{"id":188,"date":"2024-10-18T21:12:54","date_gmt":"2024-10-18T21:12:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/functions-background-youll-need-page-1\/"},"modified":"2024-10-18T21:12:54","modified_gmt":"2024-10-18T21:12:54","slug":"functions-background-youll-need-page-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/functions-background-youll-need-page-1\/","title":{"raw":"Functions: Background You'll Need - Page 2","rendered":"Functions: Background You&#8217;ll Need &#8211; Page 2"},"content":{"raw":"\n<section>\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Calculate the distance between two points<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<p>The concept of distance between two points in a Cartesian coordinate system is rooted in the Pythagorean Theorem, which is expressed as [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]. This based on a right triangle where [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs adjacent to the right angle, and [latex]c[\/latex] is the length of the hypotenuse.<\/p>\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\"><\/center>\n<p>&nbsp;<\/p>\n<p>In the context of the coordinate plane, the lengths [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] correspond to [latex]a[\/latex] and [latex]b[\/latex], respectively. The distance [latex]d[\/latex] between two points is analogous to [latex]c[\/latex], the hypotenuse. The absolute value symbols ensure that these lengths are positive, as the absolute value of any number is positive.<\/p>\n<p>To find [latex]c[\/latex], or in our case [latex]d[\/latex], the Pythagorean Theorem is rearranged as follows:<\/p>\n<p style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/p>\n<p>This leads us to the distance formula:<\/p>\n<p style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/p>\n<p>Note that the absolute value symbols are not necessary in this formula, as squaring any number results in a positive value.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>The Distance Formula<\/h3>\n<p>Given endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">Find the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].[reveal-answer q=\"639228\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"639228\"]Let us first look at the graph of the two points. Connect the points to form a right triangle.<center><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"A triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\"><\/center>\n<p>&nbsp;<\/p>\n<p>Then, calculate the length of [latex]d[\/latex] using the distance formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6201[\/ohm2_question]<\/section>\n","rendered":"<section>\n<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the distance between two points<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<p>The concept of distance between two points in a Cartesian coordinate system is rooted in the Pythagorean Theorem, which is expressed as [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]. This based on a right triangle where [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs adjacent to the right angle, and [latex]c[\/latex] is the length of the hypotenuse.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>In the context of the coordinate plane, the lengths [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] correspond to [latex]a[\/latex] and [latex]b[\/latex], respectively. The distance [latex]d[\/latex] between two points is analogous to [latex]c[\/latex], the hypotenuse. The absolute value symbols ensure that these lengths are positive, as the absolute value of any number is positive.<\/p>\n<p>To find [latex]c[\/latex], or in our case [latex]d[\/latex], the Pythagorean Theorem is rearranged as follows:<\/p>\n<p style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/p>\n<p>This leads us to the distance formula:<\/p>\n<p style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/p>\n<p>Note that the absolute value symbols are not necessary in this formula, as squaring any number results in a positive value.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>The Distance Formula<\/h3>\n<p>Given endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">Find the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q639228\">Show Answer<\/button><\/p>\n<div id=\"q639228\" class=\"hidden-answer\" style=\"display: none\">Let us first look at the graph of the two points. Connect the points to form a right triangle.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"A triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Then, calculate the length of [latex]d[\/latex] using the distance formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6201\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6201&theme=lumen&iframe_resize_id=ohm6201&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":6,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"","content_attributions":[{"type":"cc-attribution","description":"College Algebra","author":"OpenStax College Algebra","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface","project":"","license":"cc-by","license_terms":""},{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"College Algebra","author":"Abramson, Jay et al.","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2","project":"","license":"cc-by","license_terms":"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/188"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/188\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/188\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=188"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=188"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}