[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-143371-en":3,"doc-seo-143371-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},143371,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",6,"Technology","Functions MathPro by Lazizbek","This document, \"Functions MathPro by Lazizbek,\" details various types of mathematical functions, primarily focusing on their definitions, properties, and graphical representations. It begins by explaining linear functions (f(x) = kx + b), elaborating on the meaning of slope (k) and y-intercept (b) with a graphical example of g(x) = -x + 4. The document then delves into quadratic functions (f(x) = ax^2 + bx + c), describing their parabolic graph shape based on the sign of 'a', and detailing how to find the vertex, y-intercept, and axis of symmetry, illustrated with calculations for f(x) = x^2 - 3x + 1. Subsequent sections cover power functions (y = x^n), explaining symmetry for even and odd 'n', and fractional powers. Rational functions are introduced with the concept of asymptotes, using y = 1/(x - 2) as an example where x=2 is the vertical asymptote. Finally, exponential functions (y = a * b^x) are discussed, noting the constraint that b cannot equal 1, and differentiating between exponential growth (b>1) and decay (0\u003Cb\u003C1). The document serves as a concise guide to understanding fundamental function types in mathematics.","〰  \nFunctions | MathPro by Lazizbek  \n⁉Question: What is a function?  \n􀂶 Linear function f (x) = k *x+b  \n k and b are constants  \n straight line  \nf(x)=y  \nf(x) 􀗿 2x+3  \n Slope (k)􀗤 2 means that for every increase of 1 in x, y increases by 2.  \n Y-intercept (b)􀗤 3, so the line crosses the y-axis at 􀖨 0, 3􀖩 .  \nPractice  \nGraph the function:  \ng(x)= −x+4  \nFunctions | MathPro by Lazizbek 1  \nFunctions | MathPro by Lazizbek 2  \nQuadratic Function  \nf (x) = ax2 + bx + c  \n a, b, and c are constants  \n Graph of a quadratic function is parabola  \n if a>0, parabola opens upward  \n if a\u003C0, parabola opens downward  \nVertex, Y-intercept, Axis of Symmetry  \nx =-b/2a  \nx^2􀘁3x+1  \na=1; b=3; c=1  \nx = 􀗁3/2 􀗿 􀗁1.5  \nf(-1. 5)= vertex point  \nf(-1 . 5)􀗿􀖨􀗁1.5􀖩 ^2􀘁3* 􀖨􀗁1.5􀖩􀘁1 􀗿 2.25􀗁4.5􀘁1 􀗿 􀗁1.25  \n􀖨X; Y􀖩  \n􀖨􀗁1.5; 􀗁1.25􀖩  \nVertex  \nx = −b  \n2a  \nFunctions | MathPro by Lazizbek 3  \nY-intercept  \nf(x)⇒ f  \nf(x)=kx+b  \nf(x)=3x+1  \nf(0)􀗿3􀘎0􀘁1􀗿1  \nY-intercept point:􀖨0; 1􀖩  \nf (x) = > f (0)  \nPower  \ny = xn  \n if n is even, the graph is symmetrical about the y-axis and opens upwards  \n In n is odd, the graph has rotational symmetry around the origin y = x2 ; y = x3  \nFractional power  \nFunctions | MathPro by Lazizbek 4  \nRational Functions  \np (x) y = ~~  ~~q (x)  \nAsymptote  \n1 y = ~~ ~~  \nx − 2  \nDenominator 􀗿 0 􀜢 x-2􀗿0  \nAsymptote: x=2  \ny = 2x  \nExponential Functions  \ny = a ∗ bx  \n b cannot be equal to 1  \n if b>1, the graph shows exponential growth (grows rapidly)  \n 0\u003Cb\u003C1, the function shows exponential decay (decreases toward zero)  \nFunctions | MathPro by Lazizbek 5","cbCairD2EUv4LJeQ","https://ap.wps.com/l/cbCairD2EUv4LJeQ","pdf",188244,1,5,"English","en",105,"# Linear Function\n## Practice\n## Y-intercept\n# Quadratic Function\n# Power\n# Rational Functions\n# Exponential Functions","[{\"question\":\"What is a linear function and what do its components represent?\",\"answer\":\"A linear function is defined as f(x) = kx + b, where k and b are constants. The 'k' represents the slope, indicating how much y changes for every unit increase in x, while 'b' is the y-intercept, the point where the line crosses the y-axis.\"},{\"question\":\"How is the graph of a quadratic function determined?\",\"answer\":\"The graph of a quadratic function, f(x) = ax^2 + bx + c, is a parabola. If the coefficient 'a' is positive (a\\u003e0), the parabola opens upwards; if 'a' is negative (a\\u003c0), it opens downwards.\"},{\"question\":\"What are asymptotes in rational functions?\",\"answer\":\"In a rational function of the form y = p(x)/q(x), asymptotes are lines that the graph of the function approaches but never touches. A vertical asymptote can occur where the denominator q(x) equals zero, for example, in y = 1/(x - 2), the asymptote is x=2.\"}]","Functions MathPro by Lazizbek | PDF",1787694747,13,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"functions-mathpro-by-lazizbek","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/technology/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/functions-mathpro-by-lazizbek/143371/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-25",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is a linear function and what do its components represent?","Question",{"text":75,"@type":76},"A linear function is defined as f(x) = kx + b, where k and b are constants. The 'k' represents the slope, indicating how much y changes for every unit increase in x, while 'b' is the y-intercept, the point where the line crosses the y-axis.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the graph of a quadratic function determined?",{"text":80,"@type":76},"The graph of a quadratic function, f(x) = ax^2 + bx + c, is a parabola. If the coefficient 'a' is positive (a>0), the parabola opens upwards; if 'a' is negative (a\u003C0), it opens downwards.",{"name":82,"@type":73,"acceptedAnswer":83},"What are asymptotes in rational functions?",{"text":84,"@type":76},"In a rational function of the form y = p(x)/q(x), asymptotes are lines that the graph of the function approaches but never touches. A vertical asymptote can occur where the denominator q(x) equals zero, for example, in y = 1/(x - 2), the asymptote is x=2.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,109,112,117,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"comic",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":110,"slug":111},50,"technology",{"id":113,"doc_module":4,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},8,"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":21,"slug":137},19,"General","general"]