standard_description: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
problem_activity: Task
The profit, $P$ (in thousands of dollars), that a company makes selling an item is a quadratic function of the price, $x$ (in dollars), that they charge for the item. The following expressions for $P(x)$ are equivalent:
$$\begin{align}P(x)=&-2x^2+24x-54\\
P(x)=&-2(x-3)(x-9)\\
P(x)=&-2(x-6)^2+18
\end{align}$$
Which of the equivalent expressions for $P(x)$ reveals the price which gives a profit of zero without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find a price which gives a profit of zero.
Which of the equivalent expressions for $P(x)$ reveals the profit when the price is zero without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find the profit when the price is zero.
Which of the equivalent expressions for $P(x)$ reveals the price which produces the highest possible profit without changing the form of the expression?
P(x)= -2x$^2$+24x-54
P(x)= -2(x-3)(x-9)
P(x)= -2(x-6)$^2$+18
Find the price which gives the highest possible profit.
For each of parts (a), (c), and (e), students choose which form of the function is displayed, and then use that to answer the corresponding questions in parts (b), (d), and (f) respectively.  Students are assessed both on selecting the correct form of the function and on their numerical answer for each part.


aligns: no