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To determine the key features of a quadratic function, you often begin with finding its vertex. A vertex is a significant point since it represents either the maximum or minimum value of the function. Particularly for the function in the form \( h(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the vertex formula:\[h = -\frac{b}{2a}\]This formula is derived from completing the square or using calculus methods to find where the derivative of the function equals zero. Substituting the given coefficients will help in finding the exact location of the vertex along the x-axis.

After finding \( h \), substituting it back into the function \( h(x) \) provides the y-coordinate \( k \). Thus, the vertex \((h, k)\) can be determined fully, which in this case gives us the coordinate point of the vertex.

A parabola is the graphical representation of a quadratic function, and it is always a U-shaped curve. Depending on the coefficient \( a \) of \( ax^2 \) in the function \( h(x) = ax^2 + bx + c \), this curve will either open upwards or downwards.

- If \( a > 0 \), the parabola opens upwards.- If \( a < 0 \), the parabola opens downwards.

The direction the parabola opens affects whether the vertex will represent a minimum or maximum value of the function. For this particular function where \( a = \frac{1}{2} \), the parabola opens upwards, indicating that the vertex is at its minimum point. Understanding parabolas will help you visualize the nature of quadratic functions.

The minimum value of a quadratic function occurs at the vertex when the parabola opens upwards. In the given function \( h(x) = \frac{1}{2} x^{2} + 2 x - 6 \), since \( a > 0 \), the parabola is upward-facing, and thus, the vertex represents the minimum point.

To find this minimum value, you first determine the vertex coordinates using the vertex formula. For this function, the vertex is \((-2, -8)\). Hence, the minimum value of the function is \( h(-2) = -8 \).

This point \((-2, -8)\) is where the function achieves its lowest value with respect to y, telling you that it does not descend further beyond -8.

Coefficients play a critical role in shaping the behavior of a quadratic function. Specifically, in the function \( h(x) = ax^2 + bx + c \):

- \( a \) is the leading coefficient. It determines the direction (upwards or downward) and the width of the parabola.- \( b \) affects the vertex location along the x-axis.- \( c \) represents the y-intercept, indicating where the parabola crosses the y-axis.

For instance, in \( h(x) = \frac{1}{2} x^{2}+2 x-6 \), \( a = \frac{1}{2} \) being positive, confirms that the parabola opens upwards. The coefficient \( b = 2 \) is used in the vertex formula, ultimately affecting the x-coordinate of the vertex. Lastly, \( c = -6 \) shows the parabola crosses the y-axis at \( y = -6 \). Understanding coefficients helps in predicting and analyzing the function's graph properly.