{"id":14789,"date":"2018-04-22T19:00:45","date_gmt":"2018-04-22T18:00:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14789"},"modified":"2022-01-10T01:31:34","modified_gmt":"2022-01-10T01:31:34","slug":"partes-dos-graficos-de-duas-funcoes-e-um-quadrado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14789","title":{"rendered":"Partes dos gr\u00e1ficos de duas fun\u00e7\u00f5es e um quadrado"},"content":{"rendered":"<p><ul id='GTTabs_ul_14789' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14789' class='GTTabs_curr'><a  id=\"14789_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14789' ><a  id=\"14789_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_14789'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14790\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14790\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png\" data-orig-size=\"570,506\" 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\/9V2Pag121-4.png\" class=\"alignright wp-image-14790\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4-300x266.png\" alt=\"\" width=\"360\" height=\"320\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4-300x266.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png 570w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><\/a>No referencial cartesiano da figura,est\u00e3o representadas partes dos gr\u00e1ficos de duas fun\u00e7\u00f5es, <em>f<\/em> e <em>g<\/em>, e um quadrado [<em>OABC<\/em>].<br \/>\nSabe-se que:<\/p>\n<ul>\n<li>o ponto <em>O<\/em> \u00e9 a origem do referencial;<\/li>\n<li>a fun\u00e7\u00e3o <em>f<\/em> \u00e9 definida por\u00a0\\(f\\left( x \\right) = \\frac{{10}}{x}\\), com\u00a0\\(x &gt; 0\\);<\/li>\n<li>o gr\u00e1fico da fun\u00e7\u00e3o <em>g<\/em> \u00e9 uma reta que passa na origem do referencial;<\/li>\n<li>o ponto <em>A<\/em> pertence ao eixo das abcissas;<\/li>\n<li>o ponto <em>C<\/em> pertence ao eixo das ordenadas;<\/li>\n<li>o ponto <em>B<\/em> pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em>;<\/li>\n<li>o ponto <em>P<\/em> pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em> e ao gr\u00e1fico da fun\u00e7\u00e3o <em>g<\/em> e tem abcissa 5.<\/li>\n<\/ul>\n<ol>\n<li>Em qual das op\u00e7\u00f5es seguintes est\u00e3o as coordenadas de um ponto que pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em>?<br \/>\n[<strong>A<\/strong>] \\(\\left( {50,\\;2} \\right)\\)\u00a0 \u00a0 \u00a0 [<strong>B<\/strong>] \\(\\left( {20,\\;2} \\right)\\)\u00a0 \u00a0 \u00a0 [<strong>C<\/strong>] \\(\\left( {50,\\;\\frac{1}{2}} \\right)\\)\u00a0 \u00a0 \u00a0 [<strong>D<\/strong>]\u00a0\\(\\left( {20,\\;\\frac{1}{2}} \\right)\\)<\/li>\n<li>Define a fun\u00e7\u00e3o <em>g<\/em> por uma express\u00e3o alg\u00e9brica.<br \/>\nApresenta todos os c\u00e1lculos que efetuares.<\/li>\n<li>Qual \u00e9 a medida exata do comprimento do lado do quadrado [<em>OABC<\/em>]?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14789' onClick='GTTabs_show(1,14789)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14789'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14790\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14790\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png\" data-orig-size=\"570,506\" 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\/9V2Pag121-4.png\" class=\"alignright wp-image-14790\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4-300x266.png\" alt=\"\" width=\"360\" height=\"320\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4-300x266.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4.png 570w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><\/a>No referencial cartesiano da figura, est\u00e3o representadas partes dos gr\u00e1ficos de duas fun\u00e7\u00f5es, <em>f<\/em> e <em>g<\/em>, e um quadrado [<em>OABC<\/em>].<br \/>\nSabe-se que:<\/p>\n<\/blockquote>\n<ul>\n<li>\n<blockquote>\n<p>o ponto <em>O<\/em> \u00e9 a origem do referencial;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>a fun\u00e7\u00e3o <em>f<\/em> \u00e9 definida por\u00a0\\(f\\left( x \\right) = \\frac{{10}}{x}\\), com\u00a0\\(x &gt; 0\\);<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o gr\u00e1fico da fun\u00e7\u00e3o <em>g<\/em> \u00e9 uma reta que passa na origem do referencial;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>A<\/em> pertence ao eixo das abcissas;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>C<\/em> pertence ao eixo das ordenadas;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>B<\/em> pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em>;<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>o ponto <em>P<\/em> pertence ao gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em> e ao gr\u00e1fico da fun\u00e7\u00e3o <em>g<\/em> e tem abcissa 5.<\/p>\n<\/blockquote>\n<\/li>\n<\/ul>\n<p>\u00ad<\/p>\n<ol>\n<li>Como a fun\u00e7\u00e3o <em>f<\/em> \u00e9 de proporcionalidade inversa com constante de proporcionalidade 10, ent\u00e3o o produto das coordenadas dos pontos do seu gr\u00e1fico ter\u00e1 de ser exatamente o valor dessa constante.<br \/>\nPor isso, a alternativa correta \u00e9 a [<strong>D<\/strong>].<br \/>\n\u00ad<\/li>\n<li>Comecemos por determinar a ordenada do ponto <em>P<\/em>:\u00a0\\({y_P} = f\\left( 5 \\right) = \\frac{{10}}{5} = 2\\).<br \/>\nO declive da reta <em>OP<\/em> \u00e9\u00a0\\({m_{OP}} = \\frac{{2 &#8211; 0}}{{5 &#8211; 0}} = \\frac{2}{5}\\) e a ordenada na origem \u00e9 nula.<br \/>\nLogo,\u00a0a fun\u00e7\u00e3o <em>g<\/em> pode ser definida pela express\u00e3o alg\u00e9brica \\(g\\left( x \\right) = \\frac{2}{5}x\\).<br \/>\n\u00ad<\/li>\n<li>Designado por <em>a<\/em>, positivo, a medida do comprimento do lado do quadrado, o ponto <em>B<\/em> ter\u00e1 de coordenadas\u00a0\\(\\left( {a,\\;a} \\right)\\). Como o ponto <em>B<\/em> \u00e9 um ponto do gr\u00e1fico da fun\u00e7\u00e3o <em>f<\/em>, ent\u00e3o as coordenadas de <em>B<\/em> t\u00eam de verificar a express\u00e3o alg\u00e9brica de <em>f<\/em>:\\[\\begin{array}{*{20}{l}}{\\begin{array}{*{20}{c}}{a = \\frac{{10}}{a}}&amp; \\wedge &amp;{a &gt; 0}\\end{array}}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{{a^2} = 10}&amp; \\wedge &amp;{a &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{\\left( {\\begin{array}{*{20}{c}}{a = &#8211; \\sqrt {10} }&amp; \\vee &amp;{a = \\sqrt {10} }\\end{array}} \\right)}&amp; \\wedge &amp;{a &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{a = \\sqrt {10} }\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14789' onClick='GTTabs_show(0,14789)'>&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 representadas partes dos gr\u00e1ficos de duas fun\u00e7\u00f5es, f e g, e um quadrado [OABC]. Sabe-se que: o ponto O \u00e9 a origem do referencial; a fun\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14791,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,249],"tags":[426,345,503,250],"series":[],"class_list":["post-14789","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-de-proporcionalidade-inversa","tag-proporcionalidade-inversa-2"],"views":4161,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag121-4a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14789","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=14789"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14789\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14791"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14789"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14789"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14789"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}