few_shots: Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Your response:
{
  "explanation": "This problem applies trigonometric identities, including the angle addition formula for cosine.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS
options: Represent and solve problems involving multiplication and division.

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

Represent and solve equations and inequalities graphically.

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

Analyze patterns and relationships.

Multiply and divide within 100.

Perform arithmetic operations on polynomials.

Represent and solve problems involving addition and subtraction.

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

Solve systems of equations.

Use the four operations with whole numbers to solve problems.

Add and subtract within 20.

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

Work with addition and subtraction equations.

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

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

Interpret the structure of expressions.

Understand the relationship between zeros and factors of polynomials.

Gain familiarity with factors and multiples.

Represent and analyze quantitative relationships between dependent and independent variables.

Work with radicals and integer exponents.

Use properties of operations to generate equivalent expressions.

Rewrite rational expressions.

Create equations that describe numbers or relationships

Write expressions in equivalent forms to solve problems.

Generate and analyze patterns.

Solve equations and inequalities in one variable.

Use polynomial identities to solve problems.

Reason about and solve one-variable equations and inequalities.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Write and interpret numerical expressions.

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

Understand solving equations as a process of reasoning and explain the reasoning.
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: 16

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