To convert the quadratic function f(x) = x² + 12x + 6 into vertex form, we need to identify a suitable zero pair. The vertex form of a quadratic function is typically written as:
f(x) = a(x – h)² + k
Where (h, k) is the vertex of the parabola. To achieve this, we can utilize the method of completing the square.
1. **Identify the coefficients:** For the function x² + 12x + 6, the coefficient of x² is 1, and the coefficient of x is 12.
2. **Find the necessary zero pair:** To complete the square, we look at the coefficient of x, which is 12. We need to take half of this coefficient and square it:
– Half of 12 is 6.
– Squaring 6 gives us 36.
3. **Rewrite the function:** We can now rewrite the function by adding and subtracting this square. Our goal is to add 36 and simultaneously subtract it to maintain the balance:
– So, we rewrite the function as:
f(x) = x² + 12x + 36 – 36 + 6
– Which simplifies to:
f(x) = (x + 6)² – 30
4. **Final vertex form:** Now we can clearly see the function in vertex form:
f(x) = (x + 6)² – 30
From this, we can deduce that the zero pair we added is (36, -36), which allows the function to be rewritten in vertex form. Thus, the vertex of this quadratic function is at the point (-6, -30).