How do you convert the quadratic function y = x² + 2x + 5 into vertex form?

Converting to Vertex Form

The standard form of a quadratic function is given by:

y = ax² + bx + c

In this case, we have:

y = x² + 2x + 5

To convert this to vertex form, which is:

y = a(x – h)² + k

where (h, k) is the vertex of the parabola, we will complete the square.

Step 1: Factor out the coefficient of x²

In our function, the coefficient of x² is 1, so we can skip this step. We focus on the x terms:

y = x² + 2x + 5

Step 2: Complete the square

To complete the square, take the coefficient of x, which is 2, divide it by 2 to get 1, and then square it to obtain 1:

So, we can rewrite the quadratic by adding and subtracting 1:

y = (x² + 2x + 1 – 1) + 5

This simplifies to:

y = ((x + 1)² – 1) + 5

Step 3: Simplify

Now we simplify the equation:

y = (x + 1)² + 4