randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 ) A === 1 ? "" : A === -1 ? "-" : A [ parse( "y &= A( x - #{h})^2 + #{k}", [ GREEN, GREEN ] ), parse( "y &= " + A_DISP + "( x - #{" + H + "})^2 + #{" + K + "}", [ GREEN, GREEN ] ) ] [ parse( "y &= " + plus( A + "x^2", ( -2 * A * H ) + "x", ( A * H * H + K ) ) ), parse( "y + " + ( -A * H * H - K ) + " &= " + plus( A + "x^2", ( -2 * A * H ) + "x" ) ), ] [ parse( plus( "y", ( -A * H * H - K ) ) + " = " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + ")" ), ] [ parse( plus( "y", ( -A * H * H - K ) ) + " &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + ")" ), parse( plus( "y", ( -A * H * H - K ) ) + " + #{" + ( A * H * H ) + "} &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + " + #{" + ( H * H ) + "})", [ BLUE, BLUE ] ), parse( plus( "y", ( ( -A * H * H - K ) + ( A * H * H ) ) ) + " &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x", ( H * H ) ) + ")" ), ] [ parse( plus( "y", -K ) + " = " + A_DISP + "(" + plus( "x", -H ) + ")^2" ), ] [ parse( "y = " + A_DISP + "(x - " + H + ")^2 + " + K ) ]

Given the equation:

\qquad y = A_DISPx^2 + -2 * A * Hx + A * H * H + K

Find the parabola's vertex.

\large{\left(\right.}H,\text{ }K\large{\left.\right)}

a point, like (-1, 2)

When the equation is rewritten in vertex form like this, the vertex is the point (h, k):

\qquad formatGroup( GROUP1, [ 0 ] )

We can rewrite the equation in vertex form by completing the square. First, move the constant term to the left side of the equation:

\qquad formatGroup( COMP_SQR1, [ 0, 1 ] )

Next, we can factor out a A from the right side:

\qquad formatGroup( COMP_SQR2, [ 0 ] )

We can complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation. The coefficient of our x term is -2 * H, so half of it would be -H, and squaring that gives us H * H. Because we're adding the H * H inside the parentheses on the right where it's being multiplied by A, we need to add A * H * H to the left side to make sure we're adding the same thing to both sides.

\qquad formatGroup( COMP_SQR3, [ 0, 1, 2 ] )

Now we can rewrite the expression in parentheses as a squared term:

\qquad formatGroup( COMP_SQR4, [ 0 ] )

Move the constant term to the right side of the equation. Now the equation is in vertex form:

\qquad formatGroup( COMP_SQR5, [ 0 ] )

Now that the equation is written in vertex form, the vertex is the point (h, k):

\qquad formatGroup( GROUP1, [ 0 ] )

The vertex is (H, K). Be sure to pay attention to the signs when interpreting an equation in vertex form.