)² + [k]
12 = [a] (0) + [k]
12 = [k]
So, [k] = 12
Now, we have [h] = 5 and [k] = 12. Substituting these values back into the original profit function equation, we get:
p(x) = [a] (x - 5)² + 12
Since we still need to find [a], plug in the point (5, 12) to find [a]:
12 = [a] (5 - 5)² + 12
12 = 0 + 12
12 = 12
Therefore, [a] = 0
Therefore, Charlie's profit function p(x) is:
p(x) = 0(x - 5)² + 12
p(x) = 12