Find the coordinates of the vertex of the graph of the following equation: 𝑦=(𝑥−4)(𝑥+2)y=(x−4)(x+2)

1. Break Down the Problem

We need to find the coordinates of the vertex of the quadratic equation given by:

$y = (x - 4)(x + 2)$

First, we will rewrite this equation in standard form, $y = ax^2 + bx + c$.

2. Relevant Concepts

A quadratic equation in standard form can be derived from factored form. The x-coordinate of the vertex of the parabola represented by the standard form of the quadratic equation can be found using the formula:

$x = -\frac{b}{2a}$

3. Analysis and Detail

Step 3.1: Expand the equation

$y = (x - 4)(x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8$

Thus, the standard form of the equation is:

$y = x^2 - 2x - 8$

Here, $a = 1$ and $b = -2$.

Step 3.2: Find the x-coordinate of the vertex

Using the vertex formula:

$x = -\frac{b}{2a} = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1$

Step 3.3: Find the y-coordinate of the vertex

Now, plug $x = 1$ back into the original equation to find $y$:

$y = (1 - 4)(1 + 2) = (-3)(3) = -9$

4. Verify and Summarize

The vertex of the quadratic function is at the point (1, -9). This calculation is verified by substituting back into the factored form.

Final Answer

The coordinates of the vertex are:

$(1, -9)$