{"id":237,"date":"2017-04-15T03:19:58","date_gmt":"2017-04-15T03:19:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/exponential-relationships-3-of-6\/"},"modified":"2022-08-01T16:04:39","modified_gmt":"2022-08-01T16:04:39","slug":"exponential-relationships-3-of-6","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/chapter\/exponential-relationships-3-of-6\/","title":{"raw":"Exponential Relationships (3 of 6)","rendered":"Exponential Relationships (3 of 6)"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OUTCOMES<\/h3>\r\n<ul>\r\n \t<li>Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.<\/li>\r\n<\/ul>\r\n<\/div>\r\nLet\u2019s summarize what we have learned about exponential growth models:\r\n\r\nThe general form of an exponential growth model is <em>y = C \u00b7 b<sup><em>x<\/em><\/sup><\/em>.\r\n<ul>\r\n \t<li><em><strong>C <\/strong><\/em><strong>is the initial value<\/strong>. It is the <em>y<\/em>-value when <em>x<\/em> = 0. It is also the <em>y<\/em>-intercept.<\/li>\r\n \t<li><em><strong>b <\/strong><\/em><strong>is the growth factor; it will always be greater than 1 in cases of growth. <\/strong>From the growth factor, we can determine the percentage increase in <em>y<\/em> for each additional 1 unit increase in <em>x<\/em>.<\/li>\r\n<\/ul>\r\nLet\u2019s compare the general form of an exponential growth model to the general form for a <em>linear model:<\/em> <em>y<\/em> = <em>a<\/em> + <em>bx<\/em>.\r\n<ul>\r\n \t<li>In the linear model, <em>a<\/em> is the <strong>initial value<\/strong>. It is the <em>y<\/em>-value when <em>x<\/em> = 0. It is also the <em>y<\/em>-intercept.<\/li>\r\n \t<li><em>b<\/em> is the <strong>slope<\/strong>. From the slope, we can determine the amount and direction the <em>y<\/em>-value changes for each additional 1 unit increase in <em>x<\/em>.<\/li>\r\n<\/ul>\r\nNow we apply what we have learned about exponential growth to find a model for a set of data.\r\n\r\nIn this activity, you use a simulation to find an exponential model that fits the population growth of Kenya.\r\n\r\nHere are the data graphed in the scatterplot in the simulation.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031957\/m4_non_linear_models_ExponentialRelationships1of2_image24.png\" alt=\"Data on Kenya's population growth, from 1950\u20132010, in 10 year increments\" width=\"481\" height=\"239\" \/>\r\n\r\nNotice that the Kenyan population growth has a strong positive exponential form. Use the sliders in the simulation to adjust the values of <em>C<\/em> and <em>b<\/em> to find a reasonable exponential model that fits this data.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/exponential_relationships_3_of_6\/exponentialrelationships3of6.html\" target=\"new\">Click here to open this simulation in its own window.<\/a>\r\n\r\n<iframe id=\"_i_4b\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/exponential_relationships_3_of_6\/exponentialrelationships3of6.html\" width=\"500\" height=\"500\"><\/iframe>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nhttps:\/\/assess.lumenlearning.com\/practice\/710422ba-1844-4c85-9f9e-869f190a4331\r\n\r\n<\/div>\r\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1JrxakRBXk3gZmToNspgyIrrDCQUNju4txyxxN9657XA\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OUTCOMES<\/h3>\n<ul>\n<li>Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.<\/li>\n<\/ul>\n<\/div>\n<p>Let\u2019s summarize what we have learned about exponential growth models:<\/p>\n<p>The general form of an exponential growth model is <em>y = C \u00b7 b<sup><em>x<\/em><\/sup><\/em>.<\/p>\n<ul>\n<li><em><strong>C <\/strong><\/em><strong>is the initial value<\/strong>. It is the <em>y<\/em>-value when <em>x<\/em> = 0. It is also the <em>y<\/em>-intercept.<\/li>\n<li><em><strong>b <\/strong><\/em><strong>is the growth factor; it will always be greater than 1 in cases of growth. <\/strong>From the growth factor, we can determine the percentage increase in <em>y<\/em> for each additional 1 unit increase in <em>x<\/em>.<\/li>\n<\/ul>\n<p>Let\u2019s compare the general form of an exponential growth model to the general form for a <em>linear model:<\/em> <em>y<\/em> = <em>a<\/em> + <em>bx<\/em>.<\/p>\n<ul>\n<li>In the linear model, <em>a<\/em> is the <strong>initial value<\/strong>. It is the <em>y<\/em>-value when <em>x<\/em> = 0. It is also the <em>y<\/em>-intercept.<\/li>\n<li><em>b<\/em> is the <strong>slope<\/strong>. From the slope, we can determine the amount and direction the <em>y<\/em>-value changes for each additional 1 unit increase in <em>x<\/em>.<\/li>\n<\/ul>\n<p>Now we apply what we have learned about exponential growth to find a model for a set of data.<\/p>\n<p>In this activity, you use a simulation to find an exponential model that fits the population growth of Kenya.<\/p>\n<p>Here are the data graphed in the scatterplot in the simulation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031957\/m4_non_linear_models_ExponentialRelationships1of2_image24.png\" alt=\"Data on Kenya's population growth, from 1950\u20132010, in 10 year increments\" width=\"481\" height=\"239\" \/><\/p>\n<p>Notice that the Kenyan population growth has a strong positive exponential form. Use the sliders in the simulation to adjust the values of <em>C<\/em> and <em>b<\/em> to find a reasonable exponential model that fits this data.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/exponential_relationships_3_of_6\/exponentialrelationships3of6.html\" target=\"new\">Click here to open this simulation in its own window.<\/a><\/p>\n<p><iframe loading=\"lazy\" id=\"_i_4b\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/exponential_relationships_3_of_6\/exponentialrelationships3of6.html\" width=\"500\" height=\"500\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>\t<iframe id=\"assessment_practice_710422ba-1844-4c85-9f9e-869f190a4331\" class=\"resizable\" src=\"https:\/\/assess.lumenlearning.com\/practice\/710422ba-1844-4c85-9f9e-869f190a4331?iframe_resize_id=assessment_practice_id_710422ba-1844-4c85-9f9e-869f190a4331\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:300px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1JrxakRBXk3gZmToNspgyIrrDCQUNju4txyxxN9657XA\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-237\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":163,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"35509f10-cffa-479f-b6f3-dcb9a7381137, ffa973e0-23dc-40e8-9174-72aa4d64023e","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-237","chapter","type-chapter","status-publish","hentry"],"part":225,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/237","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/237\/revisions"}],"predecessor-version":[{"id":2744,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/237\/revisions\/2744"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/237\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/media?parent=237"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapter-type?post=237"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/contributor?post=237"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/license?post=237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}