Question

What is the vertex form of the quadratic equation y = x^2 + 14x + 33, and what are the coordinates of the vertex?

A) Vertex Form: y = (x + 7)^2 - 16, Vertex Coordinates: (-7, -16)
B) Vertex Form: y = (x + 7)^2 + 16, Vertex Coordinates: (-7, 16)
C) Vertex Form: y = (x - 7)^2 - 16, Vertex Coordinates: (7, -16)
D) Vertex Form: y = (x - 7)^2 + 16, Vertex Coordinates: (7, 16)

Answer 1

The vertex form of the quadratic equation y = x^2 + 14x + 33 is y = (x + 7)^2 - 16, and the vertex coordinates are (-7, -16), which corresponds to answer choice A.

Step-by-step explanation:

The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. To convert the given quadratic equation y = x^2 + 14x + 33 into vertex form, we complete the square:

1. Group the x terms together: y = (x^2 + 14x) + 33.
2. Find the number that completes the square for x^2 +14x by taking half of the coefficient of x, which is 14, and squaring it (14/2)^2 = 49.
3. Add and subtract this number inside the parenthesis: y = (x^2 + 14x + 49) - 49 + 33.
4. As (x+7)^2 expands to x^2 + 14x + 49, rewrite the equation: y = (x + 7)^2 - 16.

Therefore, the vertex form of the given quadratic equation is y = (x + 7)^2 - 16, and the coordinates of the vertex are (-7, -16).

So the correct answer is: A) Vertex Form: y = (x + 7)^2 - 16, Vertex Coordinates: (-7, -16).