Question

How do you find your vertex when the equation is y= (x+3)(x-7)

Answer 1

To find the vertex of a quadratic equation in the form \( y = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots (or x-intercepts) of the equation, you can follow these steps:

1. **Identify the Roots:** From the given equation \( y = (x + 3)(x - 7) \), the roots are \( r_1 = -3 \) and \( r_2 = 7 \).

2. **Calculate the x-coordinate of the Vertex:** The x-coordinate of the vertex is the midpoint between the roots. You can find it using the formula:
\[
x_{\text{vertex}} = \frac{r_1 + r_2}{2}
\]
Substituting the values:
\[
x_{\text{vertex}} = \frac{-3 + 7}{2} = \frac{4}{2} = 2
\]

3. **Calculate the y-coordinate of the Vertex:** To find the y-coordinate of the vertex, substitute \( x_{\text{vertex}} \) back into the original equation:
\[
y = (2 + 3)(2 - 7) = (5)(-5) = -25
\]

4. **Vertex:** Therefore, the vertex of the quadratic function \( y = (x + 3)(x - 7) \) is at the point:
\[
(2, -25)
\]

So the vertex is \( (2, -25) \).