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
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)