What are the coordinates of the vertex for the quadratic function f(x) = x² + 6x + 13?

Finding the Vertex of a Quadratic Function

The vertex of a quadratic function in the standard form f(x) = ax² + bx + c can be found using the formula:

x-coordinate of the vertex: x = -b / (2a)

In the given function f(x) = x² + 6x + 13, we can identify:

• a = 1
• b = 6
• c = 13

Step 1: Calculate the x-coordinate of the vertex

Substituting the values of a and b into the vertex formula gives:

x = -6 / (2 * 1) = -6 / 2 = -3

Step 2: Calculate the y-coordinate of the vertex

Now, we substitute x = -3 back into the function to find the y-coordinate:

f(-3) = (-3)² + 6*(-3) + 13

= 9 - 18 + 13

= 4

Conclusion

Therefore, the coordinates of the vertex of the quadratic function f(x) = x² + 6x + 13 are:

Vertex: (-3, 4)