document.write( "Question 558739: What is the vertex, focus, axis of symmetry, and directiv of the following equation, (x-2)² = y+3 and how did you get those answers? \n" ); document.write( "

Algebra.Com's Answer #363163 by KMST(5328) $\"\"$ $\"About$

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The equation is the equation of a parabola in vertex form.
\n" ); document.write( "VERTEX
\n" ); document.write( "The coordinates of the vertex are shown in the equation subtracted from the $\"x\"$ and the $\"y\"$ .
\n" ); document.write( "The vertex is (2,-3).
\n" ); document.write( "The vertex has $\"x-2=0\"$ <---> $\"x=2\"$ and $\"y%2B3=0\"$ <---> $\"y=-3\"$ , and $\"x-2=0\"$ .
\n" ); document.write( "It is a minimum, because, for any value of $\"x\"$ other than $\"x=2\"$ , $\"%28x-2%29%5E2%3E0\"$ , making $\"y%2B3%3E0\"$ <---> $\"y%3E-3\"$ .
\n" ); document.write( "AXIS OF SYMMETRY
\n" ); document.write( "For all points other than the vertex, the same value of $\"y\"$ happens for two different values of $\"x\"$ , at equal distances to the left and right of the line $\"x-2=0\"$ <---> $\"x=2\"$ . That line is the axis of symmetry.
\n" ); document.write( "FOCUS AND DIRECTRIX
\n" ); document.write( "The focus is the point (2,-3+c) above the vertex/minimum that the parabola \"wraps\" around. The directrix is the line $\"y=-3-c\"$ at the same distance on the other side of the vertex.
\n" ); document.write( "Your book will tell you that the coefficient of $\"y\"$ in the equation equals $\"4c\"$ , so in this case $\"1=4c\"$ --> $\"c=1%2F4\"$
\n" ); document.write( "The focus has $\"y=-3%2Bc=-3%2B1%2F4=-11%2F4\"$ . It is the point (2,-11/4).
\n" ); document.write( "The directrix is the line $\"y=-3-c=-3-1%2F4\"$ --> $\"y=-13%2F4\"$ . \n" ); document.write( "

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