Find the coordinates of the vertex of the graph of the following equation: 𝑦=(𝑥+1)(𝑥−1)y=(x+1)(x−1)

1. Break Down the Problem

To find the coordinates of the vertex of the graph of the equation $y = (x + 1)(x - 1)$, we first need to rewrite the equation in a more manageable form and identify the characteristics of the quadratic function.

2. Relevant Concepts

The equation $y = (x + 1)(x - 1)$ can be simplified to the standard form of a quadratic equation. The standard form of a quadratic is given by: $y = ax^2 + bx + c$ The vertex of a quadratic function can be found using the formula for the x-coordinate of the vertex: $x_v = -\frac{b}{2a}$ We can then find the y-coordinate by substituting $x_v$ back into the quadratic equation.

3. Analysis and Detail

1. Expand the equation: $y = (x + 1)(x - 1) = x^2 - 1$ Here, we can see that $a = 1$, $b = 0$, and $c = -1$.

2. Find the x-coordinate of the vertex: $x_v = -\frac{b}{2a} = -\frac{0}{2 \cdot 1} = 0$

3. Find the y-coordinate of the vertex: Substitute $x_v$ back into the equation: $y_v = (0 + 1)(0 - 1) = 1 \cdot (-1) = -1$

4. Verify and Summarize

The coordinates of the vertex are calculated as follows:

• The x-coordinate is $0$.
• The y-coordinate is $-1$.

Thus, the vertex is at the coordinates: $(0, -1)$

Final Answer

The coordinates of the vertex of the graph of the equation $y = (x + 1)(x - 1)$ are $\boxed{(0, -1)}$.