How do you write the quadratic function in vertex form for y = x² + 2x + 5?

To write the quadratic function in vertex form, we need to complete the square. The standard form of a quadratic function is given as:

y = ax² + bx + c

For the function y = x² + 2x + 5, we identify:

• a = 1
• b = 2
• c = 5

First, we focus on the quadratic and linear terms, which are x² + 2x. We need to complete the square for these terms:

1. Take the coefficient of x (which is 2), divide it by 2 giving us 1, and then square this result. This gives us 1.
2. Add and subtract this square inside the equation, giving us y = (x² + 2x + 1) – 1 + 5.
3. Now, this simplifies to y = (x + 1)² + 4.

Now, we have the function in vertex form, which is:

y = (x + 1)² + 4

In vertex form, y = a(x – h)² + k, the vertex of the parabola is located at the point (h, k). In our case, h = -1 and k = 4, so the vertex of the parabola is (-1, 4).