Final answer:
The vertex of the parabola from the function f(x) is (-6,-4) with the axis of symmetry at x = -6. The zeros can be found via symmetry and the factored form of f(x) is (x + 4)(x + 8) = 0. A parabola with a vertex of (5,12) that passes through (7,20) can be written in vertex form as y = 2(x - 5)² + 12.
Step-by-step explanation:
Vertex and Axis of Symmetry
The vertex of the quadratic function f(x) = (x + 6)² - 4 is at the point (-6, -4). The axis of symmetry is the vertical line passing through the vertex, hence the axis of symmetry is x = -6.
Finding the Zeros Using Symmetry
To find the zeros of the function, we can use the symmetry of the parabola about the axis of symmetry. Since the vertex is (-6, -4), we look for points equidistant from the axis on both sides which have y-values of zero. We solve the equation when f(x) = 0. The factored form will reveal the zeros directly.
Factored Form of f(x)
To write f(x) in factored form, we set f(x) equal to zero and find the solutions for x. The function can be written as
(x + 6 - 2)(x + 6 + 2) = 0, or in factored form as (x + 4)(x + 8) = 0.
Vertex Form of a Parabola
To write the equation of a parabola in vertex form with vertex (5,12) that passes through the point (7,20), we use the vertex form equation y = a(x - h)² + k. We substitute the vertex into h and k and plug in the point (7,20) to solve for a. The resulting equation is y = 2(x - 5)² + 12.