Find the coordinates of the vertex of the graph of the following equation: 𝑦=(𝑥+2)(𝑥+6)y=(x+2)(x+6)

1. ### Break Down the Problem

To find the coordinates of the vertex of the quadratic equation $y = (x + 2)(x + 6)$, we first need to convert it into standard form, which is $y = ax^2 + bx + c$.

2. ### Relevant Concepts

The vertex $(h, k)$ of a parabola given in standard form can be found using the formula: $h = -\frac{b}{2a}$ where $a$ and $b$ are the coefficients from $y = ax^2 + bx + c$.

3. ### Analysis and Detail

1. Expand the given equation: $y = (x + 2)(x + 6) = x^2 + 6x + 2x + 12 = x^2 + 8x + 12$ Here, we identify: $a = 1, \quad b = 8, \quad c = 12$

2. Calculate the x-coordinate of the vertex: $h = -\frac{b}{2a} = -\frac{8}{2 \cdot 1} = -4$

3. Calculate the y-coordinate of the vertex: Substitute $x = -4$ back into the equation to find $k$: $y = (-4 + 2)(-4 + 6) = (-2)(2) = -4$

4. ### Verify and Summarize

The vertex coordinates are $h = -4$ and $k = -4$. Thus, the vertex is located at $(-4, -4)$.

Final Answer

The coordinates of the vertex are $\boxed{(-4, -4)}$.