[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-605565-105":59,"doc-detail-605565-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","chapter-five-graphing-polynomials-end-behavior-finding-the-zeroes-of-the-graph","Chapter Five - Graphing Polynomials - End Behavior - Finding the Zeroes of the Graph","","Graphing polynomials focuses on two core ideas: end behavior and the behavior at zeros. End behavior is determined by the leading term, especially the highest-degree exponent and the sign of its coefficient, which dominate the function for large positive or negative x. The discussion then turns to zeros, using examples to distinguish crossing from “bouncing” based on how the polynomial touches the x-axis.",{"@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/chapter-five-graphing-polynomials-end-behavior-finding-the-zeroes-of-the-graph/605565/",{"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/chapter-five-graphing-polynomials-end-behavior-finding-the-zeroes-of-the-graph/605565.png","ImageObject",300,407,{"name":92,"@type":93},"nayy☆","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-10-16",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you determine a polynomial’s end behavior from the leading term?","Question",{"text":112,"@type":113},"Look at the leading term with the highest degree. For large |x|, that term dominates, so the exponent and its sign determine whether the graph rises or falls as x goes to the left or right.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why does an even or odd exponent affect the sign of the leading term differently?",{"text":117,"@type":113},"With an even exponent, a negative x-value multiplied even times gives a positive result, so that side tends to rise. With an odd exponent, a negative x multiplied an odd number of times stays negative, so that side tends to fall.",{"name":119,"@type":110,"acceptedAnswer":120},"What does it mean when a polynomial “bounces” at the x-axis?",{"text":121,"@type":113},"A “bounce” means the graph touches the x-axis but does not cross it. The examples show this often happens with even-power terms, where the function value remains nonnegative near the zero.","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},605565,1792502351,{"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":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":144,"read_time":145},962090893153,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Chapter Five  \nGraphing Polynomials  \nEnd Behavior  \nI mentioned end behavior in the last chapter, but almost in passing. With graphing polynomials, it becomes critical to understand what happens to the function as the domain stretches out to the right or to the left of the origin. We touched on portions of what follows in the last chapter; I wanted you to get your feet wet before jumping full on into this chapter.  \nThe key to getting end behavior right is to look at the leading term of the polynomial function, that is, the term with the highest degree, or the highest exponent. The reason is that as the exponents climb from one integer to the next, the result on large x-values overshadows any lower degree term. For example, if  \nf (x) = x 4 − 100,000x2  \nit might be tempting to think that the x 2 term, with that big number in front of it, would be the term that dictates what happens to the function as x gets large. If you think that 100 is a big number, then perhaps so because  \nf (100) = (100)4 − 100,000(100)2 = 100,000,000 − 1,000,000,000 = − 900,000,000  \nwhich is a big negative number, but think of x equal to a billion, or x = 109. Then f (109) = (109)4 − 100,000(109)2  \nwhich ends up being a number close to positive ten to the 35th power, which is a huge number, about the number of water molecules in all the oceans of the earth.  \nAll this to show you that the leading term really does matter when we talk about what happens when x gets very large.  \nNow for some examples:  \nf (x) = x 3 − 52x2 + 19  \nNote that the leading term is x 3 . Suppose x = − 20, then the leading term would be  \n(−20)3 = (−20)(−20)(−20) = − 8000  \nand I want you to notice that we had a negative number multiplied by itself three times, and ended up being negative.  \nThat gives you a clue of what to look for on the leading term; is the exponent odd or even?  \nLet’s suppose f (x) = x 6 − 12 . Then, if we place any x-value into our leading term, call it “…”, it would look like  \n(…)(…)(…)(…)(…)(…)  \nand whatever the sign on the “…”, this product is going to come out positive, because that sign is being multiplied by itself an even number of times, six in this case.  \nIf, however, the exponent on the leading term is odd, then if x-value is negative, the sign on that negative number will be multiplied by itself an odd number of times, and so it will result in a negative number.  \nPractice this and see what happens to f (x) with various combinations of exponents. For now, don’t be concerned with what is happening close to the origin; test the edges of your graph to see what is happening with the functions that you pick. The important thing is that the exponent of the leading term will tell you what your end behavior is going to be.  \nFinally, if the coefficient in front of your leading term is negative, then everything you’ve just learned gets flipped to the other side of the x-axis of your graph. For example, if  \nf (x) = x 5 + 3x3 − 2x2 + 18 ,  \nthen the leading term tells us that when x is big and x is positive, then x 5 is really big and positive, and when x is big and negative, then x 5 is really big and negative. Here’s the graph:  \nBut if I change the sign in front of the leading term to negative, watch what happens:  \nf (x) = − x 5 + 3x3 − 2x2 + 18  \nThe directions of the end behaviors has switched; what came up from the bottom on the left side of the first graph now comes from above, and what went up to the top on the right side of the graph now goes down to the bottom. All I did was change the sign on the leading term!  \nThe takeaway here is that the leading term dictates what happens to the graph as x gets large both to the left (negative) and to the right (positive). Notice also that with the odd exponent in our example, one side went down while the other went up. As you practice this, you will understand even better why this is so.  \nFinding the Zeroes of the Graph  \nOnce we understand what the end behavior of our polynomial function is, then ","cbCaikPLzwsbmISQ","https://ap.wps.com/l/cbCaikPLzwsbmISQ","pdf",3452178,13,"English","# End Behavior\n## Leading term determines end behavior\n## Odd vs even exponents\n## Sign of the leading coefficient\n# Finding the Zeroes of the Graph\n## Crossing vs bouncing (kissing) at the x-axis","[{\"question\":\"How do you determine a polynomial’s end behavior from the leading term?\",\"answer\":\"Look at the leading term with the highest degree. For large |x|, that term dominates, so the exponent and its sign determine whether the graph rises or falls as x goes to the left or right.\"},{\"question\":\"Why does an even or odd exponent affect the sign of the leading term differently?\",\"answer\":\"With an even exponent, a negative x-value multiplied even times gives a positive result, so that side tends to rise. With an odd exponent, a negative x multiplied an odd number of times stays negative, so that side tends to fall.\"},{\"question\":\"What does it mean when a polynomial “bounces” at the x-axis?\",\"answer\":\"A “bounce” means the graph touches the x-axis but does not cross it. The examples show this often happens with even-power terms, where the function value remains nonnegative near the zero.\"}]","Chapter Five - Graphing Polynomials - End Behavior - Finding the Zeroes of the Graph | PDF",1792132692,33]