{"id":90,"date":"2017-08-23T07:46:36","date_gmt":"2017-08-23T07:46:36","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=90"},"modified":"2017-09-13T07:39:58","modified_gmt":"2017-09-13T07:39:58","slug":"conic-sections","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/conic-sections\/","title":{"rendered":"Conic Sections"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/binomial-expansion\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-integer-exponents\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Conic Sections<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the basics of three different conic sections: circles, parabolas, and ellipses.<\/p>\n<h4>Previously Covered:<\/h4>\n<p>Binomials can be expanded using two methods:<\/p>\n<ul>\n<li><strong><em>Pascal\u2019s\u00a0triangle<\/em><\/strong> is a quick and efficient way to generate\u00a0the coefficients in the binomial expansion.<\/li>\n<li>The <em><strong>Binomial Theorem<\/strong><\/em> requires\u00a0knowledge of computing combinations using factorials.<\/li>\n<\/ul>\n<section>\n<h3>Conic Sections<\/h3>\n<p>Intersecting a single or double cone with a geometric plane\u00a0produces shapes called <abbr title=\"the intersection of a plane and a cone\">conic\u00a0sections<\/abbr>. In this lesson we will review three types of conic sections, or conics:\u00a0circles, parabolas, and ellipses.<\/p>\n<h3>Circles<\/h3>\n<p>The standard form of the equation for a circle is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image003.gif\" width=\"141\" height=\"17\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>where <em>r\u00a0<\/em>is the radius, (<em>x<\/em>,\u00a0<em>y<\/em>)\u00a0is the coordinate plane location for any point on the circle, and\u00a0(<em>h<\/em>,\u00a0<em>k<\/em>)\u00a0is the coordinate plane location for the center of the circle.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>If the circle is centered at the origin, then the equation simplifies\u00a0to<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image006.gif\" width=\"74\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>The following graph shows a circle centered at the origin, with\u00a0a radius of 2. The equation of the circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image014.gif\" width=\"69\" height=\"17\" name=\"graphics5\" align=\"ABSBOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20009.JPG\" alt=\"Circle with center at the (0,0)\" width=\"370\" height=\"172\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The following graph shows a circle centered at (3, 2), with a\u00a0radius of 2. The equation of the circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image017.gif\" width=\"137\" height=\"17\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20010.JPG\" alt=\"Circle with center at (3,2)\" width=\"357\" height=\"168\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the equation of a circle with a radius of 3, centered\u00a0at (\u20132, 4)?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image020.gif\" width=\"138\" height=\"17\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image023.gif\" width=\"138\" height=\"17\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image026.gif\" width=\"138\" height=\"17\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image029.gif\" width=\"138\" height=\"17\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A.\u00a0We know that <em>h\u00a0<\/em>= \u20132, <em>k\u00a0<\/em>= 4, and the radius is 3. Using the standard form, the equation\u00a0becomes\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p2_clip_image031.gif\" width=\"138\" height=\"17\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Parabolas<\/h3>\n<p>Previously, we discussed parabolas and how they graphically\u00a0reflect the behavior of quadratic equations. Now, we will look at\u00a0the actual graphs themselves and some of the characteristics of\u00a0parabolas in more depth.<\/p>\n<p>Two parts of the graph that are important to identify are the\u00a0<abbr title=\" a line located outside of the curvature of the parabola, perpendicular to the axis of symmetry. Any point located on the parabola is the same distance from the focus and the directrix. \">directrix<\/abbr> and the <abbr title=\"a fixed point on the inside of the curve of a parabola located along the axis of symmetry. Any point located on the parabola is the same distance from the focus and the directrix. \">focus<\/abbr>.\u00a0The focus is a fixed point on the inside of a parabola, and the\u00a0directrix is a fixed line on the outside of a parabola. The\u00a0directrix is a line whose distance from the vertex is the same as\u00a0the foci&#8217;s distance from the vertex. Any point on the graph of a\u00a0parabola is the same distance from the focus as from the\u00a0directrix.<\/p>\n<p>The equation for the following graph is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image003.gif\" width=\"47\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image006.gif\" width=\"42\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.The constant <em>c\u00a0<\/em>identifies the location of the focus and directrix.<\/p>\n<p>In this graph,\u00a0the focus is located at (0, 3), therefore <em>c\u00a0<\/em>= 3. The directrix is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s6_p3_html_m6424b129.gif\" width=\"48\" height=\"21\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0The equation of the parabola is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image009.gif\" width=\"58\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20011.JPG\" alt=\" Graph of y=ax2\" width=\"241\" height=\"113\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<p>One example of a focus point that can be found in everyday\u00a0objects is a flashlight. The bulb is seated at the focal point\u00a0within the reflective surface, which is sort of like a parabola.\u00a0Another example is a satellite dish. The receiver is located at\u00a0the focal point, and the actual dish forms the shape of a\u00a0parabola.<\/p>\n<p>Parabolas can have four different configurations.<\/p>\n<table width=\"75%\">\n<tbody>\n<tr>\n<td>Open upward: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image011.gif\" width=\"47\" height=\"17\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20012.JPG\" alt=\" Parabola opening up\" width=\"181\" height=\"172\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td>Open downward:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image014.gif\" width=\"56\" height=\"17\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20013.JPG\" alt=\"Parabola opening down\" width=\"181\" height=\"167\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td>Open left:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image017.gif\" width=\"54\" height=\"17\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20014.JPG\" alt=\" Parabola opening left\" width=\"178\" height=\"167\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td>Open right:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image020.gif\" width=\"45\" height=\"17\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203.2%20Art%20015.JPG\" alt=\"Parabola opening right\" width=\"183\" height=\"171\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the equation of a parabola that opens upward and has a\u00a0directrix at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image023.gif\" width=\"42\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image025.gif\" width=\"58\" height=\"34\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image028.gif\" width=\"50\" height=\"34\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image031.gif\" width=\"51\" height=\"34\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image034.gif\" width=\"51\" height=\"34\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B. If the directrix is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image036.gif\" width=\"42\" height=\"14\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0then the focus is (0, 2). Therefore,\u00a0<em>c <\/em>=\u00a02<em>.<\/em><br \/>\nThe equation for a parabola that opens upward is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p3_clip_image039.gif\" width=\"141\" height=\"34\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Ellipses<\/h3>\n<p>If a geometric plane intersects a single cone at an angle, the\u00a0result is an <abbr title=\"section of a right circular cone that is a closed curve\">ellipse<\/abbr>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>The following definitions pertain to ellipses:<\/p>\n<ul>\n<li>Focus points \u2013 Ellipses have two focus points along the major axis.<\/li>\n<li>Vertices \u2013 the endpoints of the ellipse along the major axis<\/li>\n<li>Co-vertices \u2013 the endpoints of the ellipse along the minor axis<\/li>\n<\/ul>\n<\/div>\n<p>The ellipse has two separate configurations, and therefore has\u00a0two separate standard forms.<\/p>\n<h3><strong>Major axis on the <em>x<\/em>-axis<\/strong><\/h3>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math-Mod-3.2-Art-016.gif\" alt=\"Ellipse with major axis on x-axis\" width=\"393\" height=\"215\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The standard form for an ellipse centered at zero with vertices\u00a0on the <em>x<\/em>-axis\u00a0and co-vertices on the <em>y<\/em>-axis\u00a0is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p4_clip_image003.gif\" width=\"129\" height=\"37\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3><strong>Major axis on the <em>y<\/em>-axis <\/strong><\/h3>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math-Mod-3.2-Art-017b.gif\" alt=\"Ellipse with major axis on y-axis\" width=\"393\" height=\"268\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>The standard form for an\u00a0ellipse centered at zero with co-vertices on the <em>x-<\/em>axis\u00a0and vertices on the <em>y<\/em>-axis\u00a0is<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p4_clip_image006.gif\" width=\"129\" height=\"37\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/center>For example, identify the major axis, vertices, and co-vertices\u00a0for the ellipse modeled by the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p4_clip_image009.gif\" width=\"116\" height=\"17\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<p><strong>Step 1:<\/strong> Write the equation in standard form.\u00a0To convert the equation to standard form, divide each term by 400.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s6_p4_clip_image012.gif\" width=\"72\" height=\"36\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/center><strong>Step 2:<\/strong> Since\u00a0<em>a<\/em> &gt; <em>b<\/em>, we know that <em>a\u00a0<\/em>= 5 , <em>b\u00a0<\/em>= 4, and the major\u00a0axis is the <em>y<\/em>-axis.<\/p>\n<p><strong>Step 3:<\/strong> Since\u00a0<em>y\u00a0<\/em>is the major axis, the vertices are located at (0, 5) and (0, \u20135).\u00a0The co-vertices are located at (\u20134, 0) and (4, 0).<\/p>\n<h3>Review of New Vocabulary and Terms<\/h3>\n<ul>\n<li>There are numerous ways to solve quadratic equations:\n<ul>\n<li>graphing,<\/li>\n<li>factoring,<\/li>\n<li>completing the square, and<\/li>\n<li>using the quadratic formula.<\/li>\n<\/ul>\n<\/li>\n<li>There are two forms of a\u00a0quadratic equation, <em><strong>standard form<\/strong><\/em> and\u00a0<strong><em>vertex form<\/em><\/strong>. The vertex form is used to\u00a0easily identify the coordinate location of the <strong><em>vertex<\/em><\/strong>.<\/li>\n<li>The real-number solution to a\u00a0quadratic equation is the point or points where <em>f<\/em>(<em>x<\/em>)\u00a0= 0.<\/li>\n<li><strong><em>Roots\u00a0<\/em><\/strong>of quadratic equations can be either real or complex.<\/li>\n<li>Binomials can be expanded using two methods:\n<ul>\n<li>Pascal\u2019s triangle &#8211; <strong><em>Pascal\u2019s triangle<\/em><\/strong> is an efficient way to generate the\u00a0coefficients in the binomial expansion.<\/li>\n<li>The Binomial Theorem &#8211; The <strong><em>Binomial Theorem<\/em><\/strong> requires knowledge of computing\u00a0combinations, using factorials.<\/li>\n<\/ul>\n<\/li>\n<li>Intersecting a single or double\u00a0cone with a geometric plane produces shapes called <strong><em>conic\u00a0sections<\/em><\/strong> such as circles, parabolas, and ellipses.<\/li>\n<li>The <em><strong>focus point<\/strong><\/em> and <em><strong>directrix\u00a0<\/strong><\/em>of a parabola are the same distance away from the vertex.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/binomial-expansion\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-integer-exponents\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Conic Sections Objective In this lesson, you will study the basics of three different conic sections: circles, parabolas, and ellipses. Previously Covered: Binomials can be expanded using two methods: Pascal\u2019s\u00a0triangle is a quick and efficient way to generate\u00a0the coefficients in the binomial expansion. The Binomial Theorem requires\u00a0knowledge of computing [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-90","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/90","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=90"}],"version-history":[{"count":14,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/90\/revisions"}],"predecessor-version":[{"id":688,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/90\/revisions\/688"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=90"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}