AP Precalculus Practice Exam 1 Free-Response Questions Scoring Guidelines

Sep 30, 2026

Scoring guidelines for AP Precalculus free-response questions provide point allocations and model-solution steps for Question 1 on function concepts and Question 2 on modeling a non-periodic context. It includes requirements for graphing calculator use, details for computing function compositions, solving equations from given tables, checking inverse-function existence using one-to-one reasoning, and analyzing exponential modeling via P(t)=ab^t. The document also specifies how to estimate average rate of change and apply it to predict plant counts with clear justification for limit and end-behavior reasoning.

LanguageEnglish

CategoryExam

FormatPDF

Pages7

Reading Time18 min

AccessFree to download

Scoring Guidelines for Question 1: Function Concepts

Part A

Part B

Part C

Scoring Guidelines for Question 2: Modeling a Non-Periodic Context

Part A

Part B

Part C

Free-Response Questions Scoring Guidelines Scoring Guidelines for Question 1: Function Concepts Part A: Graphing calculator required 6 points | x | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | f (x ) | 1 | 3 | 5 | 3 | 1 | The domain off consists of the five real numbers 1, 2 , 3, 4 , and 5 . The table defines the function f for these values. The function g is given by g (x ) = 2ln x . Model Solution Scoring (A) (i) The function h is defined by h (x ) = (g 􀁆 f )(x ) = g (f (x )). Find the value of h(4) as a decimal approximation, or indicate that it is not defined. (ii) Find all values of x for which f (x ) = 3 , or indicate there are no such values. | (i) h (4) = g (f (4)) = g (3) = 2ln3 = 2.197 | Value | 1 point
2.A | | --- | --- | --- | | (ii) From the table, f (x ) = 3 when x = 2 and x = 4. | Values | 1 point
2.A | Total for part (A) 2 points | (B) | (i) Find all values of x, as decimal approximations, for which g(x ) = 3, or indicate there are no such values.
(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit. | | | | --- | --- | --- | --- | | | (i) g (x ) 3 2lnx = 3
x = 4.482 (OR 4 .481) | Answer 1 point
1.A | | | | (ii) The function g is increasing. As x increases without bound, g(x) increases without bound. Therefore, xlim g (x) ~~ ~~ . | End behavior with limit
notation | 1 point
3.A | | | Total for part (B) 2 points | | | | (C) | (i) Determine if f has an inverse function.
(ii) Give a reason for your answer based on the definition of a function and the table of values off (x ). | | | | | (i) f does not have an inverse function on its domain of the five real Answer 1 point numbers 1, 2, 3, 4, and 5. 1.C | | | | | (ii) There are output values of f that are not mapped from unique input values. (The function fis not one-to-one.) Because f(1) = 1 and f (5) = 1, the inverse function on this domain does not exist.
A reason that only states "fails the horizontal line test" is not sufficient. | Reason 1 point
3.C | | | | Total for part (C) 2 points | | | Total for Question 1 6 points AP Precalculus Practice Exam 1 Free-Response Questions Scoring Guidelines V.1 | 59 Scoring Guidelines for Question 2: Modeling a Non-Periodic Context Part A: Graphing calculator required 6 points An ecologist began studying a certain type of plant species in a wetlands area in 2013. In 2015 ( t = 2), there were 59 plants. In 2021 ( t = 8), there were 118 plants. The number of plants in this species can be modeled by the function P given by P (t ) = abt , where P(t ) is the number of plants during year t , and t is the number of years since 2013. Model Solution Scoring (A) (i) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for P(t ) . (ii) Find the values for a and b as decimal approximations. | (i) Because P(2) = 59 and P(8) = 118 , two equations to find a and b are | Two equations | 1 point | | --- | --- | --- | | 2 ab = 59
ab8 = 118. | | 1.C | | (ii)

a = 46.828331
P (t) = 46.828 (1.122)t | Values of a and b | 1 point
1.C | Total for part (A) 2 points (B) (i) Use the given data to find the average rate of change of the number of plants, in plants per year, from t = 2 to t = 8 years. Express your answer as a decimal approximation. Show the computations that lead to your answer. (ii) Use the average rate of change found in (i) to estimate the number of plants for t = 10 years. Show the work that leads to your answer. (iii) The average rate of change found in (i) can be used to estimate the number of plants during year t for t > 10 years. Will these estimates, found using the average rate of change, be less than or greater than the number of plants predicted by the model P during year t for t > 10 years? Explain your reasoning. | P (8)−P(2) (118 −5

FAQs

What calculator requirement applies to these free-response questions?

Both Question 1 and Question 2 specify that a graphing calculator is required for their scoring parts.

How is h(4) found in Question 1, Part (A)(i)?

Compute h(4)=g(f(4)) using the table value f(4)=3, then evaluate g(3)=2ln3 to obtain a decimal approximation.

What modeling approach is used in Question 2?

The plant population is modeled with an exponential function P(t)=ab^t, where t is the number of years since 2013. The guidelines use the given data to form equations for a and b, then apply average rate of change reasoning to estimate later values.

Sep 30, 2026

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