Final answer:
To convert the standard form y = 7x^2 + 14x + 4 to the vertex form, complete the square around the x terms. The resulting vertex form is y = 7(x + 1)^2 - 3 and the vertex is (-1, -3).
Step-by-step explanation:
Converting Standard Form to Vertex Form
To convert the standard form of a parabola equation y = 7x2 + 14x + 4 to vertex form, we need to complete the square. The vertex form of a parabolic equation is y = a(x - h)2 + k, where (h, k) is the vertex of the parabola.
Starting with y = 7x2 + 14x + 4, we factor out the coefficient of the x2 term from the first two terms:
y = 7(x2 + 2x) + 4
We then complete the square by adding and subtracting the square of half the coefficient of x inside the parentheses:
y = 7(x2 + 2x + 1 - 1) + 4
y = 7((x + 1)2 - 1) + 4
Now distribute the 7 and combine like terms:
y = 7(x + 1)2 - 7 + 4
y = 7(x + 1)2 - 3
Therefore, the vertex form of the parabola is y = 7(x + 1)2 - 3, and the vertex is (-1, -3).