Answer :

Question:

Find the vertex of the graph of the function f(x) = −6x² + x + 8.

Answer:

The vertex of the graph is (1/12, 65/24) or approximately (0.0833, 2.7083).

Step-by-step explanation:

1. Identify the quadratic function in the form f(x) = ax² + bx + c

  Here, a = -6, b = 1, and c = 8

2. Use the formula for the x-coordinate of the vertex: x = -b / (2a)

  x = -1 / (2(-6)) = -1 / -12 = 1/12

3. Calculate the y-coordinate by plugging the x-coordinate into the original function:

  f(1/12) = -6(1/12)² + (1/12) + 8

 

4. Simplify:

  = -6(1/144) + 1/12 + 8

  = -1/24 + 1/12 + 8

  = -1/24 + 2/24 + 192/24

  = 193/24

  = 65/24

5. Therefore, the vertex is (1/12, 65/24)

Additional information:

The vertex of a parabola is the point where it reaches its maximum (for a parabola that opens downward) or minimum (for a parabola that opens upward). In this case, since the coefficient of x² is negative (-6), the parabola opens downward, and the vertex represents the maximum point of the function. The x-coordinate of the vertex also represents the axis of symmetry of the parabola.