The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
a. To write y = x^2 - 2x + 3 in vertex form, we need to complete the square. We can do this by adding and subtracting the square of half of the x-coefficient, which is (-2/2)^2 = 1.
y = x^2 - 2x + 3
y + 1 = x^2 - 2x + 1 + 3
y + 1 = (x - 1)^2 + 2
Therefore, the vertex form of y = x^2 - 2x + 3 is y = (x - 1)^2 + 2, and the vertex is (1, 2).
b. Similarly, to write y = x^2 + 6x + 25 in vertex form, we need to complete the square by adding and subtracting (6/2)^2 = 9.
y = x^2 + 6x + 25
y + 9 = x^2 + 6x + 9 + 25
y + 9 = (x + 3)^2 + 16
Therefore, the vertex form of y = x^2 + 6x + 25 is y = (x + 3)^2 + 16, and the vertex is (-3, 16).