To find the vertex of a quadratic equation in factored form, which is given as:
\[ y = a(x - s)(x - t) \]
where \( s \) and \( t \) are the roots (or x-intercepts) of the quadratic, you can follow these steps:
Step 1: Identify the Roots
From the factored form, \( s \) and \( t \) are the points where the graph of the quadratic crosses the x-axis. These points represent the two solutions where \( y = 0 \).
Step 2: Calculate the x-coordinate of the Vertex
The x-coordinate of the vertex of a parabola can be found by taking the average of the x-intercepts (roots).
The formula to find the x-coordinate of the vertex (\( x_v \)) is:
\[ x_v = \frac{s + t}{2} \]
Step 3: Calculate the y-coordinate of the Vertex
To find the y-coordinate of the vertex, substitute \( x_v \) back into the original quadratic equation.
- Substitute \( x_v \) into the equation:
\[ y_v = a\left(\frac{s + t}{2} - s\right)\left(\frac{s + t}{2} - t\right) \]
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Simplify:
- The expression \( \frac{s + t}{2} - s \) simplifies to \( \frac{t - s}{2} \)
- The expression \( \frac{s + t}{2} - t \) simplifies to \( \frac{s - t}{2} \)
Thus, we can rewrite \( y_v \) as:
\[ y_v = a\left(\frac{t - s}{2}\right)\left(\frac{s - t}{2}\right) \]
This can be combined to give:
\[ y_v = a\left(\frac{-(t - s)^2}{4}\right) \]
Step 4: The Vertex Coordinates
Combining both parts, the vertex (\( x_v, y_v \)) can be expressed as:
\[ \text{Vertex} = \left( \frac{s + t}{2}, a\left(-\frac{(t - s)^2}{4}\right) \right) \]
Conclusion
So, the vertex of the quadratic in factored form \( y = a(x - s)(x - t) \) is located at:
\[ \text{Vertex} = \left( \frac{s + t}{2}, a\left(-\frac{(t - s)^2}{4}\right) \right) \]
This gives you both the x-coordinate and y-coordinate for the vertex based on the roots \( s \) and \( t \), as well as the coefficient \( a \). If \( a \) is positive, the parabola opens upward, and if \( a \) is negative, it opens downward, affecting the position of the vertex accordingly.