Final answer:
To determine the value of b in a parabola equation with a known vertex (h, k), we start with the vertex form y = a(x - h)^2 + k and expand it to match the standard form y = ax^2 + bx + c. For the given vertex (-1, 1), after expansion, we find that b = 2.
Step-by-step explanation:
The question asks for the value of b in the parabolic equation y = ax^2 + bx + c when the vertex of the parabola is known to be (-1, 1).
To find b, we can use the fact that the vertex form of a parabolic equation is y = a(x - h)^2 + k, where (h, k) is the vertex. Rewriting the vertex form to standard form,
we would get terms involving x and x^2,
and by comparing these with the standard form y = ax^2 + bx + c, we can solve for b.
Starting with the vertex form and substituting h = -1 and k = 1 we have y = a(x + 1)^2 + 1.
Expanding this we get y = a(x^2 + 2x + 1) + 1. If we let a = 1 for simplicity,
the expanded form would be y = x^2 + 2x + 1 + 1,
which simplifies to y = x^2 + 2x + 2. Here, b corresponds to the coefficient of x, which is 2.