Profit of a company

Task

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.