To convert the quadratic function f(x) = 3x² + 6x + 9 into vertex form, which is expressed as f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola, follow these steps:
- Factor out the coefficient of x²: In this case, the coefficient is 3. We factor this out from the first two terms:
- Complete the square: To complete the square for the expression in the parentheses, take the coefficient of x (which is 2), divide it by 2, and square it. This gives us:
- Add and subtract this value inside the parentheses: We will add and subtract 1 inside the parentheses. However, since we factored out a 3 earlier, we need to account for that when adjusting the equation:
- Simplify: Distributing the 3 gives us:
- Final Vertex Form: Now we can see that the quadratic function is in the vertex form:
- Vertex (h, k) = (-1, 6)
- Coefficient a = 3
f(x) = 3(x² + 2x) + 9
(2 / 2)² = 1
f(x) = 3(x² + 2x + 1 – 1) + 9
Now we rewrite it as:
f(x) = 3((x + 1)² – 1) + 9
f(x) = 3(x + 1)² – 3 + 9
So, we simplify it further:
f(x) = 3(x + 1)² + 6
f(x) = 3(x + 1)² + 6
From this, we identify:
In conclusion, the vertex form of the quadratic function f(x) = 3x² + 6x + 9 is f(x) = 3(x + 1)² + 6, with a vertex located at (-1, 6).