options: Understand and apply properties of operations and the relationship between addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Interpret the structure of expressions.

Perform arithmetic operations on polynomials.

Work with addition and subtraction equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Write expressions in equivalent forms to solve problems.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with radicals and integer exponents.

Represent and solve problems involving addition and subtraction.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Solve systems of equations.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Create equations that describe numbers or relationships

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve equations and inequalities in one variable.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Work with equal groups of objects to gain foundations for multiplication.

Represent and analyze quantitative relationships between dependent and independent variables.

Use the four operations with whole numbers to solve problems.

Understand properties of multiplication and the relationship between multiplication and division.

Add and subtract within 20.

Use polynomial identities to solve problems.

Understand the connections between proportional relationships, lines, and linear equations.

Represent and solve equations and inequalities graphically.

Analyze and solve linear equations and pairs of simultaneous linear equations.
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.


correct_option_index: 2

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