[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-311485-105":59,"doc-detail-311485-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","the-ultimate-ap-precalculus-cheat-sheet","The Ultimate AP Precalculus Cheat Sheet","","Covers key AP Precalculus concepts in a compact, exam-focused reference organized by units: rates of change and polynomial behavior, rational functions with asymptotes and holes, and function transformations and modeling. Includes rules for average rate of change, concavity, zeros and end behavior, plus domain restrictions for rational expressions. Adds guidance for exponential functions, sequences, equivalent forms, and model selection using data patterns and residual reasoning, with FRQ answer tips and common pitfalls.",{"@graph":69,"@context":123},[70,84,106],{"@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/the-ultimate-ap-precalculus-cheat-sheet/311485/",{"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/the-ultimate-ap-precalculus-cheat-sheet/311485.png","ImageObject",300,407,{"name":92,"@type":93},"Grenda","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-05","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",35,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you compute average rate of change from two points on a graph?","Question",{"text":113,"@type":114},"Use the secant slope formula: [f(b)−f(a)]/(b−a) using the two points (a,f(a)) and (b,f(b)).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"When identifying features of a rational function, what must be done first?",{"text":118,"@type":114},"Factor both the numerator and denominator completely before determining zeros, vertical asymptotes, and holes.",{"name":120,"@type":111,"acceptedAnswer":121},"How can you justify choosing a model like a polynomial or exponential on an FRQ?",{"text":122,"@type":114},"Cite the data pattern (e.g., constant % change for exponential, multiple turning points for higher-degree polynomials) rather than only claiming the fit is good, and mention domain restrictions to address extrapolation risk.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},311485,1789894361,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":81,"preview_page_count":4},7971474921005,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","[The Ultimate AP Precalculus Cheat Sheet](The Ultimate AP Precalculus Cheat Sheet www.albert.io)[ www.albert.io](The Ultimate AP Precalculus Cheat Sheet www.albert.io)  \n\n| Unit 1A: Rates of Change & Polynomials (30–40%) |\n| --- |\n| Change in Tandem & Rates |\n| Change in tandem: as input ↑ , describe how output changes —↑ , ↓ , or neither. Use tables, graphs, eqs.\u003Cbr>Avg rate of change = [f(b)−f(a)]/(b−a) = slope of secant line b/w (a,f(a)) & (b,f(b)) .\u003Cbr>Concavity: if rate of change ↑ → concave up; if rate of change ↓→ concave down.\u003Cbr>Linear fn: constant rate of change. Quadratic: rate of change is linear (changes at constant rate) .\u003Cbr>Positive rate → fn ↑ . Negative rate → fn ↓ . Rate = 0 at local extrema.\u003Cbr>Rates from table: compute Δy/Δx for consecutive rows. If 2nd differences constant → quadratic.\u003Cbr>Polynomial Functions |\n| Degree n poly: at most n real zeros & n−1 turning pts. Leading coeff & degree → end behavior.\u003Cbr>Even degree: both ends same dir. Odd degree: ends opposite dirs. + lead coeff → rises right.\u003Cbr>Complex zeros: conjugate pairs (a+bi & a−bi) . Total zeros = degree (w/ multiplicity) .\u003Cbr>Multiplicity: even → bounces at zero. Odd → crosses. Higher mult → flatter near zero.\u003Cbr>End Behavior & Zeros |\n| Zeros from factors: (x−a) is factor → x=a is zero. Factor completely to find all real zeros.\u003Cbr>IVT: if f is continuous & f(a) and f(b) have opposite signs → at least one zero b/w a & b.\u003Cbr>End behavior: lim x→∞ and lim x→−∞ determined by leading term. State using limit notation on FRQs.\u003Cbr>Graph sketch: plot zeros, check multiplicities, find end behavior, connect smoothly thru turning pts. |\n| ✓ FRQ tip: always state end behavior using limit notation — lim x→±∞ f(x) = ±∞ . |\n| ⚠ A poly of degree n can have FEWER than n real zeros b/c some may be complex. |\n\n\n| Unit 1B: Rational Functions (30–40%) |\n| --- |\n| Rational Function Basics |\n| Rational fn = p(x)/q(x) where p & q are polynomials, q(x) ≠ 0. Domain excludes zeros of q.\u003Cbr>Vertical asymptote at x=a if a is zero of denom but NOT of numer (or mult in denom > mult in numer) .\u003Cbr>Hole at x=c if c is zero of both numer & denom w/ mult in numer ≥ mult in denom.\u003Cbr>Hole y-value: cancel common factor, evaluate simplified fn atx=c. Or: y = lim x→c r(x) .\u003Cbr>End Behavior & Asymptotes |\n| Degree Comparison End Behavior |\n| deg(num) \u003C deg(den) HA at y=0 |\n| deg(num) = deg(den) HA at y=leading coeffs ratio |\n| deg(num) = deg(den)+1 Slant asymptote (long div) |\n| deg(num) > deg(den)+1 No HA; behaves like quotient |\n| Graph crosses HA: possible for finite x-values, but not as x→±∞ . Set r(x)=HA value to find crossing.\u003Cbr>Long Division & Representations |\n| Poly long division: f(x) = g(x)·q(x) + r(x) . Quotient q(x) gives slant asymptote when applicable.\u003Cbr>Factored form → zeros, asymptotes, holes. Standard form → end behavior. Use both strategically.\u003Cbr>Binomial theorem: (a+b)^n uses Pascal's Triangle for coeffs.\u003Cbr>Row n: 1 , n, n(n−1)/2 , ...\u003Cbr>Sign of r(x): determined by signs of factors in numer & denom. Use sign chart w/ all zeros & VAs.\u003Cbr>Zeros of rational fn: set numer = 0 & solve. Exclude any x that also makes denom = 0 (those are holes) . |\n| ⚠ Hole ≠ VA: factor first! If (x−c) cancels fully → hole at x=c, NOT an asymptote. |\n| ✓ Always factor numer & denom completely before identifying features of the graph. |\n\n\n| Transformations & Modeling |\n| --- |\n| Function Transformations |\n| g(x) = f(x) + k: vertical shift ↑k (if k>0) or ↓|k| (if k\u003C0) .\u003Cbr>g(x) = f(x + h): horizontal shift LEFT h units (if h>0) or RIGHT |h|(if h\u003C0) . Note: OPPOSITE sign.\u003Cbr>g(x) = a·f(x): vertical dilation by |a| . If a\u003C0 → reflect over x-axis. g(x) = f(bx): horizontal dilation by 1/|b| . If b\u003C0 → reflect over yaxis.\u003Cbr>Order matters: apply horizontal transforms to input first, then vertical transforms to output.\u003Cbr>Domain/range shift: transformations alter domain & range of parent fn accordingly.\u003Cbr>Model Selection |\n|  |\n| Pattern in Data Model Type |\n| Constant rate of change Linear |\n|","cbCaijRoQZ37yB6W","https://ap.wps.com/l/cbCaijRoQZ37yB6W","pdf",192321,"English","# Unit 1A: Rates of Change & Polynomials (30–40%)\n## Change in Tandem & Rates\n## Polynomial Functions\n## End Behavior & Zeros\n# Unit 1B: Rational Functions (30–40%)\n## Rational Function Basics\n## Degree Comparison End Behavior\n## Long Division & Representations\n# Transformations & Modeling\n## Function Transformations\n## Model Selection\n# Unit 2A: Exponential Functions (27–40%)\n## Sequences\n## Exponential Functions\n## Manipulation & Equivalent Forms","[{\"question\":\"How do you compute average rate of change from two points on a graph?\",\"answer\":\"Use the secant slope formula: [f(b)−f(a)]/(b−a) using the two points (a,f(a)) and (b,f(b)).\"},{\"question\":\"When identifying features of a rational function, what must be done first?\",\"answer\":\"Factor both the numerator and denominator completely before determining zeros, vertical asymptotes, and holes.\"},{\"question\":\"How can you justify choosing a model like a polynomial or exponential on an FRQ?\",\"answer\":\"Cite the data pattern (e.g., constant % change for exponential, multiple turning points for higher-degree polynomials) rather than only claiming the fit is good, and mention domain restrictions to address extrapolation risk.\"}]","The Ultimate AP Precalculus Cheat Sheet | PDF"]