Question

Rewrite the quadratic function f(x) = 6x² + x + 4 in vertex form.

Answer 1

To convert the quadratic function f(x) = 6x² + x + 4 to vertex form, we first complete the square by factoring out the coefficient of x² and then adding and subtracting the square of half the coefficient of x. The function becomes f(x) = 6(x + 1/12)² + 95/24.

Step-by-step explanation:

Converting a Quadratic Function to Vertex Form

To rewrite the quadratic function f(x) = 6x² + x + 4 in vertex form, we need to complete the square.

1. First, factor out the coefficient of the x² term from the x terms: f(x) = 6(x² + (1/6)x) + 4.
2. Next, find the square of half the coefficient of x, which we will add and subtract inside the brackets to complete the square, ensuring to keep the equation balanced: (1/12)² = 1/144.
3. Add and subtract this value inside the brackets: f(x) = 6((x² + (1/6)x + 1/144) - 1/144) + 4.
4. Rewrite the perfect square trinomial inside the brackets and simplify: f(x) = 6((x + 1/12)² - 1/144) + 4.
5. Finally, distribute the 6 and combine like terms to get the vertex form: f(x) = 6(x + 1/12)² + 4 - 6/144, which simplifies to f(x) = 6(x + 1/12)² + 95/24.

The vertex form of f(x) = 6x² + x + 4 is f(x) = 6(x + 1/12)² + 95/24, where the vertex of the parabola is at (-1/12, 95/24).