document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206260>Question 206260</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all numbers less than or equal to 4. Determine the expression for the function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #155843 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=155843\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your x intercepts are -1 and 3 and this is a quadratic equation.<BR>\n" );
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document.write( "this means that (x+1)*(x-3) = 0 because that is the quadratic equation that would yield these 2 x intercepts.<BR>\n" );
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document.write( "multiplying them out we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+-+3+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+-+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "that's our equation.<BR>\n" );
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document.write( "a limitation is that the range of our function can't be greater than 4.<BR>\n" );
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document.write( "Since y represents the range of our function, this means y has to be less than or equal to 4.<BR>\n" );
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document.write( "since our equation is  y = f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+-+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-2x+-+3+=+4 ALIGN=MIDDLE  ALT=\"x%5E2+-2x+-+3+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > would represent the maximum value that y could be.<BR>\n" );
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document.write( "in order to restrict the range, we have to restrict the domain because the range is dependent on the domain.<BR>\n" );
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document.write( "we need to find the x values for when y = 4<BR>\n" );
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document.write( "if we subtract 4 from both sides of the equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-2x+-+3+=+4 ALIGN=MIDDLE  ALT=\"x%5E2+-2x+-+3+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-2x+-+7+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-2x+-+7+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "if we find the roots of this equation, we should be able to find the x values for when y = 4.<BR>\n" );
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document.write( "solving the equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-2x+-+7+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-2x+-+7+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the quadratic formula of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > yields the following answers:<BR>\n" );
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document.write( "x = 3.828427125 or x = -1.828427125<BR>\n" );
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document.write( "we got this in the following manner:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -2<BR>\n" );
document.write( "c = -7<BR>\n" );
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document.write( "2a = 2<BR>\n" );
document.write( "-b = 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28b%5E2+-+4ac%29 ALIGN=MIDDLE  ALT=\"sqrt%28b%5E2+-+4ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284-%284%2A1%2A%28-7%29%29%29 ALIGN=MIDDLE  ALT=\"sqrt%284-%284%2A1%2A%28-7%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2B28%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2B28%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2832%29 ALIGN=MIDDLE  ALT=\"sqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+%2B-+sqrt%2832%29%29%2F2 ALIGN=MIDDLE  ALT=\"%282+%2B-+sqrt%2832%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes x = 3.828427125 or x = -1.828427125.<BR>\n" );
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document.write( "The graph of our equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+-+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the restriction that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.828427125+%3C=+x+%3C=+3.828427125 ALIGN=MIDDLE  ALT=\"-1.828427125+%3C=+x+%3C=+3.828427125\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "a graph of our equation would look like this:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-3%2C5%2C-6%2C6%2Cx%5E2-2x-3%2C4%2C1000%2F%28x%2B2%29%2C1000%2F%28x-4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "you can see that the lower limit for x will be around -1.828... and the upper limit for x will be around +3.828... and that anything in between is good.<BR>\n" );
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document.write( "those horizontal and vertical lines are just there to let you see the x values and the y values easier.  they are not part of the quadratic equation. </SPAN>\n" );
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