{"id":56,"date":"2022-04-19T20:52:29","date_gmt":"2022-04-19T20:52:29","guid":{"rendered":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/?page_id=56"},"modified":"2022-04-20T01:45:51","modified_gmt":"2022-04-20T01:45:51","slug":"lines","status":"publish","type":"page","link":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/home\/lines\/","title":{"rendered":"Lines and Linear Relationships"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-63 size-large\" src=\"http:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686-1024x682.jpg\" alt=\"\" width=\"1024\" height=\"682\" srcset=\"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686-1024x682.jpg 1024w, https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686-300x200.jpg 300w, https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686-768x512.jpg 768w, https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686-1536x1023.jpg 1536w, https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-content\/uploads\/sites\/46\/2022\/04\/100_8686.jpg 2032w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>This course begins with a review of lines.\u00a0 Why should we care about lines?\u00a0 For one thing, linear relationships are all over the place.\u00a0 They appear in just about every field of study that puts numbers to things.\u00a0 For example,<\/p>\n<p style=\"text-align: center\">[pictures]<\/p>\n<p>Of course, [this relationship isn&#8217;t exactly linear it&#8217;s stochastic and that&#8217;s ok, we can do regression]<\/p>\n<p style=\"text-align: center\">[pictures]<\/p>\n<p>\u00a0Another reason we care about lines is that they&#8217;re among the simplest of mathematical models.\u00a0 The key idea of calculus is that we can approximate difficult, nonlinear things using simpler, linear things.<\/p>\n<p style=\"text-align: center\">[pictures]<\/p>\n<h2>Key Features of Linear Relationships<\/h2>\n<p>Let&#8217;s look at the line\u00a0<em>y =\u00a0<\/em>\u00bd\u00a0<em>x<\/em> + 3<em>.\u00a0\u00a0<\/em>For this line:<\/p>\n<ul>\n<li>Draw a graph. How do the numbers \u00bd and 3 correspond to features in your graph?<\/li>\n<li>Create a table of values.\u00a0 What patterns can you find in the table of values?<\/li>\n<li>Find ways to articulate connections between the graph, the equation, and the table.\u00a0 Explain your connections to someone who&#8217;s willing to listen.\u00a0 For example: a friend; a tutor; Marc Boucher; or me.<\/li>\n<\/ul>\n<p style=\"text-align: center\">[diagram]<\/p>\n<p>Let&#8217;s look at the line drawn below:<\/p>\n<p style=\"text-align: center\">[picture]<\/p>\n<p>For this line:<\/p>\n<ul>\n<li>Create a table of values.<\/li>\n<li>Create an equation.\u00a0 How do features in the graph correspond to numbers in your equation?<\/li>\n<li>Look for the same connections you found above. Are they still there? In what ways are they the same? In what ways are they different?<\/li>\n<\/ul>\n<p>Find the point (<em>x<\/em>,<em>y<\/em>) at which these two lines meet.\u00a0 For this problem, don&#8217;t use any technology &#8211; just algebra. What are some ways you can make sense of your answer?<\/p>\n<h2>Moving On<\/h2>\n<p>Make sure that you can solve linear equations like \u00be <em>x<\/em> + 3 = 2<em>x + <\/em>4 and that you understand these connections:<\/p>\n<p style=\"text-align: center\">[diagram]<\/p>\n<p>Linear relationships are so ubiquitous that this will be worth your time. It&#8217;s also the foundation on which we&#8217;re going to build things.<\/p>\n<p>When you&#8217;re ready, let&#8217;s move on to <a href=\"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/quadratics-the-simplest-nonlinearity\/\">quadratics<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This course begins with a review of lines.\u00a0 Why should we care about lines?\u00a0 For one thing, linear relationships are all over the place.\u00a0 They appear in just about every field of study that puts numbers to things.\u00a0 For example, [pictures] Of course, [this relationship isn&#8217;t exactly linear it&#8217;s stochastic and that&#8217;s ok, we can [&hellip;]<\/p>\n","protected":false},"author":41,"featured_media":0,"parent":13,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"episode_type":"","audio_file":"","cover_image":"","cover_image_id":"","duration":"","filesize":"","filesize_raw":"","date_recorded":"","explicit":"","block":"","itunes_episode_number":"","itunes_title":"","itunes_season_number":"","itunes_episode_type":"","footnotes":""},"class_list":["post-56","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/pages\/56","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/users\/41"}],"replies":[{"embeddable":true,"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/comments?post=56"}],"version-history":[{"count":0,"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/pages\/56\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/pages\/13"}],"wp:attachment":[{"href":"https:\/\/lakerblogs.lssu.edu\/intermediatealgebra\/wp-json\/wp\/v2\/media?parent=56"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}