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In quadratic functions, the "vertex" is a key point that defines many properties of the function's graph. For the function given, which is in the form \( ax^2 \) with \( a = 2 \), the vertex is simply the point at which the function reaches its minimum value. Since there are no linear or constant terms (i.e., \( b = 0 \) and \( c = 0 \)), the vertex can be quickly identified at the origin of the coordinate plane. That is, the vertex is at the point (0, 0).

The vertex plays a crucial role as it represents either a peak or a trough of the parabola. For functions of the form like \( H(x) = ax^2 \), when \( a > 0 \), the vertex represents a local minimum, and the parabola opens upward. The coordinates of the vertex also help in determining other attributes like its axis of symmetry, making it essential in sketching the graph of the quadratic function.

The axis of symmetry in a quadratic function is an imaginary line that vertically slices through the graph of the parabola, dividing it into two mirror image halves. It is essentially the x-coordinate of the vertex. For the function \( H(x) = 2x^2 \), this line can be determined by using the formula \( x = -\frac{b}{2a} \).

Because both \( b \) and \( c \) are zero in the given function, the calculation simplifies directly to \( x = 0 \). This indicates that the axis of symmetry is simply the y-axis itself. As you sketch the graph, you'll see that this line of symmetry serves as a guide, helping you locate other points on the parabola since they will reflect equally on either side of this axis.

The graph of a quadratic function results in a U-shaped curve known as a "parabola." How this parabola looks depends significantly on the coefficient \( a \) in the function \( ax^2 + bx + c \).

For the function \( H(x) = 2x^2 \), since \( a = 2 \) is a positive number, the parabola opens upwards. Also, the larger the absolute value of \( a \), the "narrower" the parabola will appear. Therefore, compared to a standard parabola like \( x^2 \), \( H(x) = 2x^2 \)'s parabola will be more narrow.

• The direction the parabola opens (upwards here) is determined by the sign of \( a \).
• The width of the parabola (narrower in this case) is influenced by the magnitude of \( a \).

Understanding these properties helps in effectively sketching the quadratic function graph.

Graphing quadratic functions like \( H(x) = 2x^2 \) involves plotting specific points and understanding the properties of the parabola. Here's how you approach this:

First, identify and mark out the vertex on your graph. For this equation, it's at (0, 0), the origin. Next, draw the axis of symmetry, which you previously found, as a vertical line through the vertex, in this case, \( x = 0 \). Use a dashed line for clarity.

Now, sketch the parabola by plotting additional points. Given \( H(x) = 2x^2 \), you can create a table of values to help. For instance, if \( x = 1 \) or \( x = -1 \), \( H(x) = 2 \). Plot these points as (1, 2) and (-1, 2). These points help you accurately sketch the curve, showing its narrow upward opening.

• Always label key features of the graph, including the vertex and axis of symmetry.
• Use reflection across the axis of symmetry to confirm the shape and position of the parabola.

These steps ensure a comprehensive graph that highlights all vital features of the quadratic function.