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The vertex of a parabola is a critical point that either represents the highest or lowest point of the graph, depending on the parabolic direction. It's essential for sketching and understanding the graph of a quadratic function. The vertex can be calculated using the standard form of the quadratic equation, which is given by:\[ y = ax^2 + bx + c \]Here, the vertex formula helps determine its coordinates. The x-coordinate of the vertex is given by:\[-\frac{b}{2a}\]To find the y-coordinate, substitute the x-coordinate back into the quadratic function. For instance, in a function like \( y = -\frac{1}{2}x^2 - 4x + 6 \), with coefficients \( a = -\frac{1}{2} \), \( b = -4 \), and \( c = 6 \), the x-coordinate is calculated as:\[-\frac{-4}{2 \times -\frac{1}{2}} = 4\]The y-coordinate is found by substituting \( x = 4 \) back into the equation:\[y = -\frac{1}{2} \times 4^2 - 4 \times 4 + 6 = -8 - 16 + 6 = -18\]Therefore, the vertex of this parabola is at the point \( (4, -18) \), marking the lowest point since the parabola opens downward.

Graphing parabolas is an essential skill for understanding quadratic functions. A parabola's shape is U-shaped and symmetrical, where the vertex acts as the line of symmetry. To properly graph a parabola, here are the general steps:

• Start by calculating the vertex, as it's a pivotal reference point for your graph.
• If possible, determine the y-intercept \((0, c)\) as a secondary point.
• Consider additional points on both sides of the vertex by plugging various x-values into the quadratic equation.

In the example function \( y = -\frac{1}{2}x^2 - 4x + 6 \), once the vertex \( (4, -18) \) is identified, plot this point accurately. The parabola opens downwards because the coefficient \( a = -\frac{1}{2} \) is negative.
To help visualize, find two additional points. Let's choose \( x = 2 \) and \( x = 6 \) and calculate the corresponding \( y \)-values:

• For \( x = 2 \), \( y = -\frac{1}{2}\times2^2 - 4\times2 + 6 = 0 \)
• For \( x = 6 \), \( y = -\frac{1}{2}\times6^2 - 4\times6 + 6 = -18 \)

These points help you sketch a smoother curve. Remember, the structure will mirror evenly across the vertex.

The standard form of a quadratic function is crucial as it forms the foundation for finding key features like the vertex and axis of symmetry. This form is represented as:\[ y = ax^2 + bx + c \]Where:

• \( a \) is the coefficient determining the direction (upward if \( a > 0 \), downward if \( a < 0 \)).
• \( b \) is the linear coefficient influencing the line's slope.
• \( c \) stands as the constant term, indicating the y-intercept.

In a function like \( y = -\frac{1}{2}x^2 - 4x + 6 \), identify coefficients as follows:

• \( a = -\frac{1}{2} \), indicating the graph opens downwards.
• \( b = -4 \), contributing to the slope.
• \( c = 6 \), providing the y-intercept.

With this form, you not only determine the vertex but also easily recognize if additional transformations like shifts or stretches have modified the parabola. Each part of the standard form gives insight into the parabola’s symmetry and overall position upon the graph.