document.write( "Question 1060914: What is the equation of the parabola passing through the points A(1,1) , B(2,2) , C(-1,5) and axis parallel to the y-axis \n" ); document.write( "

Algebra.Com's Answer #675805 by KMST(5328) $\"\"$ $\"About$

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The equation would be of the form $\"y=ax%5E2%2Bbx%2Bc\"$ .
\n" ); document.write( "Substituting the coordinates of $\"A%281%2C1%29\"$ , we get
\n" ); document.write( " $\"1=a%2A1%5E2%2Bb%2A1%2Bc\"$ --> $\"a%2Bb%2Bc=1\"$ .
\n" ); document.write( "Substituting the coordinates of $\"B%282%2C2%29\"$ , we get
\n" ); document.write( " $\"2=a%2A2%5E2%2Bb%2A2%2Bc\"$ --> $\"4a%2B2b%2Bc=2\"$ .
\n" ); document.write( "Substituting the coordinates of $\"C%28-1%2C5%29\"$ , we get
\n" ); document.write( " $\"5=a%2A%28-1%29%5E2%2Bb%2A%28-1%29%2Bc\"$ --> $\"a-b%2Bc=5\"$ .
\n" ); document.write( "The 3 equations above give us the system
\n" ); document.write( " $\"system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc=2%2Ca-b%2Bc=5%29\"$
\n" ); document.write( "Subtracting the first equation from each ot the other two equations, we get the equivalent system
\n" ); document.write( " $\"system%28a%2Bb%2Bc=1%2C4a%2B2b%2Bc-a-b-c=2-1%2Ca-b%2Bc-a-b-c=5-1%29\"$ ---> $\"system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2C-2b=4%29\"$ --> $\"system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2Cb=4%2F%28-2%29%29\"$ --> $\"system%28a%2Bb%2Bc=1%2C3a%2Bb=1%2Chighlight%28b=-2%29%29\"$ .
\n" ); document.write( "Substituting $\"-2\"$ for $\"b\"$ in $\"3a%2Bb=1\"$ we get
\n" ); document.write( " $\"3a-2=1\"$ --> $\"3a=1%2B2\"$ --> $\"3a=3\"$ --> $\"a=3%2F3\"$ --> $\"highlight%28a=1%29\"$ .
\n" ); document.write( "Then substituting $\"-2\"$ for $\"b\"$ and $\"1\"$ for $\"a\"$ in $\"a%2Bb%2Bc=1\"$ , we get
\n" ); document.write( " $\"1-2%2Bc=1\"$ --> $\"-1%2Bc=1\"$ --> $\"c=1%2B1\"$ --> $\"highlight%28c=2%29\"$ .
\n" ); document.write( "So, the equation of the parabola is
\n" ); document.write( " $\"highlight%28y=x%5E2-2x%2B2%29\"$ .
\n" ); document.write( "
\n" ); document.write( "We could also write it as
\n" ); document.write( " $\"y=x%5E2-2x%2B1%2B1\"$ --> $\"y=%28x%5E2-2x%2B1%29%2B1\"$ --> $\"highlight%28y=%28x-1%29%5E2%2B1%29\"$ . \n" ); document.write( "

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