# Profit of a company

Alignments to Content Standards: A-SSE.B.3

The proﬁt that a company makes selling an item (in thousands of dollars) depends on the price of the item (in dollars). If $p$ is the price of the item, then three equivalent forms for the proﬁt are: \begin{align} \text{Standard form:}& -2p^2 + 24p - 54 \\ \text{Factored form:}& -2(p - 3)(p - 9) \\ \text{Vertex form:}& -2(p - 6)^2 + 18. \end{align} Which form is most useful for ﬁnding

1. The prices that give a proﬁt of zero dollars?

2. The proﬁt when the price is zero?

3. The price that gives the maximum proﬁt?

## IM Commentary

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From Algebra: Form and Function, McCallum et al., Wiley 2010 )

## Solution

1. The factored form gives the values of $p$ that make the profit zero. Since factored form is $-2(p-3)(p-9)$, the profit is zero when $p = 3$ or $p = 9$. The company breaks even if the price charged for the product is $\$3$or$\$9.$

2. The standard form is the easiest one to use to find the profit when the price is zero. Substituting $p = 0$ into the standard form $-2p^2 + 24p - 54$, we see that the profit is $-54$ (in thousands of dollars) when the price is zero. If the company gives the product away for free, it loses $\$54,\!000$. 3. The vertex form shows us what price maximizes profit. From the expression$-2(p-6)^2 + 18$, we see that the maximum profit is$18$thousand dollars, and it occurs when$p = 6$. The company should charge a price of$\$6$ for this product.