Certainly! Let's delve into the factored form of a quadratic equation and explain how to find the vertex.
### Step-by-Step Process
1. Identify the Factored Form of the Quadratic Equation:
The given quadratic equation has the form:
[tex]\[
y = a(x - s)(x - t)
\][/tex]
where [tex]\( s \)[/tex] and [tex]\( t \)[/tex] are the roots of the equation, and [tex]\( a \)[/tex] is a coefficient.
2. Understand the Properties of the Vertex in Quadratics:
For any quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex], the vertex is the highest or lowest point on the graph, depending on the sign of [tex]\( a \)[/tex]. In the factored form, the vertex lies exactly midway between the roots [tex]\( s \)[/tex] and [tex]\( t \)[/tex].
3. Calculate the x-Coordinate of the Vertex:
Since the vertex lies at the midpoint of the roots [tex]\( s \)[/tex] and [tex]\( t \)[/tex], the x-coordinate of the vertex [tex]\( h \)[/tex] can be found by averaging the roots:
[tex]\[
h = \frac{s + t}{2}
\][/tex]
Let’s use example values for [tex]\( s \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[
s = 2, \quad t = 6
\][/tex]
Then,
[tex]\[
h = \frac{2 + 6}{2} = \frac{8}{2} = 4
\][/tex]
4. Calculate the y-Coordinate of the Vertex:
To find the y-coordinate [tex]\( k \)[/tex] of the vertex, we substitute the value of [tex]\( h \)[/tex] back into the original equation [tex]\( y = a(x - s)(x - t) \)[/tex].
Let’s assume a value for [tex]\( a \)[/tex]:
[tex]\[
a = 1
\][/tex]
Substituting [tex]\( h \)[/tex] into the equation:
[tex]\[
k = a(h - s)(h - t)
\][/tex]
With [tex]\( h = 4 \)[/tex], [tex]\( s = 2 \)[/tex], and [tex]\( t = 6 \)[/tex], we get:
[tex]\[
k = 1 \cdot (4 - 2) \cdot (4 - 6) = 1 \cdot 2 \cdot (-2) = -4
\][/tex]
### Conclusion
Thus, the vertex of the quadratic equation [tex]\( y = a(x - s)(x - t) \)[/tex] is located at:
[tex]\[
(h, k) = (4, -4)
\][/tex]
This systematic approach shows how the midpoint of the roots provides the x-coordinate of the vertex and how substituting this back in gives the y-coordinate, ultimately identifying the vertex point.
