document.write( "Question 88331: Identify the axis of symmetry, create a suitable table of values, then sketch the graph (including the axis of symmetry).\r
\n" ); document.write( "\n" ); document.write( "y = –x^2 + 3x – 3\r
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Algebra.Com's Answer #64154 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " $\"y=-1+x%5E2%2B3+x-3\"$ Start with the given equation
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\n" ); document.write( " $\"y%2B3=-1+x%5E2%2B3+x\"$ Add $\"3\"$ to both sides
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\n" ); document.write( " $\"y%2B3=-1%28x%5E2-3x%29\"$ Factor out the leading coefficient $\"-1\"$
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\n" ); document.write( " Take half of the x coefficient $\"-3\"$ to get $\"-3%2F2\"$ (ie $\"%281%2F2%29%28-3%29=-3%2F2\"$ ).
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\n" ); document.write( " Now square $\"-3%2F2\"$ to get $\"9%2F4\"$ (ie $\"%28-3%2F2%29%5E2=%28-3%2F2%29%28-3%2F2%29=9%2F4\"$ )
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\n" ); document.write( " $\"y%2B3=-1%28x%5E2-3x%2B9%2F4-9%2F4%29\"$ Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of $\"9%2F4\"$ does not change the equation
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\n" ); document.write( " $\"y%2B3=-1%28%28x-3%2F2%29%5E2-9%2F4%29\"$ Now factor $\"x%5E2-3x%2B9%2F4\"$ to get $\"%28x-3%2F2%29%5E2\"$
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\n" ); document.write( " $\"y%2B3=-1%28x-3%2F2%29%5E2%2B1%289%2F4%29\"$ Distribute
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\n" ); document.write( " $\"y%2B3=-1%28x-3%2F2%29%5E2%2B9%2F4\"$ Multiply
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\n" ); document.write( " $\"y=-1%28x-3%2F2%29%5E2%2B9%2F4-3\"$ Now add $\"%2B3\"$ to both sides to isolate y
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\n" ); document.write( " $\"y=-1%28x-3%2F2%29%5E2-3%2F4\"$ Combine like terms
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\n" ); document.write( " Now the quadratic is in vertex form $\"y=a%28x-h%29%5E2%2Bk\"$ where $\"a=-1\"$ , $\"h=3%2F2\"$ , and $\"k=-3%2F4\"$ . Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
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\n" ); document.write( " Check:
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\n" ); document.write( " Notice if we graph the original equation $\"y=-1x%5E2%2B3x-3\"$ we get:
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\n" ); document.write( " $\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2%2B3x-3%29\"$ Graph of $\"y=-1x%5E2%2B3x-3\"$ . Notice how the vertex is ( $\"3%2F2\"$ , $\"-3%2F4\"$ ).
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\n" ); document.write( " Notice if we graph the final equation $\"y=-1%28x-3%2F2%29%5E2-3%2F4\"$ we get:
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\n" ); document.write( " $\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%28x-3%2F2%29%5E2-3%2F4%29\"$ Graph of $\"y=-1%28x-3%2F2%29%5E2-3%2F4\"$ . Notice how the vertex is also ( $\"3%2F2\"$ , $\"-3%2F4\"$ ).
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\n" ); document.write( " So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.
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\n" ); document.write( "\n" ); document.write( "Since we know the vertex is ( $\"3%2F2\"$ , $\"-3%2F4\"$ ) or (1.5,-0.75), this is one point on the graph. \r
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\n" ); document.write( "\n" ); document.write( "Now lets pick any point after $\"x=1.5\"$ . Lets evaluate $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=-x%5E2%2B3x-3\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"f%282%29=-%282%29%5E2%2B3%282%29-3\"$ Plug in $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%282%29=-%284%29%2B3%282%29-3\"$ Raise 2 to the second power to get 4\r
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\n" ); document.write( "\n" ); document.write( " $\"f%282%29=-%284%29%2B6-3\"$ Multiply 3 by 2 to get 6\r
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\n" ); document.write( "\n" ); document.write( " $\"f%282%29=-1\"$ Now combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So we get the point (2,-1)\r
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\n" ); document.write( "\n" ); document.write( "Lets pick another value $\"x=3\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=-x%5E2%2B3x-3\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"f%283%29=-%283%29%5E2%2B3%283%29-3\"$ Plug in $\"x=3\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%283%29=-%289%29%2B3%283%29-3\"$ Raise 3 to the second power to get 9\r
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\n" ); document.write( "\n" ); document.write( " $\"f%283%29=-%289%29%2B9-3\"$ Multiply 3 by 3 to get 9\r
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\n" ); document.write( "\n" ); document.write( " $\"f%283%29=-3\"$ Now combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So another point is (3,-3)\r
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\n" ); document.write( "\n" ); document.write( "Now since the graph is symmetrical with respect to the axis of symmetry, this means x-values on the other side of the vertex will have the same y-values as their respective counterparts. For instance, the counterpart to $\"x=2\"$ is $\"x=1\"$ and the counterpart to $\"x=3\"$ is $\"x=0\"$ (notice they are the same distance away from the vertex along the x-axis)\r
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\n" ); document.write( "\n" ); document.write( "So here's the table of suitable values\r
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document.write( " x y
 0 -3
 1 -1
 1.5 -0.75
 2 -1
 3 -3\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice if we graph the equation $\"y=-x%5E2%2B3x-3\"$ and the table of points we get\r
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\n" ); document.write( "\n" ); document.write( "Since the points lie on the curve, this verifies our answer. \n" ); document.write( "

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