How do you calculate the vertex of a function 

To calculate the vertex of a quadratic function in the standard form:

\[
f(x) = ax^2 + bx + c
\]

### Steps:
1. **Find the x-coordinate** of the vertex using the formula:

   \[
   x_{\text{vertex}} = \frac{-b}{2a}
   \]

2. **Find the y-coordinate** of the vertex by substituting the x-coordinate back into the original quadratic function:

   \[
   y_{\text{vertex}} = f\left(\frac{-b}{2a}\right) = a\left(\frac{-b}{2a}\right)^2 + b\left(\frac{-b}{2a}\right) + c
   \]

Thus, the vertex \((x_{\text{vertex}}, y_{\text{vertex}})\) is the point:

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

Alternatively, if the quadratic is given in **vertex form**:

\[
f(x) = a(x - h)^2 + k
\]

The vertex is simply at \((h, k)\), where \(h\) and \(k\) are constants.

### Example:
For the quadratic function:

\[
f(x) = 2x^2 - 4x + 1
\]

1. Find \(x_{\text{vertex}}\):

   \[
   x_{\text{vertex}} = \frac{-(-4)}{2(2)} = \frac{4}{4} = 1
   \]

2. Find \(y_{\text{vertex}}\) by substituting \(x = 1\) into \(f(x)\):

   \[
   y_{\text{vertex}} = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1
   \]

The vertex is \((1, -1)\).

Would you like more details or have any questions?

### Related Questions:
1. How do you convert a quadratic function to vertex form?
2. What does the vertex of a parabola represent in real-world applications?
3. How can you find the axis of symmetry of a quadratic function?
4. How does the value of \(a\) in the quadratic function affect the graph's shape?
5. Can the vertex of a quadratic function be outside the visible part of the graph?

### Tip:
The vertex gives the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.

How do you calculate the vertex of a function?

Learn how to calculate the vertex of a quadratic function using formulas like x = -b / 2a and vertex form. This guide provides a step-by-step explanation for students in Grades 9-11, covering key concepts like the vertex of a parabola and algebraic calculations.

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