options: Perform arithmetic operations on polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

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

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

Represent and solve problems involving multiplication and division.

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

Solve systems of equations.

Multiply and divide within 100.

Solve equations and inequalities in one variable.

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

Add and subtract within 20.

Interpret the structure of expressions.

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

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.

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

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

Generate and analyze patterns.

Rewrite rational expressions.

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

Represent and solve equations and inequalities graphically.

Represent and solve problems involving addition and subtraction.

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

Work with addition and subtraction equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze patterns and relationships.

Use polynomial identities to solve problems.

Understand the relationship between zeros and factors of polynomials.

Write and interpret numerical expressions.

Gain familiarity with factors and multiples.

Create equations that describe numbers or relationships

Reason about and solve one-variable equations and inequalities.

Use the four operations with whole numbers to solve problems.
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: 11

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