{"id":11611,"date":"2013-02-21T12:22:28","date_gmt":"2013-02-21T12:22:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11611"},"modified":"2022-01-11T21:59:25","modified_gmt":"2022-01-11T21:59:25","slug":"mais-retas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11611","title":{"rendered":"Mais retas"},"content":{"rendered":"<p><ul id='GTTabs_ul_11611' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11611' class='GTTabs_curr'><a  id=\"11611_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11611' ><a  id=\"11611_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11611'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere os seguintes casos de pontos e declives:<\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" border=\"0\" cellspacing=\"8\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Caso 1<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>\u00a0<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Caso 2<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>\u00a0<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Caso 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$A\\left( {0, &#8211; 3} \\right)$ e $m = 2$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">$B\\left( {0,4} \\right)$ e $m =\u00a0 &#8211; 1$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">$C\\left( {1,4} \\right)$ e $m = 0$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Para cada caso, desenhe a reta a que pertence o ponto indicado e tem como declive o valor de $m$ apresentado. Defina as correspondentes fun\u00e7\u00f5es afins.<\/li>\n<li>Para cada caso, defina a fun\u00e7\u00e3o afim cujo gr\u00e1fico \u00e9 a reta paralela \u00e0 reta considerada a que pertence o ponto $P\\left( {1,2} \\right)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11611' onClick='GTTabs_show(1,11611)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11611'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" border=\"0\" cellspacing=\"8\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Caso 1<\/strong><\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><strong>Caso 2<\/strong><\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><strong>Caso 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$A\\left( {0, &#8211; 3} \\right)$ e $m = 2$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">$B\\left( {0,4} \\right)$ e $m =\u00a0 &#8211; 1$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">$C\\left( {1,4} \\right)$ e $m = 0$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11624\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11624\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2.png\" data-orig-size=\"851,490\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"10-pag37-2-V2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2.png\" class=\"aligncenter  wp-image-11624\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2.png\" alt=\"10-pag37-2-V2\" width=\"596\" height=\"343\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2.png 851w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2-300x172.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2-150x86.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10-pag37-2-V2-400x230.png 400w\" sizes=\"auto, (max-width: 596px) 100vw, 596px\" \/><\/a>\u00ad<\/p>\n<ol>\n<li>Os vetores $\\overrightarrow u\u00a0 = \\left( {1,2} \\right)$, $\\overrightarrow v\u00a0 = \\left( {1, &#8211; 1} \\right)$ e $\\overrightarrow w\u00a0 = \\left( {1,0} \\right)$ s\u00e3o diretores das retas consideradas nos casos 1, 2 e 3, respetivamente.\n<p>No caso 1, a ordenada na origem \u00e9 $b =\u00a0 &#8211; 3$, pelo que $y = 2x &#8211; 3$ \u00e9 a equa\u00e7\u00e3o reduzida da reta considerada.<\/p>\n<p>No caso 2, a ordenada na origem \u00e9 $b = 4$, pelo que $y =\u00a0 &#8211; x + 4$ \u00e9 a equa\u00e7\u00e3o reduzida da reta considerada.<\/p>\n<p>No caso 3, como o declive \u00e9 $m = 0$, a reta \u00e9 paralela ao eixo Ox. Logo, a sua equa\u00e7\u00e3o reduzida \u00e9 $y = 4$.<\/p>\n<p>Portanto, as correspondentes fun\u00e7\u00f5es afins s\u00e3o: $$\\begin{array}{*{20}{l}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 2x &#8211; 3}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{g:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to\u00a0 &#8211; x + 4}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{h:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 4}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$\u00ad<\/p>\n<\/li>\n<li>Como se sabe, retas paralelas t\u00eam iguais declives.\n<p>No caso 1, como $P\\left( {1,2} \\right)$ pertence a essa reta, a ordenada na origem \u00e9: $2 = 2 \\times 1 + b \\Leftrightarrow b = 0$.<\/p>\n<p>No caso 2, como $P\\left( {1,2} \\right)$ pertence a essa reta, a ordenada na origem \u00e9: $2 =\u00a0 &#8211; 1 \\times 1 + b \\Leftrightarrow b = 3$.<\/p>\n<p>No caso 3, como $P\\left( {1,2} \\right)$ pertence a essa reta, a ordenada na origem \u00e9: $2 = 0 \\times 1 + b \\Leftrightarrow b = 2$.<\/p>\n<p>Portanto, as correspondentes fun\u00e7\u00f5es afins s\u00e3o: $$\\begin{array}{*{20}{l}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{i:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 2x}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{j:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to\u00a0 &#8211; x + 3}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{k:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 2}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11611' onClick='GTTabs_show(0,11611)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere os seguintes casos de pontos e declives: Caso 1 \u00a0 Caso 2 \u00a0 Caso 3 $A\\left( {0, &#8211; 3} \\right)$ e $m = 2$ $B\\left( {0,4} \\right)$ e $m =\u00a0&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19342,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,343],"tags":[429,346,345,344,347],"series":[],"class_list":["post-11611","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-funcoes-e-graficos","tag-10-o-ano","tag-declive","tag-funcao-afim","tag-grafico","tag-ordenada-na-origem"],"views":2119,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/Mais_retas.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11611","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11611"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11611\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19342"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11611"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11611"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11611"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11611"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}