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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.