To convert the quadratic function y = x² + 4x + 7 into vertex form, we need to complete the square.
Starting with the given equation:
y = x² + 4x + 7
First, we focus on the x² + 4x part. To complete the square, we look for a number that, when added and subtracted inside the equation, will allow us to form a perfect square trinomial.
Take half of the coefficient of x, which is 4. Half of 4 is 2, and squaring it gives us 4.
Now, we rewrite the equation:
y = (x² + 4x + 4) – 4 + 7
The expression in parentheses (x² + 4x + 4) can be factored as (x + 2)².
Thus, we can rewrite the function as:
y = (x + 2)² – 4 + 7
Simplifying gives us:
y = (x + 2)² + 3
Now we have the quadratic function in vertex form:
y = (x + 2)² + 3
The vertex of this parabola is at the point (-2, 3), which is derived from the form (x – h)² + k where (h, k) is the vertex.