{"id":4437,"date":"2022-06-07T18:25:40","date_gmt":"2022-06-07T16:25:40","guid":{"rendered":"https:\/\/mathe-lernen.net\/?p=4437"},"modified":"2022-06-12T10:28:33","modified_gmt":"2022-06-12T08:28:33","slug":"die-funktion-yax%c2%b2c","status":"publish","type":"post","link":"https:\/\/mathe-lernen.net\/?p=4437","title":{"rendered":"Die Funktion y=ax\u00b2+c"},"content":{"rendered":"\n<p>Der Graph der Funktion <strong>y=<span style=\"color:#005cfa\" class=\"has-inline-color\">a<\/span>x\u00b2 +<span style=\"color:#0055ff\" class=\"has-inline-color\"> <\/span><span style=\"color:#81cb00\" class=\"has-inline-color\">c<\/span><\/strong> ist eine Parabel. Der<strong><span style=\"color:#002ea3\" class=\"has-inline-color\"> <\/span><span style=\"color:#0048ff\" class=\"has-inline-color\">Faktor a<\/span><span style=\"color:#002ea3\" class=\"has-inline-color\"> <\/span><\/strong>sorgt dabei f\u00fcr Streckung, Stauchung oder gar Spiegelung. Der <strong><span style=\"color:#52b004\" class=\"has-inline-color\">Summand c<\/span><\/strong><span style=\"color:#73ff00\" class=\"has-inline-color\"> <\/span> in der Funktionsvorschrift verschiebt den Graphen auf der y-Achse.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#ffffc5\">\u25ba<a rel=\"noreferrer noopener\" href=\"https:\/\/www.geogebra.org\/m\/wvnjnae9\" target=\"_blank\">externer Link zur hier folgenden Geogebra Anwendung<\/a><\/p>\n\n\n\n<iframe loading=\"lazy\" scrolling=\"no\" title=\"Quadratische Funktion der Form f(x) = ax\u00b2+c\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/wvnjnae9\/width\/1066\/height\/517\/border\/888888\/sfsb\/true\/smb\/false\/stb\/false\/stbh\/false\/ai\/false\/asb\/false\/sri\/false\/rc\/false\/ld\/false\/sdz\/false\/ctl\/false\" width=\"1066px\" height=\"517px\" style=\"border:0px; allowfullscreen=\"\"> <\/iframe>\n\n\n\n<p>geogebra.org, Anton Bruckner<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#c8efff\"><strong>Der Scheitelpunkt  S<\/strong><\/p>\n\n\n\n<p>Der Scheitelpunkt der Funktion liegt bei <strong>S( 0| <span style=\"color:#004cff\" class=\"has-inline-color\">c<\/span>)<\/strong>. Er ist also direkt von <span style=\"color:#003cff\" class=\"has-inline-color\">c<\/span> abh\u00e4ngig.<br><br>Beispiel:<br>Die Funktion y= 2x\u00b2 <span style=\"color:#0048ff\" class=\"has-inline-color\"><strong>\u2013 5<\/strong><\/span> hat ihren Scheitelpunkt also bei  <strong>S( 0| <span style=\"color:#004cff\" class=\"has-inline-color\">\u20135<\/span>)<\/strong>. <\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#fffbad\"><strong>Die Nullstelle x<sub>0<\/sub> \/ die Nullstellen<\/strong> der Funktion<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#feffda\">Die Funktion hat maximal 2 Nullstellen &#8211; Schnittpunkte mit der x-Achse.<br>Diese liegen bei x<sub>01<\/sub>= + \\( \\sqrt{\\frac{-c}{a}} \\) und  x<sub>02<\/sub> = \u2013 \\( \\sqrt{\\frac{-c}{a}} \\) <br><\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#feffda\">Beispiel Nullstellenberechnung:<\/p>\n\n\n\n<p><strong>y = 2x\u00b2 &#8211; 5<\/strong><\/p>\n\n\n\n<p><strong>Setze <span style=\"color:#ff0004\" class=\"has-inline-color\">y = 0 <\/span>und forme die Gleichung danach nach x um!<\/strong><\/p>\n\n\n\n<p><span style=\"color:#ff0004\" class=\"has-inline-color\">0 <\/span>= 2x\u00b2 &#8211; 5 | <strong>+ 5<\/strong><\/p>\n\n\n\n<p><strong>5<\/strong> = 2x\u00b2 | : 2<\/p>\n\n\n\n<p>2,5 = x\u00b2 | \\( \\sqrt{} \\)<\/p>\n\n\n\n<p>x<sub>01 <\/sub>= + \\( \\sqrt{2,5} \\) = 1,58<\/p>\n\n\n\n<p> x<sub>02<\/sub> = \u2013 \\( \\sqrt{2,5} \\) = \u20131,58<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#caff8d\"><strong>Die Wertetabelle<\/strong><\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#e6ffc5\"><strong>y=4x\u00b2 \u20131 <\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">x<\/td><td class=\"has-text-align-center\" data-align=\"center\"> <span class=\"has-inline-color has-primary-color\"><strong>\u20132<\/strong><\/span><\/td><td class=\"has-text-align-center\" data-align=\"center\"><span class=\"has-inline-color has-primary-color\"><strong>\u20131<\/strong><\/span><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong><span class=\"has-inline-color has-primary-color\">0<\/span><\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><span class=\"has-inline-color has-primary-color\"><strong>5<\/strong><\/span><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">y<\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"has-background\" style=\"background-color:#e2ffbd\"><strong>y-Werte i<\/strong>n einer Wertetabelle werden <strong>durch Einsetzen<\/strong> der gegebenen x-Werte in die Funktionsvorschrift errechnet.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">x<\/td><td class=\"has-text-align-center\" data-align=\"center\"> <span class=\"has-inline-color has-primary-color\"><strong>\u20132<\/strong><\/span><\/td><td class=\"has-text-align-center\" data-align=\"center\"><span class=\"has-inline-color has-primary-color\"><strong>\u20131<\/strong><\/span><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong><span class=\"has-inline-color has-primary-color\">0<\/span><\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><span class=\"has-inline-color has-primary-color\"><strong>5<\/strong><\/span><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">y<\/td><td class=\"has-text-align-center\" data-align=\"center\"> = 4*(<span class=\"has-inline-color has-primary-color\"><strong>\u20132<\/strong><\/span>)\u00b2  <strong>\u2013<\/strong> 1 <br>= 15<\/td><td class=\"has-text-align-center\" data-align=\"center\">=4*(<strong><span class=\"has-inline-color has-primary-color\">\u20131<\/span><\/strong>)\u00b2 <strong>\u2013<\/strong> 1<br>= 3<\/td><td class=\"has-text-align-center\" data-align=\"center\">=4*(<strong><span class=\"has-inline-color has-primary-color\">0<\/span><\/strong>)\u00b2 <strong>\u2013<\/strong> 1 <br>= \u20131<\/td><td class=\"has-text-align-center\" data-align=\"center\">=4*(<strong><span class=\"has-inline-color has-primary-color\">5<\/span><\/strong>)\u00b2 <strong>\u2013<\/strong> 1<br>= 99<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator is-style-wide\"\/>\n\n\n\n<p class=\"has-background\" style=\"background-color:#d1fff1\"><strong>y=2x\u00b2+1<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">x<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">y<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">100<\/td><td class=\"has-text-align-center\" data-align=\"center\">-12<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"has-background\" style=\"background-color:#ccfff0\">Sind allerdings<strong> x-Werte gesucht <\/strong>und y-Werte gegeben, so wird &#8230;<br><br>&#8230;der y-Wert in die Funktionsvorschrift eingesetzt <br>und <br>&#8230;die Gleichung dann durch Umstellen nach x gel\u00f6st.<strong> <\/strong><br><br><strong>Achtung: Hier sind 2 L\u00f6sungen m\u00f6glich!<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">x<\/td><td class=\"has-text-align-left\" data-align=\"left\">y=5<br>5=2x\u00b2+1 |  \u2013 1<br>4= 2x\u00b2 | :2<br>2 = x\u00b2 | \u221a<br>x<sub>1<\/sub>=1,41<br>x<sub>2<\/sub> = \u20131,41<\/td><td class=\"has-text-align-left\" data-align=\"left\"> y=8<br>8=2x\u00b2+1 |  \u2013 1<br>7= 2x\u00b2 | :2<br>3,5 = x\u00b2 | \u221a<br>x<sub>1<\/sub>=1,87<br>x<sub>2<\/sub> = \u20131,87<\/td><td class=\"has-text-align-left\" data-align=\"left\"> y=100<br>100=2x\u00b2+1 |  \u2013 1<br>99= 2x\u00b2 | :2<br>49,5 = x\u00b2 | \u221a<br>x<sub>1<\/sub>=7,04<br>x<sub>2<\/sub> = \u20137,04 <\/td><td class=\"has-text-align-left\" data-align=\"left\">  y=\u201312<br>\u201312=2x\u00b2+1 |  \u2013 1<br>\u201313= 2x\u00b2 | :2<br>\u20136,5 = x\u00b2 | \u221a<br>x<sub>1<\/sub> = n.d. <sup><span style=\"color:#d10b0b\" class=\"has-inline-color\">(1)<\/span><\/sup><br>x<sub>2<\/sub> = n.d.  <sup><span style=\"color:#d10b0b\" class=\"has-inline-color\">(1)<\/span><\/sup>  <\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">y<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td class=\"has-text-align-left\" data-align=\"left\">8<\/td><td class=\"has-text-align-left\" data-align=\"left\">100<\/td><td class=\"has-text-align-left\" data-align=\"left\">-12<\/td><\/tr><\/tbody><\/table><figcaption> <sup><span style=\"color:#d10b0b\" class=\"has-inline-color\">(1)<\/span><\/sup> <strong>Die Wurzel aus einer negativen Zahl ist nicht definiert (n.d.)! <\/strong>Es existiert kein Punkt der Funktion mit y=\u201312 also A(__|\u201312)<\/figcaption><\/figure>\n\n\n\n<p class=\"has-background\" style=\"background-color:#a4ffd9\">Vereinfachend tr\u00e4gt man in die Wertetabelle die Ergebnisse mit einem Plusminus-Zeichen(\u00b1) ein. <\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">x<\/td><td class=\"has-text-align-left\" data-align=\"left\"><strong>\u00b11,41<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>\u00b11,87<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>\u00b17,04<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>n.d.<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">y<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">100<\/td><td class=\"has-text-align-center\" data-align=\"center\">-12<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><\/p>\n<div class=\"pvc_clear\"><\/div><p id=\"pvc_stats_4437\" class=\"pvc_stats all  \" data-element-id=\"4437\" style=\"\"><i class=\"pvc-stats-icon small\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/mathe-lernen.net\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p><div class=\"pvc_clear\"><\/div> ","protected":false},"excerpt":{"rendered":"<p>Der Graph der Funktion y=ax\u00b2 + c ist eine Parabel. Der Faktor a sorgt dabei f\u00fcr Streckung, Stauchung oder gar Spiegelung. Der Summand c in der Funktionsvorschrift verschiebt den Graphen auf der y-Achse. \u25baexterner Link zur hier folgenden Geogebra Anwendung<\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_4437\" class=\"pvc_stats all  \" data-element-id=\"4437\" style=\"\"><i class=\"pvc-stats-icon small\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/mathe-lernen.net\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[67,54],"tags":[127,128,491,1086],"class_list":["post-4437","post","type-post","status-publish","format-standard","hentry","category-grundwissen","category-quadratische-funktionen","tag-quadratisch","tag-quadratische-funktion","tag-funktion","tag-axc"],"_links":{"self":[{"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=\/wp\/v2\/posts\/4437","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4437"}],"version-history":[{"count":0,"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=\/wp\/v2\/posts\/4437\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4437"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4437"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathe-lernen.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}