# Khan/khan-exercises

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 Vertex of a parabola
randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) A === 1 ? "" : A === -1 ? "-" : A

Given the equation:

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

Find the parabola's vertex.

( H , K )

When the equation is rewritten in vertex form like this, the vertex is the point (\green{h}, \blue{k}):

\qquad y = A(x - \green{h})^2 + \blue{k}

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 \begin{eqnarray} y &=& A_DISPx^2 + -2 * A * Hx + A * H * H + K \\ \\ y - A * H * H + K &=& A_DISPx^2 + -2 * A * Hx \end{eqnarray}

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

\qquad y - A * H * H + K = A(x^2 + -2 * Hx)

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 \pink{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 \pink{A * H * H} to the left side to make sure we're adding the same thing to both sides.

\qquad \begin{eqnarray} y - A * H * H + K &=& A(x^2 + -2 * Hx) \\ \\ y - A * H * H + K + \pink{A * H * H} &=& A(x^2 + -2 * Hx + \pink{H * H}) \\ \\ y - K &=& A(x^2 + -2 * Hx + H * H) \end{eqnarray}

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

\qquad y - K = A(x - H)^2

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

\qquad y = A(x - H)^2 + K

Now that the equation is written in vertex form, the vertex is the point (\green{h}, \blue{k}):

\qquad y = A(x - \green{h})^2 + \blue{k}

\qquad y = A(x - \green{(H)})^2 + \blue{(K)}

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