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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.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ap-precalculus-practice-exam-1-free-response-questions-scoring-guidelines/449185/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ap-precalculus-practice-exam-1-free-response-questions-scoring-guidelines/449185.png","ImageObject",300,407,{"name":92,"@type":93},"Riley West","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-04","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What calculator requirement applies to these free-response questions?","Question",{"text":112,"@type":113},"Both Question 1 and Question 2 specify that a graphing calculator is required for their scoring parts.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is h(4) found in Question 1, Part (A)(i)?",{"text":117,"@type":113},"Compute h(4)=g(f(4)) using the table value f(4)=3, then evaluate g(3)=2ln3 to obtain a decimal approximation.",{"name":119,"@type":110,"acceptedAnswer":120},"What modeling approach is used in Question 2?",{"text":121,"@type":113},"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.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},449185,1791057423,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},1099523885074,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Free-Response Questions Scoring Guidelines  \nScoring Guidelines for Question 1: Function Concepts  \nPart A: Graphing calculator required 6 points  \n\n| x | 1 | 2 | 3 | 4 | 5 |\n| --- | --- | --- | --- | --- | --- |\n| f (x ) | 1 | 3 | 5 | 3 | 1 |\n\nThe 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 .  \nModel Solution Scoring  \n(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.  \n(ii) Find all values of x for which f (x ) = 3 , or indicate there are no such values.  \n\n| (i) h (4) = g (f (4)) = g (3) = 2ln3 = 2.197 | Value | 1 point\u003Cbr> 2.A  |\n| --- | --- | --- |\n| (ii) From the table, f (x ) = 3 when x = 2 and x = 4. | Values | 1 point\u003Cbr> 2.A  |\n\nTotal for part (A) 2 points  \n\n| (B) | (i) Find all values of x, as decimal approximations, for which g(x ) = 3, or indicate there are no such values.\u003Cbr>(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit. |  |  |\n| --- | --- | --- | --- |\n|  | (i) g (x )  3  2lnx = 3\u003Cbr>x = 4.482 (OR 4 .481) | Answer 1 point\u003Cbr> 1.A  |  |\n|  | (ii) The function g is increasing. As x increases without bound, g(x) increases without bound. Therefore, xlim g (x) ~~ ~~  . | End behavior with limit\u003Cbr>notation | 1 point\u003Cbr> 3.A  |\n|  | Total for part (B) 2 points |  |  |\n| (C) | (i) Determine if f has an inverse function.\u003Cbr>(ii) Give a reason for your answer based on the definition of a function and the table of values off (x ). |  |  |\n|  | (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  |  |  |\n|  | (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.\u003Cbr>A reason that only states \"fails the horizontal line test\" is not sufficient. | Reason 1 point\u003Cbr> 3.C  |  |\n|  | Total for part (C) 2 points |  |  |\n\nTotal for Question 1 6 points  \nAP Precalculus Practice Exam 1 Free-Response Questions Scoring Guidelines V.1 | 59  \nScoring Guidelines for Question 2: Modeling a Non-Periodic Context  \nPart A: Graphing calculator required 6 points  \nAn 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.  \nThe 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.  \nModel Solution Scoring  \n(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 ) .  \n(ii) Find the values for a and b as decimal approximations.  \n\n| (i) Because P(2) = 59 and P(8) = 118 , two equations to find a and b are | Two equations | 1 point |\n| --- | --- | --- |\n| 2 ab = 59\u003Cbr>ab8 = 118. |  |  1.C  |\n| (ii)\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>a = 46.828331\u003Cbr>P (t) = 46.828 (1.122)t | Values of a and b | 1 point\u003Cbr> 1.C  |\n\nTotal for part (A) 2 points  \n(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.  \n(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.  \n(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.  \n\n| P (8)−P(2) (118 −5","cbCaiiPnYfbtxBs1","https://ap.wps.com/l/cbCaiiPnYfbtxBs1","pdf",649747,"English","# Scoring Guidelines for Question 1: Function Concepts\n## Part A\n## Part B\n## Part C\n# Scoring Guidelines for Question 2: Modeling a Non-Periodic Context\n## Part A\n## Part B\n## Part C","[{\"question\":\"What calculator requirement applies to these free-response questions?\",\"answer\":\"Both Question 1 and Question 2 specify that a graphing calculator is required for their scoring parts.\"},{\"question\":\"How is h(4) found in Question 1, Part (A)(i)?\",\"answer\":\"Compute h(4)=g(f(4)) using the table value f(4)=3, then evaluate g(3)=2ln3 to obtain a decimal approximation.\"},{\"question\":\"What modeling approach is used in Question 2?\",\"answer\":\"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.\"}]","AP Precalculus Practice Exam 1 Free-Response Questions Scoring Guidelines | PDF",1790729492,18]