The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) can be found using a specific formula derived from completing the square or using calculus. 

### The Vertex Formula
For the quadratic function \( f(x) = ax^2 + bx + c \), the vertex \((x, y)\) has its x-coordinate given by:

\[
x = -\frac{b}{2a}
\]

Once you have the x-coordinate, the corresponding y-coordinate (which is the value of the function at that x) can be found by substituting \( x = -\frac{b}{2a} \) back into the original function:

\[
y = f\left(-\frac{b}{2a}\right)
\]

### Derivation of the Vertex Formula

1. **Starting with the quadratic function**:
   \[
   f(x) = ax^2 + bx + c
   \]

2. **Complete the square**:
   To convert the quadratic function into vertex form \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex, we first complete the square.
   
   - Factor out \( a \) from the quadratic and linear terms:
     \[
     f(x) = a\left(x^2 + \frac{b}{a}x\right) + c
     \]

   - To complete the square, add and subtract \( \left(\frac{b}{2a}\right)^2 \) inside the parentheses:
     \[
     f(x) = a\left[x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c
     \]

   - Simplify:
     \[
     f(x) = a\left[\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c
     \]
     \[
     f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c
     \]

   - Combine the constants outside the square:
     \[
     f(x) = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)
     \]

   This gives the vertex form:
   \[
   f(x) = a(x-h)^2 + k
   \]
   where:
   \[
   h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}
   \]

So, the vertex \((h, k)\) is located at:
\[
\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)
\]

### Summary
- **X-coordinate of the vertex**: \( x = -\frac{b}{2a} \)
- **Y-coordinate of the vertex**: \( y = f\left(-\frac{b}{2a}\right) \)

This formula helps find the highest or lowest point on the graph of a quadratic function, depending on the sign of \( a \).

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Do you want to delve deeper into this formula or any of the derivations?

Here are 5 related questions you might consider:
1. How can you use the vertex formula to determine if a quadratic function opens upwards or downwards?
2. What is the relationship between the vertex and the axis of symmetry of a parabola?
3. How would you derive the vertex form directly from the standard form of a quadratic function?
4. Can the vertex formula be applied to functions other than quadratics?
5. How does the vertex change if the quadratic function is shifted horizontally or vertically?

**Tip:** Understanding the vertex formula is crucial for graphing parabolas quickly, as it gives you the key point around which the parabola is symmetric.

Explore the vertex formula of quadratic functions and learn how to find the vertex using completing the square or calculus methods. This detailed explanation covers the x-coordinate formula -b/(2a) and its derivation, providing insights into graphing parabolas and solving quadratic equations.

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