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 Equation of a parabola
randRange( -5, 5 ) randRange( -5, 5 ) randRange( -5, 5 ) -2 * A * H A * H * H + K

The equation of a parabola P is y = Ax^2 + Bx + C.

What are its vertex (h, k) and its y-intercept?

(h, k) = (H, K)

y-intercept = C

The y-intercept is the point on the y-axis where x = 0.

If x = 0, we have y = A \cdot 0^2 + B \cdot 0 + C = C, so the y-intercept is C.

The equation of a parabola with vertex (h, k) is y = a(x - h)^2 + k.

We can rewrite the given equation as Ax^2 + A \cdot - 2 * Hx + H * A * H + K, in order to get the form a(x - h)^2

i

We factor out A, giving y = A ( x^2 + 2 * Hx + H * H ) + K

The equation in the parentheses is of the form ( a + b )^2, because ( x^2 + 2 * H + H * H ) = ( x^2 + 2 \cdot H + H^2 )

Therefore, y = A( x - H)^2 + K.

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

Thus, the center (h, k) is (H, K)