How can Marcus rewrite the function f(x) = x² + 6x + 4 in vertex form?

Rewriting the Quadratic in Vertex Form

To convert the quadratic function f(x) = x² + 6x + 4 into vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola.

Steps to Complete the Square:

1. Start with the original function:

f(x) = x² + 6x + 4

2. Focus on the quadratic and linear terms, x² + 6x. To complete the square, we take the coefficient of the x-term (which is 6), divide it by 2 (getting 3), and then square it (resulting in 9).
3. Add and subtract this square value inside the equation:

f(x) = x² + 6x + 9 - 9 + 4

4. This allows us to rewrite the expression as a perfect square:

f(x) = (x + 3)² - 5

5. Now, we have the function in vertex form, where:
• a = 1,
• h = -3,
• k = -5.

Final Answer:

The vertex form of the function f(x) = x² + 6x + 4 is:

f(x) = (x + 3)² - 5

From this, we can see that the vertex of the parabola is at the point (-3, -5).