document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170844>Question 1170844</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a parabola with a length of 72 feet and a height of 64 ft, the goal is to find the vertex as well as the x-intercepts and then write the parabola in standard, vertex and factored form; thanks a ton! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851091 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Set up a Coordinate System**\r<BR>\n" );
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document.write( "To make the calculations easier, let's place the parabola in a coordinate system. We'll place the vertex on the y-axis, which will simplify the equations.\r<BR>\n" );
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document.write( "* Let the vertex be at (0, 64).<BR>\n" );
document.write( "* The parabola opens downwards since it has a maximum height of 64 ft.<BR>\n" );
document.write( "* The length of 72 feet means the x-intercepts are at (-36, 0) and (36, 0).\r<BR>\n" );
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document.write( "**2. Vertex Form**\r<BR>\n" );
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document.write( "The vertex form of a parabola is:\r<BR>\n" );
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document.write( "y = a(x - h)² + k\r<BR>\n" );
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document.write( "where (h, k) is the vertex.\r<BR>\n" );
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document.write( "* We know the vertex is (0, 64), so h = 0 and k = 64.<BR>\n" );
document.write( "* y = a(x - 0)² + 64<BR>\n" );
document.write( "* y = ax² + 64\r<BR>\n" );
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document.write( "**3. Find the Value of 'a'**\r<BR>\n" );
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document.write( "We can use one of the x-intercepts to find 'a'. Let's use (36, 0):\r<BR>\n" );
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document.write( "* 0 = a(36)² + 64<BR>\n" );
document.write( "* 0 = 1296a + 64<BR>\n" );
document.write( "* -64 = 1296a<BR>\n" );
document.write( "* a = -64 / 1296<BR>\n" );
document.write( "* a = -1 / 20.25<BR>\n" );
document.write( "* a = -4 / 81\r<BR>\n" );
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document.write( "Therefore, the vertex form is:\r<BR>\n" );
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document.write( "y = (-4/81)x² + 64\r<BR>\n" );
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document.write( "**4. Standard Form**\r<BR>\n" );
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document.write( "The standard form of a parabola is:\r<BR>\n" );
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document.write( "y = ax² + bx + c\r<BR>\n" );
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document.write( "We already have the vertex form:\r<BR>\n" );
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document.write( "y = (-4/81)x² + 64\r<BR>\n" );
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document.write( "Since there's no 'x' term, b = 0. So the standard form is:\r<BR>\n" );
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document.write( "y = (-4/81)x² + 64\r<BR>\n" );
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document.write( "**5. Factored Form**\r<BR>\n" );
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document.write( "The factored form of a parabola is:\r<BR>\n" );
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document.write( "y = a(x - r₁)(x - r₂)\r<BR>\n" );
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document.write( "where r₁ and r₂ are the x-intercepts.\r<BR>\n" );
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document.write( "* We know the x-intercepts are (-36, 0) and (36, 0), so r₁ = -36 and r₂ = 36.<BR>\n" );
document.write( "* We know a = -4/81.\r<BR>\n" );
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document.write( "Therefore, the factored form is:\r<BR>\n" );
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document.write( "y = (-4/81)(x - (-36))(x - 36)<BR>\n" );
document.write( "y = (-4/81)(x + 36)(x - 36)\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **Vertex:** (0, 64)<BR>\n" );
document.write( "* **X-intercepts:** (-36, 0) and (36, 0)<BR>\n" );
document.write( "* **Vertex Form:** y = (-4/81)x² + 64<BR>\n" );
document.write( "* **Standard Form:** y = (-4/81)x² + 64<BR>\n" );
document.write( "* **Factored Form:** y = (-4/81)(x + 36)(x - 36)<BR>\n" );
document.write( " </SPAN>\n" );
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