{"id":14704,"date":"2018-04-19T21:56:37","date_gmt":"2018-04-19T20:56:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14704"},"modified":"2022-01-09T22:54:13","modified_gmt":"2022-01-09T22:54:13","slug":"determina-uma-expressao-algebrica-para-cada-uma-das-funcoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14704","title":{"rendered":"Determina uma express\u00e3o alg\u00e9brica para cada uma das fun\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_14704' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14704' class='GTTabs_curr'><a  id=\"14704_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14704' ><a  id=\"14704_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_14704'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14705\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14705\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\" data-orig-size=\"524,476\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Gr\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\" class=\"alignright size-medium wp-image-14705\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6-300x273.png\" alt=\"\" width=\"300\" height=\"273\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6-300x273.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png 524w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>No referencial cartesiano da figura est\u00e3o representados os gr\u00e1ficos de duas fun\u00e7\u00f5es <em>f<\/em> e <em>g<\/em>, respetivamente, a par\u00e1bola de v\u00e9rtice\u00a0\\(\\left( {0,\\;0} \\right)\\) que passa pelo ponto\u00a0\\(A\\left( { &#8211; 1,\\; &#8211; 1} \\right)\\) e a reta <em>DE<\/em> em que\u00a0\\(D\\left( {0,\\; &#8211; 2} \\right)\\) e\u00a0\\(E\\left( {2,\\;0} \\right)\\).<\/p>\n<ol>\n<li>Determina uma express\u00e3o alg\u00e9brica para cada uma das fun\u00e7\u00f5es.<\/li>\n<li>Determina as coordenadas dos pontos de interse\u00e7\u00e3o dos dois gr\u00e1ficos.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14704' onClick='GTTabs_show(1,14704)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14704'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Os gr\u00e1ficos das fun\u00e7\u00f5es do tipo\u00a0\\(f\\left( x \\right) = a{x^2}\\), com\u00a0\\(a \\ne 0\\), s\u00e3o <strong>par\u00e1bolas de eixo vertical e v\u00e9rtice na origem<\/strong>.<\/p>\n<\/blockquote>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14705\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14705\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\" data-orig-size=\"524,476\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Gr\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png\" class=\"alignright size-medium wp-image-14705\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6-300x273.png\" alt=\"\" width=\"300\" height=\"273\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6-300x273.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6.png 524w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Como o ponto\u00a0\\(A\\left( { &#8211; 1,\\; &#8211; 1} \\right)\\) pertence ao gr\u00e1fico de <em>f<\/em>, vem:\\[\\begin{array}{*{20}{l}}{f\\left( { &#8211; 1} \\right) = &#8211; 1}&amp; \\Leftrightarrow &amp;{a \\times {{\\left( { &#8211; 1} \\right)}^2} = &#8211; 1}\\\\{}&amp; \\Leftrightarrow &amp;{a = &#8211; 1}\\end{array}\\]\u00a0Logo, \\(f\\left( x \\right) = &#8211; {x^2}\\) \u00e9 uma express\u00e3o alg\u00e9brica de <em>f<\/em>.\n<p>A reta <em>DE<\/em> tem declive \\({m_{DE}} = \\frac{{ &#8211; 2 &#8211; 0}}{{0 &#8211; 2}} = 1\\) e ordenada na origem \\(b = &#8211; 2\\).<br \/>\nLogo, \\(g\\left( x \\right) = x &#8211; 2\\) \u00e9 uma express\u00e3o alg\u00e9brica da fun\u00e7\u00e3o <em>g<\/em>.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>As abcissas dos pontos de interse\u00e7\u00e3o dos dois gr\u00e1ficos, est\u00e3o calculadas seguidamente.<br \/>\n\\[\\begin{array}{*{20}{l}}{f\\left( x \\right) = g\\left( x \\right)}&amp; \\Leftrightarrow &amp;{ &#8211; {x^2} = x &#8211; 2}\\\\{}&amp; \\Leftrightarrow &amp;{{x^2} + x &#8211; 2 = 0}\\\\{}&amp; \\Leftrightarrow &amp;{x = \\frac{{ &#8211; 1 \\mp \\sqrt {1 + 8} }}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{x = \\frac{{ &#8211; 1 \\mp 3}}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = &#8211; 2}&amp; \\vee &amp;{x = 1}\\end{array}}\\end{array}\\]<br \/>\nLogo, as coordenadas dos pontos de interse\u00e7\u00e3o s\u00e3o\u00a0\\(\\left( { &#8211; 2,g\\left( { &#8211; 2} \\right)} \\right) = \\left( { &#8211; 2, &#8211; 4} \\right)\\) e\u00a0\\(\\left( {1,g\\left( 1 \\right)} \\right) = \\left( {1, &#8211; 1} \\right)\\).<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14704' onClick='GTTabs_show(0,14704)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado No referencial cartesiano da figura est\u00e3o representados os gr\u00e1ficos de duas fun\u00e7\u00f5es f e g, respetivamente, a par\u00e1bola de v\u00e9rtice\u00a0\\(\\left( {0,\\;0} \\right)\\) que passa pelo ponto\u00a0\\(A\\left( { &#8211; 1,\\; &#8211; 1}&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14707,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,249],"tags":[426,345,499],"series":[],"class_list":["post-14704","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-proporcionalidade-inversa-e-funcoes-algebricas","tag-9-o-ano","tag-funcao-afim","tag-funcao-quadratica"],"views":10354,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag113-6a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14704","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=14704"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14704\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14707"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14704"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14704"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14704"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}