document.write( "Question 83111: Rewrite each equation in vertex form. Then sketch the graph.\r
\n" ); document.write( "\n" ); document.write( "1. a. y=x^2+4x-6 \r
\n" ); document.write( "\n" ); document.write( " b.y=4x^2+4x+1\r
\n" ); document.write( "\n" ); document.write( " c.y=-3x^2+3x-1\r
\n" ); document.write( "\n" ); document.write( " d. y=-2x^2+4x+3
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\n" ); document.write( " e.y=6x^2-12x+1\r
\n" ); document.write( "\n" ); document.write( "I dont know how to do these they are really confusing
\n" ); document.write( "HELP!!!Please and Thanks \n" ); document.write( "
Algebra.Com's Answer #59676 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
a.
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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%2B4+x-6\" Start with the given equation
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\n" ); document.write( " \"y%2B6=1+x%5E2%2B4+x\" Add \"6\" to both sides
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\n" ); document.write( " \"y%2B6=1%28x%5E2%2B4x%29\" Factor out the leading coefficient \"1\"
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\n" ); document.write( " Take half of the x coefficient \"4\" to get \"2\" (ie \"%281%2F2%29%284%29=2\").
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\n" ); document.write( " Now square \"2\" to get \"4\" (ie \"%282%29%5E2=%282%29%282%29=4\")
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\n" ); document.write( " \"y%2B6=1%28x%5E2%2B4x%2B4-4%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"4\" does not change the equation
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\n" ); document.write( " \"y%2B6=1%28%28x%2B2%29%5E2-4%29\" Now factor \"x%5E2%2B4x%2B4\" to get \"%28x%2B2%29%5E2\"
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\n" ); document.write( " \"y%2B6=1%28x%2B2%29%5E2-1%284%29\" Distribute
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\n" ); document.write( " \"y%2B6=1%28x%2B2%29%5E2-4\" Multiply
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\n" ); document.write( " \"y=1%28x%2B2%29%5E2-4-6\" Now add \"%2B6\" to both sides to isolate y
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\n" ); document.write( " \"y=1%28x%2B2%29%5E2-10\" 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=-2\", and \"k=-10\". 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%2B4x-6\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2%2B4x-6%29\" Graph of \"y=1x%5E2%2B4x-6\". Notice how the vertex is (\"-2\",\"-10\").
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\n" ); document.write( " Notice if we graph the final equation \"y=1%28x%2B2%29%5E2-10\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x%2B2%29%5E2-10%29\" Graph of \"y=1%28x%2B2%29%5E2-10\". Notice how the vertex is also (\"-2\",\"-10\").
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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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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " \"y=4+x%5E2%2B4+x%2B1\" Start with the given equation
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\n" ); document.write( " \"y-1=4+x%5E2%2B4+x\" Subtract \"1\" from both sides
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\n" ); document.write( " \"y-1=4%28x%5E2%2B1x%29\" Factor out the leading coefficient \"4\"
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\n" ); document.write( " Take half of the x coefficient \"1\" to get \"1%2F2\" (ie \"%281%2F2%29%281%29=1%2F2\").
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\n" ); document.write( " Now square \"1%2F2\" to get \"1%2F4\" (ie \"%281%2F2%29%5E2=%281%2F2%29%281%2F2%29=1%2F4\")
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\n" ); document.write( " \"y-1=4%28x%5E2%2B1x%2B1%2F4-1%2F4%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"1%2F4\" does not change the equation
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\n" ); document.write( " \"y-1=4%28%28x%2B1%2F2%29%5E2-1%2F4%29\" Now factor \"x%5E2%2B1x%2B1%2F4\" to get \"%28x%2B1%2F2%29%5E2\"
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\n" ); document.write( " \"y-1=4%28x%2B1%2F2%29%5E2-4%281%2F4%29\" Distribute
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\n" ); document.write( " \"y-1=4%28x%2B1%2F2%29%5E2-1\" Multiply
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\n" ); document.write( " \"y=4%28x%2B1%2F2%29%5E2-1%2B1\" Now add \"1\" to both sides to isolate y
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\n" ); document.write( " \"y=4%28x%2B1%2F2%29%5E2%2B0\" 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=4\", \"h=-1%2F2\", and \"k=0\". 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( "
\n" ); document.write( " Notice if we graph the original equation \"y=4x%5E2%2B4x%2B1\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B4x%2B1%29\" Graph of \"y=4x%5E2%2B4x%2B1\". Notice how the vertex is (\"-1%2F2\",\"0\").
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\n" ); document.write( " Notice if we graph the final equation \"y=4%28x%2B1%2F2%29%5E2%2B0\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C4%28x%2B1%2F2%29%5E2%2B0%29\" Graph of \"y=4%28x%2B1%2F2%29%5E2%2B0\". Notice how the vertex is also (\"-1%2F2\",\"0\").
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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( "c.
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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=-3+x%5E2%2B3+x-1\" Start with the given equation
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\n" ); document.write( " \"y%2B1=-3+x%5E2%2B3+x\" Add \"1\" to both sides
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\n" ); document.write( " \"y%2B1=-3%28x%5E2-1x%29\" Factor out the leading coefficient \"-3\"
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\n" ); document.write( " Take half of the x coefficient \"-1\" to get \"-1%2F2\" (ie \"%281%2F2%29%28-1%29=-1%2F2\").
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\n" ); document.write( " Now square \"-1%2F2\" to get \"1%2F4\" (ie \"%28-1%2F2%29%5E2=%28-1%2F2%29%28-1%2F2%29=1%2F4\")
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\n" ); document.write( " \"y%2B1=-3%28x%5E2-1x%2B1%2F4-1%2F4%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"1%2F4\" does not change the equation
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\n" ); document.write( " \"y%2B1=-3%28%28x-1%2F2%29%5E2-1%2F4%29\" Now factor \"x%5E2-1x%2B1%2F4\" to get \"%28x-1%2F2%29%5E2\"
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\n" ); document.write( " \"y%2B1=-3%28x-1%2F2%29%5E2%2B3%281%2F4%29\" Distribute
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\n" ); document.write( " \"y%2B1=-3%28x-1%2F2%29%5E2%2B3%2F4\" Multiply
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\n" ); document.write( " \"y=-3%28x-1%2F2%29%5E2%2B3%2F4-1\" Now add \"%2B1\" to both sides to isolate y
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\n" ); document.write( " \"y=-3%28x-1%2F2%29%5E2-1%2F4\" Combine like terms
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\n" ); document.write( "
\n" ); document.write( " Now the quadratic is in vertex form \"y=a%28x-h%29%5E2%2Bk\" where \"a=-3\", \"h=1%2F2\", and \"k=-1%2F4\". Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
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\n" ); document.write( " Notice if we graph the original equation \"y=-3x%5E2%2B3x-1\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3x%5E2%2B3x-1%29\" Graph of \"y=-3x%5E2%2B3x-1\". Notice how the vertex is (\"1%2F2\",\"-1%2F4\").
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\n" ); document.write( " Notice if we graph the final equation \"y=-3%28x-1%2F2%29%5E2-1%2F4\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3%28x-1%2F2%29%5E2-1%2F4%29\" Graph of \"y=-3%28x-1%2F2%29%5E2-1%2F4\". Notice how the vertex is also (\"1%2F2\",\"-1%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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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " \"y=-2+x%5E2%2B4+x%2B3\" Start with the given equation
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\n" ); document.write( " \"y-3=-2+x%5E2%2B4+x\" Subtract \"3\" from both sides
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\n" ); document.write( " \"y-3=-2%28x%5E2-2x%29\" Factor out the leading coefficient \"-2\"
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\n" ); document.write( " Take half of the x coefficient \"-2\" to get \"-1\" (ie \"%281%2F2%29%28-2%29=-1\").
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\n" ); document.write( " Now square \"-1\" to get \"1\" (ie \"%28-1%29%5E2=%28-1%29%28-1%29=1\")
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\n" ); document.write( " \"y-3=-2%28x%5E2-2x%2B1-1%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"1\" does not change the equation
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\n" ); document.write( " \"y-3=-2%28%28x-1%29%5E2-1%29\" Now factor \"x%5E2-2x%2B1\" to get \"%28x-1%29%5E2\"
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\n" ); document.write( " \"y-3=-2%28x-1%29%5E2%2B2%281%29\" Distribute
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\n" ); document.write( " \"y-3=-2%28x-1%29%5E2%2B2\" Multiply
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\n" ); document.write( " \"y=-2%28x-1%29%5E2%2B2%2B3\" Now add \"3\" to both sides to isolate y
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\n" ); document.write( " \"y=-2%28x-1%29%5E2%2B5\" 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=-2\", \"h=1\", and \"k=5\". Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
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\n" ); document.write( " Notice if we graph the original equation \"y=-2x%5E2%2B4x%2B3\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%5E2%2B4x%2B3%29\" Graph of \"y=-2x%5E2%2B4x%2B3\". Notice how the vertex is (\"1\",\"5\").
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\n" ); document.write( " Notice if we graph the final equation \"y=-2%28x-1%29%5E2%2B5\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2%28x-1%29%5E2%2B5%29\" Graph of \"y=-2%28x-1%29%5E2%2B5\". Notice how the vertex is also (\"1\",\"5\").
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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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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " \"y=6+x%5E2-12+x%2B1\" Start with the given equation
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\n" ); document.write( " \"y-1=6+x%5E2-12+x\" Subtract \"1\" from both sides
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\n" ); document.write( " \"y-1=6%28x%5E2-2x%29\" Factor out the leading coefficient \"6\"
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\n" ); document.write( " Take half of the x coefficient \"-2\" to get \"-1\" (ie \"%281%2F2%29%28-2%29=-1\").
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\n" ); document.write( " Now square \"-1\" to get \"1\" (ie \"%28-1%29%5E2=%28-1%29%28-1%29=1\")
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\n" ); document.write( " \"y-1=6%28x%5E2-2x%2B1-1%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"1\" does not change the equation
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\n" ); document.write( " \"y-1=6%28%28x-1%29%5E2-1%29\" Now factor \"x%5E2-2x%2B1\" to get \"%28x-1%29%5E2\"
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\n" ); document.write( " \"y-1=6%28x-1%29%5E2-6%281%29\" Distribute
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\n" ); document.write( " \"y-1=6%28x-1%29%5E2-6\" Multiply
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\n" ); document.write( " \"y=6%28x-1%29%5E2-6%2B1\" Now add \"1\" to both sides to isolate y
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\n" ); document.write( " \"y=6%28x-1%29%5E2-5\" 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=6\", \"h=1\", and \"k=-5\". 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=6x%5E2-12x%2B1\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C6x%5E2-12x%2B1%29\" Graph of \"y=6x%5E2-12x%2B1\". Notice how the vertex is (\"1\",\"-5\").
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\n" ); document.write( " Notice if we graph the final equation \"y=6%28x-1%29%5E2-5\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C6%28x-1%29%5E2-5%29\" Graph of \"y=6%28x-1%29%5E2-5\". Notice how the vertex is also (\"1\",\"-5\").
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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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