[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-605563-105":59,"doc-detail-605563-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-two-linear-functions-and-graphs-slope-and-intercepts-coordinate-plane-and-axes","Chapter Two - Linear Functions and Graphs - Slope and Intercepts - Coordinate Plane and Axes","","Chapter Two explains how the coordinate plane uses an x-y system to locate points and uniquely determine lines from two points. It introduces slope as the change in y divided by the change in x (rise over run), showing how slope links to dependent y-values. The chapter then covers how vertical shifts occur when a constant is added to y, including function notation f(x)+5. It concludes with line equations such as y=2x and the general rule f(x)=mx for lines through the origin.",{"@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-two-linear-functions-and-graphs-slope-and-intercepts-coordinate-plane-and-axes/605563/",{"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-two-linear-functions-and-graphs-slope-and-intercepts-coordinate-plane-and-axes/605563.png","ImageObject",300,407,{"name":92,"@type":93},"jayni","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 does the coordinate plane help represent points and lines?","Question",{"text":112,"@type":113},"The coordinate plane locates points using x-y coordinates, and two distinct points determine a unique line through them.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is slope and how is it calculated?",{"text":117,"@type":113},"Slope is defined as the change in y divided by the change in x, commonly described as rise over run.",{"name":119,"@type":110,"acceptedAnswer":120},"What happens to a graph when you add a constant to the y-value?",{"text":121,"@type":113},"Adding a constant to every y-value shifts all points vertically by that amount while keeping x the same.","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},605563,1792502351,{"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},3985747859343,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Chapter Two  \nLinear Functions and Graphs, Slope, Intercepts  \nCoordinate Plane and Axes  \nIf you skipped the last chapter, and I hope you didn’t because we covered a bunch of neat stuff in it, I will just point out here that we have this amazing invention of Rene Descartes called the coordinate plane. This is a mathematical device in which we can plot or locate points within the plane by a simple x-y coordinate system. As you may recall from geometry,“two points determine a line”, and this is so with the coordinate plane as well.  \nSuppose we have two points on the plane, (−2,5) and (3,7) . I can plot these points, and then draw a straight line from one point to the other:  \nThere is nothing special about these two points; I could have picked anything, both positive or negative, both x or y. As long as both points are not equal in both x and y, they will be two separate points, and a unique line connects the two, as I have demonstrated above.  \nSlope  \nIf you look again at the two points and their line above, you will notice that as x moves from left to right, from −2 to 3, the line rises (the height changes from 5 to 7). We call this slope. We define slope as the change in y divided by the change in x, and we commonly call this “rise over run”.  \nYou will see that we talk about slope a lot. Invariably, we speak of slope when we talk of straight lines, but I will casually mention that curved lines have slope too, and save further talk for a calculus class.  \nImagine that we continued to let x run beyond the x-value of 3, and simply continued the straight line. The height would change as well, increasing beyond the height of 7, and the height would depend upon the x-value.  \nPerhaps for the first time, we have explained why y-values are called dependent values, because they depend on the x-value. Since the slope is the change in height, or also the change in y-values (divided by the change in the xvalues), let’s stretch out our line to a new x-value of x = ~~ ~~212 (or x = 10.5):  \nNotice that our slope just continues on to the new height that corresponds with that x-value of 10.5. What I want you to get from this is that the slope in a straight line relates to the x-value.  \nYou will use this a lot when calculating just about anything; you know how long it takes for you to drive to Phoenix…how long will it take you to drive to New York? You know how much M&Ms to buy when having a party for eight people…how many will you need for just four people?  \nAdding a constant to the y-value of a coordinate pair.  \nNext, I want you to consider what happens when we add a constant number to the y-value in a coordinate pair. Suppose that I start with the pair (2,3) . If for example I add 5 to the 3, and don’t change the x-value, I would get the pair (2,8). Graphically, it looks like this:  \nThis is very important: by adding 5 to the y-value of the point (2,3), I shifted it vertically by 5 units to become (2,8) . I’m going to do the same now with several points (−1, − 3), (0,4), (2,3), and (3,5):  \nSee how by adding 5 to each y-value, I shifted the point to a new point exactly 5 units above it? In this case, my new points are (−1,2), (0,9), (2,8) and (3,10).  \nNow, remember that for a function of x, called f (x) , the y-value that corresponds to the x-value is f (x) . That means that we can write the coordinate pair (x , y) as (x , f (x)). Then, if I change my coordinate pair by adding 5 to every yvalue, I’m going to shift all my coordinate pairs upward by 5. We could describe what we did by writing the coordinate point (x , f (x) + 5).  \nYou perhaps see the flexibility of this coordinate system? I have just inserted a function into the y-coordinate spot.  \nLines on the Plane  \nIn the last two sections, I probably got too simplistic in your mind, and was talking about various basic ideas within the coordinate plane. In my defense, these two matters have been stumbling blocks with students who perhaps skipped over these, only to switch x and","cbCaigRZyAsZ1qrA","https://ap.wps.com/l/cbCaigRZyAsZ1qrA","pdf",4453356,22,"English","# Chapter Two\n## Linear Functions and Graphs, Slope, Intercepts\n## Coordinate Plane and Axes\n## Slope\n## Adding a Constant to the y-Value\n## Lines on the Plane","[{\"question\":\"How does the coordinate plane help represent points and lines?\",\"answer\":\"The coordinate plane locates points using x-y coordinates, and two distinct points determine a unique line through them.\"},{\"question\":\"What is slope and how is it calculated?\",\"answer\":\"Slope is defined as the change in y divided by the change in x, commonly described as rise over run.\"},{\"question\":\"What happens to a graph when you add a constant to the y-value?\",\"answer\":\"Adding a constant to every y-value shifts all points vertically by that amount while keeping x the same.\"}]","Chapter Two - Linear Functions and Graphs - Slope and Intercepts - Coordinate Plane and Axes | PDF",1792132681,55]