FRQs 1-2 – Graphing Calculator Required

Overview

• Section II, Part A contains 2 free-response questions in 35 minutes
• Part A makes up 18.75% of the total exam score
• A graphing calculator is required
• Calculators should be in radian mode
• FRQ 1 and FRQ 2 are each worth 6 points
• The AP Precalculus free-response section has 4 six-point questions, each weighted equally and scored on an analytic scale
• Two of the four free-response questions on the full exam incorporate a real-world function modeling context
• Exam content comes from Units 1, 2, and 3

The graphing-calculator FRQs are:

• FRQ 1: Function Concepts - Unit 1 and Unit 2 focus, no real-world context
• FRQ 2: Modeling a Non-Periodic Context - Unit 1 and Unit 2 focus, real-world function modeling context

The official task-model overview places FRQ 1 and FRQ 2 together in Section II, Part A. FRQ 1 is the graphing-calculator function-concepts question without a real-world context, and its defining setup is multiple representations: graphical, numerical, and analytical. FRQ 2 is the graphing-calculator non-periodic real-world modeling question and is one of the two free-response questions on the exam that incorporate a real-world function modeling context.

Official Part A calculator instructions:

• Students are expected to use a graphing calculator for tasks such as producing graphs and tables, evaluating functions, solving equations, and performing computations.
• For Part A, the calculator must be in radian mode.
• It may be helpful to use the graphing calculator to store computed values for constants, functions you are working with, solutions to equations, and intermediate values so computations maintain precision.
• Avoid rounding intermediate computations; unless otherwise specified, decimal approximations should be accurate to three places after the decimal point.

Technology capabilities relevant to calculator-required Precalculus exam tasks include graphing functions and analyzing graphs, generating tables, finding real zeros, finding points of intersection, finding minima and maxima, finding numerical solutions to one-variable equations, and using regression or modeling support when a task provides data.

Show all work. Work is scored for correctness and completeness, including supporting work and answers; answers without supporting work may not receive credit in cases where supporting work is requested. Unless otherwise specified, the domain of a function $f$ is the set of all real numbers $x$ for which $f(x)$ is a real number.

FRQ 1: Function Concepts

FRQ 1 presents functions expressed graphically, numerically, and analytically. The official task model includes three parts and can ask students to work across these representations with several function concepts, including:

• function composition
• inverse functions
• function input-output values
• zeros of a function
• end behavior of a function
• identifying an appropriate function type to construct a function model

Official point count by skill for FRQ 1:

For inverse functions, focus on the relationship between input and output. If $f^{-1}(a)=b$, then $f(b)=a$. Depending on the representation, that may mean reading a graph, using a table, solving with technology, or interpreting an equation.

FRQ 2: Modeling a Non-Periodic Context

FRQ 2 presents a real-world, non-periodic modeling context. The CED describes the task model this way:

• FRQ 2 has three parts: Part A, Part B, and Part C.
• Part A asks students to construct a function model by building a system of equations and finding parameters using a method of choice.
• Possible function types include polynomial, piecewise-defined, exponential, and logarithmic.
• Part B asks students to calculate, apply, and reason with average rates of change and their units.
• Part C asks students to justify a conclusion about assumptions, limitations, or appropriate scope of the model.

Official point count by skill for FRQ 2:

When interpreting a model, base conclusions on the specific function, parameters, domain, and context. Do not assume all exponential models increase or all logarithmic models decrease; the behavior depends on the actual model.

Task Verbs

Common AP Precalculus free-response task verbs include:

• Construct/write a function/expression/equation/model: Develop an analytical representation, with and without technology, consistent with a scenario, data set, or criteria.
• Describe: Develop a verbal representation consistent with a scenario, data set, function representation, or criteria.
• Determine/find/identify: Apply appropriate methods or processes.
• Estimate/compare: Use a function representation to find approximate values and/or compare results.
• Explain/give a reason/provide a rationale/justify: Use information from the scenario or function representation to support conclusions.
• Express/indicate: Provide information or a result in a requested form, including units when needed.
• Interpret: Describe the connection between a mathematical expression or solution and its contextual meaning.
• Plot and label, sketch and label: Develop a graphical representation consistent with a scenario, data set, or other criteria.
• Rewrite: Apply appropriate methods to determine equivalent analytical representations of an expression.
• Solve: Apply appropriate methods to determine solutions to an equation or inequality.

Calculator Use

Use the graphing calculator as an official Part A tool, especially for graphs, tables, function evaluation, equation solving, and computations. Still show supporting work when requested. Calculator output should support the mathematical claim; it should not replace the explanation.

Store computed values for constants, functions you are working with, solutions to equations, and intermediate values when useful so later calculations preserve precision. Computations that use stored information help maintain accuracy in final answers. Round final decimal approximations to three places unless the prompt says otherwise.

Final Notes

FRQs 1 and 2 reward accurate use of representations, strategic technology, and contextual reasoning. Stay close to the prompt, use units in real-world contexts, and justify conclusions from the given model or representation.