document.write( "Question 207353: Compare the parabola defined by each equation with the standard parabola defined by the equation y = x^2. Describe the corresponding transformations, and include the position of the vertex and the equation of the axis os symmetry.\r
\n" ); document.write( "\n" ); document.write( "a.) y = 3x^2 - 8\r
\n" ); document.write( "\n" ); document.write( "b.) y = (x - 6)^2 + 4\r
\n" ); document.write( "\n" ); document.write( "c.) y = -4(x + 3)^2 - 7 \n" ); document.write( "

Algebra.Com's Answer #156809 by Edwin McCravy(20077) $\"\"$ $\"About$

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Compare the parabola defined by each equation with the standard parabola defined by the equation y = x^2. Describe the corresponding transformations, and include the position of the vertex and the equation of the axis os symmetry.
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document.write( "Order of transformations:\r\n" );
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document.write( "1. Replace x by -x (reflects graph across y-axis)\r\n" );
document.write( "2. Replace x by x±k (shifts graph k units horizontally, \r\n" );
document.write( "   left if x-k, and right if x+k)  \r\n" );
document.write( "3. Multiply right side by a positive number k\r\n" );
document.write( "   Stretches vertically by factor of k if k>1\r\n" );
document.write( "   Shrinks vertically by a factor of k if k<1  \r\n" );
document.write( "4. Multiply right side by -1 (reflects graph across the x-axis) \r\n" );
document.write( "5. Add to or subtract from the right side (shifts graph up if\r\n" );
document.write( "   adding a positive number, and down if adding a negative.) \r\n" );
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document.write( "a.)  y = 3x^2 - 8\r\n" );
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document.write( "transformation              effect on graph:\r\n" );
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document.write( " y = x^2             <--- original equation, has vertex (0,0)\r\n" );
document.write( "                          axis of symmetry x=0 \r\n" );
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document.write( "                          Multiplying right side by 3, which is >1,\r\n" );
document.write( " y = 3x^2            <--- stretches vertically by factor of 3, still\r\n" );
document.write( "                          has vertex (0,0), axis of symmetry x=0\r\n" );
document.write( "                                  \r\n" );
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document.write( "                          Subtracting 8 from right side\r\n" );
document.write( " y = 3x^2 - 8        <--- shift 8 units downward, vertex (0,-9) \r\n" );
document.write( "                          axis of symmetry x=0\r\n" );
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document.write( "===========================================================\r\n" );
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document.write( "b.)  y = (x - 6)^2 + 4\r\n" );
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document.write( "transformation              effect on graph:\r\n" );
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document.write( "y = x^2               <--- original equation, has vertex (0,0)\r\n" );
document.write( "                           axis of symmetry x=0 \r\n" );
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document.write( "                           Replacing x by (x-6)\r\n" );
document.write( " y = (x-6)^2          <--- shifts 6 units right, has vertex (6,0)\r\n" );
document.write( "                           axis of symmetry x=6                           \r\n" );
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document.write( "                           Adding 4 to right side\r\n" );
document.write( " y = (x-6)^2 + 4      <--- shifts 4 units upward, vertex (6,4) \r\n" );
document.write( "                           axis of symmetry x=6\r\n" );
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document.write( "============================================\r\n" );
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document.write( "c.)  y = -4(x + 3)^2 - 7 \r\n" );
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document.write( "transformation              effect on graph:\r\n" );
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document.write( "y = x^2               <--- original equation, has vertex (0,0)\r\n" );
document.write( "                           axis of symmetry x=0 \r\n" );
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document.write( "                           Replacing x by (x+3)\r\n" );
document.write( " y = (x+3)^2          <--- shifts 3 units left, has vertex (-3,0)\r\n" );
document.write( "                           axis of symmetry x=-3                           \r\n" );
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document.write( "                           Multiplying right side by 4, which is >1,\r\n" );
document.write( " y = 4(x+3)^2         <--- stretches vertically by factor of 4, still\r\n" );
document.write( "                           has vertex (-3,0), axis of symmetry x=-3\r\n" );
document.write( "                           \r\n" );
document.write( "                           Multiplying right side by -1\r\n" );
document.write( " y = -4(x+3)^2        <--- reflects across x-axis, vertex (-3,0)\r\n" );
document.write( "                           axis of symmetry x=-3\r\n" );
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document.write( "                           Subtracting 7 from right side\r\n" );
document.write( " y = -4(x+3)^2 - 7    <--- shift 7 units downward, vertex (3,-7) \r\n" );
document.write( "                           axis of symmetry x=-3\r\n" );
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document.write( "--------------------\r\n" );
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document.write( "Edwin

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