document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1037252>Question 1037252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The parabola has a turning point at (z, -8). It intersects the y-axis at y=10 and one of the x-intercepts is x=5. Find:<BR>\n" );
document.write( "the value of z </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #652592 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=652592\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the quadratic function be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+a%28x-r%29%28x-5%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+a%28x-r%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where x=5 is one of the roots (as given via the x-intercept), and x = r is the other root.\r<BR>\n" );
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document.write( "Since y = 10 is the value of the y-intercept, the constant term in the quadratic function must be equal to 5ar, and moreover, 5ar = 10, by hypothesis.\r<BR>\n" );
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document.write( "==> ar = 2.\r<BR>\n" );
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document.write( "Since the turning point (or the vertex of the parabola) is always midway between the two roots, it follows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28r%2B5%29%2F2 ALIGN=MIDDLE  ALT=\"z+=+%28r%2B5%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28%28r%2B5%29%2F2%29+=+a%28%28r%2B5%29%2F2-r%29%28%28r%2B5%29%2F2-5%29+=+-a%28r-5%29%5E2%2F4+=+-8 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28r-5%29%5E2+=+32 ALIGN=MIDDLE  ALT=\"a%28r-5%29%5E2+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Divide this equation by the equation ar = 2, giving\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-5%29%5E2%2Fr+=+16 ALIGN=MIDDLE  ALT=\"%28r-5%29%5E2%2Fr+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-5%29%5E2+=+16r ALIGN=MIDDLE  ALT=\"%28r-5%29%5E2+=+16r\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-26r%2B25+=+%28r-25%29%28r-1%29+=+0 ALIGN=MIDDLE  ALT=\"r%5E2-26r%2B25+=+%28r-25%29%28r-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after expanding and simplifying.\r<BR>\n" );
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document.write( "==> r = 25 or r = 1.<BR>\n" );
document.write( "If r = 25, then a = 2/25.<BR>\n" );
document.write( "If r = 1, then a = 2.\r<BR>\n" );
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document.write( "Hence there are two possible answers:<BR>\n" );
document.write( "(i) The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%282%2F25%29%28x-25%29%28x-5%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%282%2F25%29%28x-25%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%2825%2B5%29%2F2+=+15 ALIGN=MIDDLE  ALT=\"z+=+%2825%2B5%29%2F2+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while, <BR>\n" );
document.write( "(ii) The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2%28x-1%29%28x-5%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2%28x-1%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%281%2B5%29%2F2+=+3 ALIGN=MIDDLE  ALT=\"z+=+%281%2B5%29%2F2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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