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A quadratic function is a second-degree polynomial of the form \[ f(x) = ax^{2} + bx + c \]. The graph of a quadratic function is a parabola, which can open upwards or downwards. The direction of the parabola is determined by the sign of the coefficient \(a\).

If \(a\) is positive, the parabola opens upwards, creating a U-shape. If \(a\) is negative, the parabola opens downwards, forming an inverted U-shape.

The quadratic function given in the exercise is \[ f(x) = -2x^{2} - 12x - 13 \], where \(a = -2\), \(b = -12\), and \(c = -13\). Because \(a\) is negative, the parabola will open downwards. Understanding the general shape and direction of the quadratic function helps us better visualize and graph the equation.

Graphing a quadratic function involves several key steps to ensure accuracy and clarity. Here are the main techniques:

• Identify the Vertex: Use the vertex form \[ f(x) = a(x-h)^{2} + k \]. Calculate the vertex (h, k). In the solved exercise, the vertex was \[ (-3,5) \]. Plot this point first.
• Determine the Direction: Check the sign of \(a\). If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• Find Additional Points: Choose x-values around the vertex and calculate corresponding y-values to get more points for the graph.
• Axis of Symmetry: The line \(x = h \) represents the axis of symmetry, where the parabola is mirrored.
• Plot the Parabola: Connect the points smoothly to draw the shape of the parabola, making sure it extends in the correct direction.

Utilizing these techniques systematically helps in accurately plotting a quadratic function.

The vertex form of a quadratic function is a useful format for graphing and understanding the properties of the parabola. The vertex form is given by \[ f(x) = a(x-h)^{2} + k \], where \( (h, k) \) is the vertex.

• Finding the Vertex: The vertex \( (h, k) \) is the highest or lowest point of the parabola depending on the sign of \(a\). For our function \[ f(x) = -2(x + 3)^{2} + 5 \], the vertex is \( (-3, 5) \). This represents the maximum point because the parabola opens downwards.
• Advantages of Vertex Form: It directly shows the vertex, making graphing easier. Also, transformations like shifts and stretches can be recognized immediately.
• Converting to Vertex Form: Completing the square is a common method to convert a quadratic function from standard form \[ f(x) = ax^{2} + bx + c \] to vertex form. Follow steps to complete the square, and factor the quadratic expression so it fits the vertex form.

Understanding and using the vertex form allows for efficient graphing and analysis of the quadratic function.