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A quadratic function forms a curve called a parabola. All quadratic functions are comparable to a bowl shape, either opening upwards or downwards. This configuration is determined by the value of the coefficient \(a\) in the general quadratic equation \(f(x) = ax^2 + bx + c\).
When \(a > 0\), the parabola opens upwards, resembling a U-shape, which means it will have a minimum point. Conversely, if \(a < 0\), the parabola opens downwards, like an upside-down U, indicating a maximum point.
Parabolas are symmetrical around a vertical line, called the axis of symmetry, which helps in easily locating the vertex of the parabola.

The vertex of a parabola is the central point of the curve. It represents the minimum point for an upward-opening parabola and the maximum point for a downward-opening one.
The vertex can be easily calculated from the quadratic equation using the vertex formula for the x-coordinate: \(x = -\frac{b}{2a}\). Knowing this x-coordinate helps you determine the position of the vertex along the x-axis. In our exercise, this is how we determined that the x-coordinate is 2.
To find the complete vertex point, substitute this x-coordinate back into the quadratic function to calculate the corresponding y-coordinate. This provides the exact location and the value of the vertex, giving insight into the function's behavior at its extremities.

For a quadratic function that forms an upward-opening parabola, such as in our exercise, the minimum value occurs at the vertex.
Finding the minimum value involves two steps:

• Calculate the x-coordinate of the vertex using \(x = -\frac{b}{2a}\).
• Substitute this x-value back into the function \(f(x)\) to determine the y-coordinate, which represents the minimum value.

This approach works efficiently, as demonstrated in the exercise where substituting \(x = 2\) led to finding the minimum value \(-19\). It's essential for accurately describing the function's lowest point and is especially useful in real-life applications where optimization is needed.