document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111615>Question 1111615</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The vertex of this parabola is at (2, -4). When the y-value is -3, the x-value is -3. What is the coefficient of the squared term in the parabola's equation? </SPAN>\n" );
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document.write( "The problem is badly presented.  The statement \"when the y value is -3, the x value is -3\" suggests that y is the independent variable, making the parabola open in the horizontal direction instead of the vertical direction.<br><BR>\n" );
document.write( "So the way I would interpret the problem, the equation of the parabola (vertex form) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2+=+a%28y%2B4%29%5E2 ALIGN=MIDDLE  ALT=\"x-2+=+a%28y%2B4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then we can find the value of a (the coefficient of the squared term, which is what the problem asks for) using the given point (-3,-3) on the parabola.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3-2+=+a%28-3%2B4%29%5E2 ALIGN=MIDDLE  ALT=\"-3-2+=+a%28-3%2B4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+=+a ALIGN=MIDDLE  ALT=\"-5+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The coefficient on the squared term is -5; and the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2+=+-5%28y%2B4%29%5E2 ALIGN=MIDDLE  ALT=\"x-2+=+-5%28y%2B4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
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document.write( "Now let's suppose that in fact the intent of the problem was for x to be the independent variable, making the parabola open in the vertical direction.<br><BR>\n" );
document.write( "So now we have the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B4+=+a%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"y%2B4+=+a%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "As above, we can find the value of a using the given point (-3,-3):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2B4+=+a%28-3-2%29%5E2 ALIGN=MIDDLE  ALT=\"-3%2B4+=+a%28-3-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+25a ALIGN=MIDDLE  ALT=\"1+=+25a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F25 ALIGN=MIDDLE  ALT=\"a+=+1%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If you understand parabolas, there is another way to find the value of a which is, I think, much easier.<br><BR>\n" );
document.write( "In the parabola y=x^2, when you are 1 unit either side of the line of symmetry the function value is 1 more than at the vertex; when you are 2 units either side the function value is 4 more than at the vertex; and so on.<br><BR>\n" );
document.write( "In the parabola y=ax^2, when you are 1 unit either side of the line of symmetry the function value is 1*a=a more than at the vertex; when you are 2 units either side the function value is 4*a=4a more than at the vertex; and so on.<br><BR>\n" );
document.write( "So in this problem, we have the vertex at (2,-4) and a point on the parabola at (-3,-3).  The point is 5 units to the left of the vertex; if the coefficient of the squared term were 1, the y value at the given point would be 25 away from the y value at the vertex.  But the y value at the given point is only 1 away from the y value at the vertex; that means the coefficient a is 1/25.<BR>\n" );
document.write( " </SPAN>\n" );
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