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The vertex of a quadratic function is a crucial point where the parabola either reaches its maximum or minimum value. It's the turning point of the curve. To find the vertex, use the formula for the x-coordinate: \(x = -\frac{b}{2a}\). For the given quadratic function \( h(x) = \frac{1}{2} x^2 + 4 x + \frac{19}{3} \), the coefficients are \( a = \frac{1}{2} \), \( b = 4 \), and \( c = \frac{19}{3} \). Plug in the values of \( a \) and \( b \) to find the x-coordinate: \( x = -\frac{4}{2(\frac{1}{2})} = -4 \). Once the x-coordinate is found, substitute it back into the function to find the y-coordinate: \( h(-4) = \frac{1}{2}(-4)^2 + 4(-4) + \frac{19}{3} = 8 - 16 + \frac{19}{3} = \frac{-5}{3} \). Thus, the vertex is at \((-4, -\frac{5}{3})\).

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It runs through the vertex. For any quadratic function \( h(x) = a x^2 + b x + c \), the axis of symmetry can be found directly from the x-coordinate of the vertex. As we found earlier, the x-coordinate of the vertex for our function is \( x = -4 \). Therefore, the axis of symmetry is the line \( x = -4 \). This line is useful when graphing the parabola as it helps maintain the balance of the curve on either side of the vertex.

In a quadratic function with a positive leading coefficient \( a \), the parabola opens upwards, and the vertex represents the minimum point of the function. For the function \( h(x) = \frac{1}{2} x^2 + 4 x + \frac{19}{3} \), since \( a = \frac{1}{2} \) is positive, the vertex gives us the minimum value of the function. We've already calculated the vertex to be at \( (-4, -\frac{5}{3}) \). Therefore, the minimum value of the function is \( h(-4) = -\frac{5}{3} \). This indicates that \( -\frac{5}{3} \) is the lowest y-value the function will reach.

Graphing a quadratic function involves a few steps. Start by plotting the vertex. For our function \( h(x) = \frac{1}{2} x^2 + 4 x + \frac{19}{3} \), the vertex is \( (-4, -\frac{5}{3}) \). Next, draw the axis of symmetry, which is the vertical line \( x = -4 \). This line helps you ensure each side of the parabola is symmetrical.
To form a more accurate graph, find additional points by plugging in x-values around the vertex. For example, calculate \( h(-3) \) and \( h(-5) \): \( h(-3) = \frac{1}{2}(-3)^2 + 4(-3) + \frac{19}{3} = \frac{-1}{6} \) and \( h(-5) = \frac{1}{2}(-5)^2 + 4(-5) + \frac{19}{3} = -\frac{1}{6} \). Plot these points, then draw a smooth curve through them, making sure it opens upwards because of the positive \( a \)-value. Remember, accurate plotting of additional points ensures a precise graph of the parabola.