document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=124130>Question 124130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the function f(x)=x^2+4x+1<BR>\n" );
document.write( "Find h, the x-coordinate of the vertex of this parabola<BR>\n" );
document.write( "So I took -b/2a or -4/2(1)^2=-4/2<BR>\n" );
document.write( "So I believe h=-2<BR>\n" );
document.write( "The next part of the question is: Substitute the two integers immediately to the left and right of h into the function to find the corresponding y.  Fill in the following table. Make sure your x-values are in increasing order in your table.<BR>\n" );
document.write( "x	y<BR>\n" );
document.write( "_________	<BR>\n" );
document.write( "_________	<BR>\n" );
document.write( "h =__	<BR>\n" );
document.write( "_________	<BR>\n" );
document.write( "_________	<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #91013 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=91013\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One minor comment to your work as noted below:<BR>\n" );
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document.write( "Consider the function f(x)=x^2+4x+1<BR>\n" );
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document.write( "Find h, the x-coordinate of the vertex of this parabola<BR>\n" );
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document.write( "So I took -b/2a or -4/2(1)^2=-4/2 <=== don't square the (1). The denominator is just 2*(1)<BR>\n" );
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document.write( "So I believe h=-2 <=== correct<BR>\n" );
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document.write( "The next part of the question is: Substitute the two integers immediately to the left and <BR>\n" );
document.write( "right of h into the function to find the corresponding y. Fill in the following table. Make <BR>\n" );
document.write( "sure your x-values are in increasing order in your table.<BR>\n" );
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document.write( "(I'm going to use periods as \"space holders only\" in the table. They are just to keep the table formatted.<BR>\n" );
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document.write( "The two integers to the left of h = -2 are x = -4 and x = -3. The two integers to the right<BR>\n" );
document.write( "of h = -2 are x = -1 and x = 0<BR>\n" );
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document.write( "You are to substitute these 5 values (-4, -3, -2, -1, and 0) for the value of x in the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2%2B4x%2B1 ALIGN=MIDDLE  ALT=\"y+=+x%5E2%2B4x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Table:<BR>\n" );
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document.write( "... x .... y<BR>\n" );
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document.write( ".. -4 ... +1 <=== Note that y = (-4)^2 + 4*(-4) + 1 = +16 - 16 + 1 = +1<BR>\n" );
document.write( ".. -3 ... -2 <=== Note that y = (-3)^2 + 4*(-3) + 1 = +9 - 12 + 1 = -2<BR>\n" );
document.write( ".. -2 ... -3 <=== x is h  and therefore y = (-2)^2 + 4*(-2) + 1 = 4 - 8 + 1 = -3<BR>\n" );
document.write( ".. -1 ... -2 <=== y = (-1)^2 + 4*(-1) + 1 = 1 - 4 +1 = -2<BR>\n" );
document.write( "... 0 ... +1 <=== y = 0^2 + 4*0 + 1 = 0 + 0 + 1 = +1<BR>\n" );
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document.write( "Notice the values of y for values of x that are equal distance from x = -2. When x = -4 and x = 0<BR>\n" );
document.write( "the values of x are both 2 units away from x = -2. And the values of y at these two distances<BR>\n" );
document.write( "are both +1.<BR>\n" );
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document.write( "Similarly when x = -1 and x = -3 both these values are 1 unit away from x = -2. And in both<BR>\n" );
document.write( "these cases y = -2. <BR>\n" );
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document.write( "This is also shown in the graph of y = x^2 + 4x + 1 which is shown below. The vertical green<BR>\n" );
document.write( "line through x = -2 represents the line of symmetry, which in this case means that if you<BR>\n" );
document.write( "printed this graph out and then folded it along the vertical green line, the two sides of the <BR>\n" );
document.write( "graph would be exactly on top of each other. <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C+5%2C+-5%2C+5%2C+x%5E2+%2B+4x+%2B+1%2C+6000%28x+%2B+2%29%2C-2%2C-1%2C1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to understand what I think you are being taught ... namely what symmetry<BR>\n" );
document.write( "means. Notice the horizontal lines I put on the graph. If you look at each horizontal line<BR>\n" );
document.write( "and the points where the graph intercepts them you will see that the distance between those <BR>\n" );
document.write( "intersections and the green line are equal on both sides of the green line of symmetry.<BR>\n" );
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document.write( "That's sort of what symmetry means. The graph on one side of the green line is the \"mirror image\"<BR>\n" );
document.write( "of the graph on the other side of the green line.<BR>\n" );
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