{"id":226,"date":"2023-07-10T22:50:09","date_gmt":"2023-07-10T22:50:09","guid":{"rendered":"https:\/\/mathority.org\/pt\/equacao-ponto-inclinacao-da-formula-da-linha\/"},"modified":"2023-07-10T22:50:09","modified_gmt":"2023-07-10T22:50:09","slug":"equacao-ponto-inclinacao-da-formula-da-linha","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/equacao-ponto-inclinacao-da-formula-da-linha\/","title":{"rendered":"Ponto de equa\u00e7\u00e3o \u2013 inclina\u00e7\u00e3o da linha"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula da equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta e, tamb\u00e9m, as diferentes formas que existem para calcul\u00e1-la. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e praticar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-ecuacion-punto-pendiente-de-la-recta\"><\/span> F\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta \u00e9 uma forma de expressar matematicamente uma reta. Em particular, voc\u00ea s\u00f3 precisa da inclina\u00e7\u00e3o e das coordenadas de um ponto na reta para encontrar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta. <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> A <strong>f\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta<\/strong> \u00e9 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left;\"> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a inclina\u00e7\u00e3o da linha e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ef0a49d51c8cfd13e37df1b9090418b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_0, y_0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: -4px;\"><\/p>\n<p> s\u00e3o as coordenadas de um ponto na linha<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf65a6c806bfd1aa7b641e609d9d4490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x_0,y_0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<p> Vamos ver <strong>como a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9 calculada<\/strong> usando um exemplo:<\/p>\n<ul>\n<li> Escreva a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa pelo ponto\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2b2d9d38efcc8624f90eef92208f4e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<p> e inclina\u00e7\u00e3o m=3.<\/li>\n<\/ul>\n<p> A f\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, a afirma\u00e7\u00e3o nos diz que a inclina\u00e7\u00e3o da reta \u00e9 m=3, ent\u00e3o a equa\u00e7\u00e3o da reta ser\u00e1 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4063450c1658fa496145cc18a36ff1bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=3(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, sabemos tamb\u00e9m que a reta passa pelo ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2b2d9d38efcc8624f90eef92208f4e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<p> , devemos, portanto, substituir as coordenadas deste ponto na equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2b2d9d38efcc8624f90eef92208f4e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5cd50c0d1730e27ca68d517d5c9adc76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=3(x-x_0) \\ \\xrightarrow{x_0=2 \\ ; \\ y_0=-1} \\ y-(-1)=3(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"419\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6ea4a8565fcf350fb588d891a0cf9fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y+1=3(x-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Tenha em mente que al\u00e9m da equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o, existem outras maneiras de expressar analiticamente uma reta: a equa\u00e7\u00e3o vetorial, as equa\u00e7\u00f5es param\u00e9tricas, a equa\u00e7\u00e3o cont\u00ednua, a equa\u00e7\u00e3o impl\u00edcita (ou geral) e a equa\u00e7\u00e3o expl\u00edcita de uma reta. Se voc\u00ea estiver mais interessado, pode conferir o que \u00e9 cada um deles em nosso site. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-significa-la-pendiente-de-una-recta\"><\/span> O que significa a inclina\u00e7\u00e3o de uma linha?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Como vimos na defini\u00e7\u00e3o da equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta, o par\u00e2metro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a inclina\u00e7\u00e3o da linha. Mas realmente\u2026 o que significa a inclina\u00e7\u00e3o de uma linha? Vamos ver isso a partir da representa\u00e7\u00e3o gr\u00e1fica de uma linha: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/point-pente-equation.webp\" alt=\"Qual \u00e9 a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta?\" class=\"wp-image-1590\" width=\"317\" height=\"318\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A <strong>inclina\u00e7\u00e3o da linha<\/strong> indica sua inclina\u00e7\u00e3o. Como voc\u00ea pode ver na linha do gr\u00e1fico,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 igual a 2, pois a linha aumenta 2 unidades verticais para 1 unidade horizontal.<\/p>\n<p> Obviamente, se a inclina\u00e7\u00e3o for positiva a fun\u00e7\u00e3o est\u00e1 aumentando (subindo), por outro lado se a inclina\u00e7\u00e3o for negativa a fun\u00e7\u00e3o est\u00e1 diminuindo (descendo).<\/p>\n<h4 class=\"wp-block-heading\"> Como calcular a inclina\u00e7\u00e3o de uma linha<\/h4>\n<p> Al\u00e9m disso, existem 3 maneiras diferentes de determinar numericamente a inclina\u00e7\u00e3o de uma linha:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Dados dois pontos diferentes na linha\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-99906702500e51b12e2859cc804a7b57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(x_1,y_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-460a66d684215738da922dc45a35aed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_2(x_2,y_2),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<p> A inclina\u00e7\u00e3o da linha \u00e9 igual a:<\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ca826248e812d4f19056960777cb00f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{\\Delta y}{\\Delta x} = \\cfrac{y_2-y_1}{x_2-x_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"150\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Sim\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-867fb10d1409b3d95ff447f6a095219d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= (\\text{v}_1,\\text{v}_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> \u00e9 o vetor de dire\u00e7\u00e3o da reta, sua inclina\u00e7\u00e3o \u00e9:<\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60d899a76c2b7588e60dc3734a47019f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{\\text{v}_2}{\\text{v}_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"59\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Sim\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> \u00e9 o \u00e2ngulo formado pela reta com o eixo das abcissas (eixo X), a inclina\u00e7\u00e3o da reta \u00e9 equivalente \u00e0 tangente desse \u00e2ngulo: <\/li>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c76cc82b1d172b2b5af3b053752befac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}(\\alpha )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/ol>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-l-equation-explicite-d-une-ligne.webp\" alt=\"f\u00f3rmula para a equa\u00e7\u00e3o expl\u00edcita da reta\" class=\"wp-image-1465\" width=\"288\" height=\"356\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h4 class=\"wp-block-heading\"> Posi\u00e7\u00e3o relativa das linhas<\/h4>\n<p> Finalmente, a inclina\u00e7\u00e3o de uma linha tamb\u00e9m \u00e9 usada para conhecer a rela\u00e7\u00e3o entre v\u00e1rias linhas. Como duas retas <strong>paralelas<\/strong> t\u00eam a mesma inclina\u00e7\u00e3o e, por outro lado, se a inclina\u00e7\u00e3o de uma reta \u00e9 o inverso negativo da inclina\u00e7\u00e3o de outra reta, isso significa que essas duas retas s\u00e3o <strong>perpendiculares<\/strong> . <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-161\">\n<div class=\"wp-block-column is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/lignes-paralleles-pente-dune-ligne.webp\" alt=\"linhas paralelas com a mesma inclina\u00e7\u00e3o\" class=\"wp-image-1550\" width=\"200\" height=\"201\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/lignes-perpendiculaires-pente-d-une-ligne.webp\" alt=\"\" class=\"wp-image-1551\" width=\"176\" height=\"260\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-la-ecuacion-punto-pendiente-de-la-recta-que-pasa-por-dos-puntos\"><\/span> Calcule a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa por dois pontos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Um problema muito comum \u00e9 determinar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o a partir de dois pontos pertencentes \u00e0 reta. Vamos ver como isso \u00e9 resolvido atrav\u00e9s de um exemplo:<\/p>\n<ul>\n<li> Encontre a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa pelos dois pontos a seguir:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2eedda183214849aae2330021a513130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(5,2) \\qquad P_2(3,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para determinar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta, precisamos determinar qual \u00e9 o declive da reta. Portanto, calculamos a inclina\u00e7\u00e3o da reta usando a f\u00f3rmula dos dois pontos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-355e91b09882111b8826f4bfe5d33698_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =\\cfrac{\\Delta y}{\\Delta x}=\\cfrac{y_2-y_1}{x_2-x_1} = \\cfrac{6-2}{3-5} = \\cfrac{4}{-2}= -2\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"308\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> Assim, a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta ser\u00e1 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bce5bf62dac701d1295a97495bafcac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=-2(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, precisamos apenas substituir as coordenadas cartesianas de um ponto da reta na equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a03a686f78d6b7326bf8047caa76b8f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0ef1c00c03b4a297e716ee315f99504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=-2(x-x_0) \\ \\xrightarrow{x_0=5 \\ ; \\ y_0=2} \\ y-2=-2(x-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"408\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b11cf352f48c55c068de6d1d36f9f51f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-2=-2(x-5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tamb\u00e9m \u00e9 bom colocarmos o outro ponto da afirma\u00e7\u00e3o na equa\u00e7\u00e3o da reta: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e806d8b8c196642821eee52b8c199107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(3,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-988654f7c8782517dc9dabe3fca4a206_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-6=-2(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"hallar-la-ecuacion-punto-pendiente-de-una-recta-a-partir-de-la-grafica\"><\/span> Encontre a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma linha no gr\u00e1fico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como vimos nas se\u00e7\u00f5es acima, existem v\u00e1rias maneiras de determinar numericamente a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta. No entanto, tamb\u00e9m pode ser encontrado graficamente. Vamos ver como isso \u00e9 feito atrav\u00e9s de um exemplo:<\/p>\n<ul>\n<li> Determine a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta mostrada no gr\u00e1fico a seguir: <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/point-pente-ligne-equation.webp\" alt=\"representa\u00e7\u00e3o gr\u00e1fica de uma linha\" class=\"wp-image-1606\" width=\"314\" height=\"315\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Para determinar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta desenhada, precisamos de determinar o seu declive e um ponto na reta.<\/p>\n<p> Neste caso, a inclina\u00e7\u00e3o da reta \u00e9 igual a 3, pois a reta sobe 3 unidades verticais para cada unidade horizontal.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0082d45adbb746641eb28f250a819459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Em seguida, precisamos de um ponto na reta. Para fazer isso, podemos escolher qualquer ponto do gr\u00e1fico por onde passa a reta, por exemplo o ponto (1,1).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc3450ac38bb3acc76eae31215c49b82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, podemos agora encontrar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta aplicando sua f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-829ac7a79e34c45087cbfb472eacb856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-1=3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/determiner-lequation-point-pente-dune-ligne-graphiquement.webp\" alt=\"determinar graficamente a inclina\u00e7\u00e3o do ponto de equa\u00e7\u00e3o de uma reta\" class=\"wp-image-1607\" width=\"314\" height=\"315\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-punto-pendiente-de-la-recta\"><\/span> Problemas resolvidos de equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Escreva a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa pelo ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e4a894322ba599f7554e658df9395ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> e sua inclina\u00e7\u00e3o \u00e9 <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e3505d96cc978cbd5e9fefda94605f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=-2.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A f\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso, a afirma\u00e7\u00e3o nos diz que a inclina\u00e7\u00e3o da reta \u00e9 m=-2, ent\u00e3o a equa\u00e7\u00e3o da reta ser\u00e1 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bce5bf62dac701d1295a97495bafcac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=-2(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, tamb\u00e9m sabemos pela afirma\u00e7\u00e3o que a reta passa pelo ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e4a894322ba599f7554e658df9395ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> , \u00e9 portanto suficiente substituir as coordenadas do ponto na equa\u00e7\u00e3o da reta: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e4a894322ba599f7554e658df9395ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e8f25bcb8e24f35f3ad9e64dbeab604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-4=-2(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Qual \u00e9 a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa pelos dois pontos a seguir? <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d87f163111868ca87a1e0d33e7373e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(1,6) \\qquad P_2(4,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para determinar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta, precisamos determinar qual \u00e9 o declive da reta. Portanto, calculamos a inclina\u00e7\u00e3o da reta com sua f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6587ef8551123fada18e66a19b120672_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =\\cfrac{\\Delta y}{\\Delta x}=\\cfrac{y_2-y_1}{x_2-x_1} = \\cfrac{0-6}{4-1} = \\cfrac{-6}{3}= -2\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"316\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta ser\u00e1 a seguinte: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bce5bf62dac701d1295a97495bafcac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=-2(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, precisamos apenas substituir as coordenadas de um ponto da reta na equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bcbff883ed57a03d2b8c0ad3d99fbd8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(1,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65df3bba32eb4d9dc32e16c8c26fd81f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-6=-2(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tamb\u00e9m teria sido correto colocar o outro ponto da afirma\u00e7\u00e3o na equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e45755fe2812135fe1f4e39b3157fcf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2(x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Encontre a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta que passa pelos dois pontos a seguir: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd4450de7b302ee03cef55687c1e7b01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(1,-2) \\qquad P_2(2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta, voc\u00ea deve primeiro calcular sua inclina\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-194cde1d2e3ba81c4bfce5bf0244c51a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =\\cfrac{\\Delta y}{\\Delta x}=\\cfrac{y_2-y_1}{x_2-x_1} = \\cfrac{3-(-2)}{2-1} = \\cfrac{3+2}{1}=\\cfrac{5}{1}= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"373\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta ser\u00e1 a seguinte: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d740b976c2c72700bfcd9562e1d6236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=5(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, precisamos apenas substituir as coordenadas de um ponto da reta na equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd29b3293380effcfb027a6b7aec2252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P_1(1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba331d69618f15f523d0bef9b28673f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-(-2)=5(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b7256b1d7a60559de4cbad1ffe360a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y+2=5(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tamb\u00e9m \u00e9 correto colocar o outro ponto da afirma\u00e7\u00e3o na equa\u00e7\u00e3o da reta: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66fd02f8e38dc1b0a598a1a384e697c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-3=5(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Calcule a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o para a reta que forma um \u00e2ngulo de 45\u00ba com o eixo X e passa pela origem das coordenadas. <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Se a linha fizer um \u00e2ngulo de 45 graus com o eixo OX, sua inclina\u00e7\u00e3o ser\u00e1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51d113bc9a4b67f4c35c31f08baa7ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}(45\u00ba) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7831f83b7ed770006cac5c20921bfa0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=1(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d02fbbf78fbaf1ec868c2555515cad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=x-x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"120\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez que conhecemos o declive da reta, podemos determinar a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o substituindo um ponto da reta na equa\u00e7\u00e3o. Al\u00e9m disso, a afirma\u00e7\u00e3o nos diz que a reta passa pela origem das coordenadas, o que significa que ela passa pelo ponto (0,0). Ainda: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-048c1ceefde62a60b4cf2420a67d7f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f032721e5e51bcb17f57204a1991d5c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=x-x_0 \\ \\xrightarrow{x=0 \\ ; \\ y=0} \\ y-0=x-0\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"323\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4909df7491ef54f0df1e922bc29417f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=x}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Encontre a equa\u00e7\u00e3o da inclina\u00e7\u00e3o do ponto da linha paralela \u00e0 linha<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> e o que acontece al\u00e9m do ponto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37ecf61896ad8378d5128916dfa0db58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,-3).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> ser direto <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5986f031932e6b3512dc564514c34b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13e021539041421cca99c04a0add9516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\; y-1=2(x+5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A inclina\u00e7\u00e3o da linha<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 igual a 2 (n\u00famero antes dos par\u00eanteses), e para que duas retas sejam paralelas, elas devem ter a mesma inclina\u00e7\u00e3o, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3439899d4781058b8eb19021ac25e27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8579f7a4e6d6535f68742555a229e8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=2(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez que conhecemos a inclina\u00e7\u00e3o da reta, simplesmente substitu\u00edmos as coordenadas de um ponto que pertence \u00e0 reta na f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-398c2be999567c477ccd7a55d5c66ced_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ae615335baed1209de396f5a98fe6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-(-1)=2(x-(-3))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-45304f06d37b44e4b468dcea500c1689_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y+1=2(x+3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 6<\/h3>\n<p> Determine a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o para cada linha mostrada no gr\u00e1fico a seguir: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-dequation-de-la-ligne-explicite.webp\" alt=\"equa\u00e7\u00e3o expl\u00edcita da linha exerc\u00edcio resolvido passo a passo\" class=\"wp-image-1500\" width=\"377\" height=\"405\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> <strong>azul \u00e0 direita<\/strong><\/p>\n<p class=\"has-text-align-left\"> A linha azul aumenta um Y para cada X, portanto sua inclina\u00e7\u00e3o \u00e9 igual a 1. Por outro lado, ela passa pelo ponto (2,4), portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2da731e16044d6c6ac573d0f5ecdc8b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4 =x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>verde direito<\/strong><\/p>\n<p class=\"has-text-align-left\"> A linha verde aumenta tr\u00eas Y para cada X, ent\u00e3o sua inclina\u00e7\u00e3o \u00e9 3. Al\u00e9m disso, um de seus pontos \u00e9 (2,2), ent\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cda4faf329c19618c240327d8a427333_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2 =3(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Linha Vermelha<\/strong><\/p>\n<p class=\"has-text-align-left\"> A linha vermelha diminui em dois Y para cada X, ent\u00e3o sua inclina\u00e7\u00e3o \u00e9 -2. E o ponto (0,-2) pertence a esta reta, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de1091d5afd35c2063ab769ade6bf7ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y =-2(x+2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula da equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta e, tamb\u00e9m, as diferentes formas que existem para calcul\u00e1-la. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e praticar com exerc\u00edcios resolvidos passo a passo. F\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta A equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma reta \u00e9 uma forma de expressar matematicamente uma &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/equacao-ponto-inclinacao-da-formula-da-linha\/\"> <span class=\"screen-reader-text\">Ponto de equa\u00e7\u00e3o \u2013 inclina\u00e7\u00e3o da linha<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[20],"tags":[],"class_list":["post-226","post","type-post","status-publish","format-standard","hentry","category-pontos-retas-e-planos"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Ponto da equa\u00e7\u00e3o \u2013 inclina\u00e7\u00e3o da reta \u2013 Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/equacao-ponto-inclinacao-da-formula-da-linha\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Ponto da equa\u00e7\u00e3o \u2013 inclina\u00e7\u00e3o da reta \u2013 Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula da equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o da reta e, tamb\u00e9m, as diferentes formas que existem para calcul\u00e1-la. Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos e praticar com exerc\u00edcios resolvidos passo a passo. 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