document.write( "Question 1045379: What is the equation of a parabola with vertex on the line y=x, axis parallel to Ox, and passing through (6,-2) and (3,4)? \n" ); document.write( "

Algebra.Com's Answer #660864 by KMST(5328) $\"\"$ $\"About$

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Let the vertex be $\"V%28k%2Ck%29\"$ .
\n" ); document.write( "The equation of a parabola with that vertex and axis parallel to the x-axis is
\n" ); document.write( " $\"x-k=a%28x-k%29%5E2\"$ .
\n" ); document.write( "Substituting the coordinates of the given points, we have
\n" ); document.write( " $\"system%28x=6%2Cy=-2%29\"$ ---> $\"6-k=a%28-2-k%29%5E2\"$
\n" ); document.write( " $\"system%28x=3%2Cy=4%29\"$ ---> $\"3-k=a%284-k%29%5E2\"$
\n" ); document.write( "Those two equations form a system of equation we have to solve to find $\"a\"$ and $\"k\"$ .
\n" ); document.write( "Subtracting one equation from the other, we have
\n" ); document.write( " $\"6-k-%283-k%29=a%28-2-k%29%5E2-a%284-k%29%5E2\"$
\n" ); document.write( " $\"6-k-3%2Bk%29=a%28%28-2-k%29%5E2-%284-k%29%5E2%29\"$
\n" ); document.write( " $\"3=a%28-2-k%2B%284-k%29%29%28-2-k-%284-k%29%29\"$
\n" ); document.write( " $\"3=a%28-2-k%2B4-k%29%28-2-k-4%2Bk%29\"$
\n" ); document.write( " $\"3=a%28-2k%2B2%29%28-6%29\"$
\n" ); document.write( " $\"3=a%28-2%29%28k-1%29%28-6%29\"$
\n" ); document.write( " $\"3=12a%28k-1%29\"$
\n" ); document.write( " $\"3%2F12%28k-1%29=a\"$
\n" ); document.write( " $\"1%2F4%28k-1%29=a\"$
\n" ); document.write( "Substituting the expression $\"1%2F4%28k-1%29\"$ for $\"a\"$ in $\"3-k=a%284-k%29%5E2\"$ we get
\n" ); document.write( " $\"3-k=%284-k%29%5E2%2F4%28k-1%29\"$ --> $\"4%28k-1%29%283-k%29=%284-k%29%5E2\"$ -->
\n" ); document.write( " $\"4%28-k%5E2%2B4k-3%29=%284-k%29%5E2\"$ --> $\"-4k%5E2%2B16k-12=16-8k%2Bk%5E2\"$
\n" ); document.write( "--> $\"0=16-16k%2Bk%5E2%2B4k%5E2-16k%2B12\"$ --> $\"0=5k%5E2-24k%2B28\"$
\n" ); document.write( "The solutions to that equation are $\"k=2\"$ and $\"k=24%2F5\"$ .
\n" ); document.write( "Only one of those solutions should work, because
\n" ); document.write( "through any 3 points, (such as (6,-2) , (3,4), and $\"V%282%2C2%29\"$ )
\n" ); document.write( "passes only one parabola with axis parallel to the x-axis.
\n" ); document.write( "Substituting $\"2\"$ for $\"k\"$ in the original equation
\n" ); document.write( " $\"3-k=a%284-k%29%5E2\"$ , we get
\n" ); document.write( " $\"3-2=a%284-2%29%5E2\"$ ---> $\"1=4a\"$ ---> $\"a=1%2F4\"$
\n" ); document.write( "The values $\"system%28k=2%2Ca=1%2F4%29\"$ also satisfy equation $\"6-k=a%28-2-k%29%5E2\"$ .
\n" ); document.write( "So, $\"highlight%28x-2=%28y-2%29%5E2%2F4%29\"$ is the equation of a parabola with vertex on the line $\"y=x\"$ , axis parallel to the x-axis, and passing through (6,-2) and (3,4).
\n" ); document.write( "The equation can be written in other, equivalent forms.
\n" ); document.write( " $\"highlight%28x-2=%28y-2%29%5E2%2F4%29\"$ <--> $\"highlight%28x=%28y-2%29%5E2%2F4%2B2%29\"$ <--> $\"highlight%28x=%281%2F4%29y%5E2-x%2B3%29\"$
\n" ); document.write( "
\n" ); document.write( "NOTE: The solution $\"k=14%2F5\"$ does not yield the same a value when substituted in the two original equations. \n" ); document.write( "

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