91Ó°ÊÓ

In a quadratic function, the maximum or minimum value represents the highest or lowest point on the graph of the parabola. Whether the parabola has a maximum or minimum is determined by the sign of the coefficient \( a \) in the function's standard form \( f(x) = ax^2 + bx + c \). When \( a < 0 \), it indicates the parabola opens downwards, resulting in a maximum value. Conversely, when \( a > 0 \), the parabola opens upwards, resulting in a minimum value.

For the function \( f(x) = -\frac{x^2}{3} + 2x + 7 \), the coefficient \( a = -\frac{1}{3} \) is less than zero. Thus, we know that this function will have a maximum value. This maximum value occurs at the vertex of the parabola, providing critical information about the highest point of the function's graph.

The vertex of a parabola is a significant point that gives us crucial information about the graph. For a quadratic function in the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the formula \( x = -\frac{b}{2a} \).

In our case, with the function \( f(x) = -\frac{x^2}{3} + 2x + 7 \), substituting the coefficients, we find that \( x = -\frac{2}{2(-\frac{1}{3})} = 3 \). The x-coordinate of the vertex is 3.

To find the vertex's y-coordinate, we evaluate the function at \( x = 3 \), yielding \( f(3) = 10 \). Hence, the vertex of the parabola is at the point \((3, 10)\). This vertex represents the maximum value of the function since the parabola opens downward.

The orientation of a parabola refers to the direction in which it opens. This is primarily determined by the sign of the quadratic coefficient \( a \) in the function \( f(x) = ax^2 + bx + c \).

If \( a > 0 \), the parabola opens upwards, resembling a "U" shape. When \( a < 0 \), the parabola opens downwards, similar to an upside-down "U".

For the quadratic function \( f(x) = -\frac{x^2}{3} + 2x + 7 \), the coefficient \( a = -\frac{1}{3} \) is negative. This negative value confirms that the parabola opens downward, a crucial aspect when identifying if the function possesses a maximum or minimum. Since our function opens downward, we know the function's graph can reach a maximum point but not a minimum.