Find the coordinates of the vertex of the graph of the following equation: 𝑦=(𝑥−3)(𝑥−7)y=(x−3)(x−7)

1. Break Down the Problem

To find the vertex of the quadratic equation given in factored form $y = (x - 3)(x - 7)$, we need to first identify the roots and then find the x-coordinate of the vertex.

2. Relevant Concepts

For a quadratic equation in factored form $y = a(x - r_1)(x - r_2)$, the x-coordinate of the vertex can be calculated using the formula: $x_{vertex} = \frac{r_1 + r_2}{2}$ where $r_1$ and $r_2$ are the roots of the equation.

3. Analysis and Detail

1. Identify the roots from the equation:

   • From $y = (x - 3)(x - 7)$, the roots are $r_1 = 3$ and $r_2 = 7$.

2. Calculate the x-coordinate of the vertex: $x_{vertex} = \frac{3 + 7}{2} = \frac{10}{2} = 5$

3. To find the y-coordinate of the vertex, substitute $x_{vertex} = 5$ back into the equation: $y = (5 - 3)(5 - 7) = (2)(-2) = -4$

4. Verify and Summarize

We have calculated the x-coordinate of the vertex to be $5$ and the y-coordinate to be $-4$. Therefore, the vertex of the graph is located at the point $(5, -4)$.

Final Answer

The coordinates of the vertex are $(5, -4)$.