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The vertex of a parabola is a key point that defines its shape and position. In the vertex form of the quadratic function, which is written as \( h(x) = a(x-h)^2 + k \), the vertex is given by the coordinates \( (h, k) \). It's the point where the parabola changes direction. In our exercise, the quadratic function is \( h(x) = -\frac{2}{7}(x+6)^2 + 11 \). By comparing it with the standard vertex form, we can see that \( h = -6 \) and \( k = 11 \). Therefore, the vertex of the given parabola is \( (-6, 11) \). The vertex tells us both the highest or lowest point of the parabola and its location along the x-axis.

The axis of symmetry is an imaginary vertical line that passes through the vertex of the parabola. This line divides the parabola into two mirror-image halves. For any quadratic function in vertex form \( h(x) = a(x-h)^2 + k \), the axis of symmetry is the line \( x = h \). In the given function \( h(x) = -\frac{2}{7}(x+6)^2 + 11 \), the value of \( h \) is -6. Thus, the axis of symmetry for this parabola is the vertical line \( x = -6 \). This means that if you fold the graph along this line, both halves of the parabola would align perfectly.

The maximum value of a quadratic function occurs at its vertex when the parabola opens downwards. This is because the highest point of the graph is at the vertex. For the given function \( h(x) = -\frac{2}{7}(x+6)^2 + 11 \), notice that the coefficient of the \( (x+6)^2 \) term is \(-\frac{2}{7} \). Since it is negative, the parabola opens downward, indicating the vertex represents a maximum point. The maximum value of the function is the y-coordinate of the vertex, which in this case is 11. Therefore, the function reaches its highest value at \( y = 11 \).

A quadratic function is a type of polynomial function that can be written in the standard form \( ax^2 + bx + c \) or in vertex form \( a(x-h)^2 + k \). It describes a parabolic graph, which is a U-shaped curve. Depending on the sign of the coefficient ‘a’, the parabola can open upwards (when ‘a’ is positive) or downwards (when ‘a’ is negative). The general properties of a quadratic function include the vertex, axis of symmetry, and direction of opening. In our example, the function is \( h(x) = -\frac{2}{7}(x+6)^2 + 11 \). This formula provides all necessary information to identify the vertex \((-6, 11)\), the axis of symmetry \(x=-6\), and the maximum value since ‘a’ is negative.